Finite Group Databasebeta

← #34344309

Subgroups — #34344309

Order 20224.

1046 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098962C211yesmaximal, characteristic, complemented
2C2×#19098863C211yesmaximal, characteristic
3#23768788C211yesmaximal, characteristic, complemented
4#23764735C211yesmaximal, characteristic, complemented
5#23764735C211yesmaximal, characteristic, complemented
6#23764730C211yesmaximal, characteristic, complemented
7#23764725C211yesmaximal, characteristic, complemented
8#23764358C211yesmaximal, characteristic, complemented
9#23764337C211yesmaximal, characteristic, complemented
10#23763661C211yesmaximal, characteristic
11#23763661C211yesmaximal, characteristic
12#23763647C211yesmaximal, characteristic
13#23763647C211yesmaximal, characteristic
14C4×#14579082C211yesmaximal, characteristic
15C2×#554×C79C211yesmaximal, characteristic
16C2×#14579153C2211yescharacteristic
17#19098962—24no—
18#19098863—24no—
19C2×#14579094C2211yescharacteristic
20C2×#14579094C2211yescharacteristic
21C2×#14579094C2211yescharacteristic
22C2×#14579092C2211yescharacteristic
23C2×#14579092C2211yescharacteristic
24C2×#14579092C2211yescharacteristic
25C2×C4⋊(C79⋊C8)C2211yescharacteristic
26C2×#14579082C2211yescharacteristic
27C2×#14579082C2211yescharacteristic
28C2×#14579082C2211yescharacteristic
29C2×#14579082C2211yescharacteristic
30C2×C4×C79⋊C8C2211yescharacteristic
31#19098494—21no—
32#19098482—21no—
33#19098461C2211yescharacteristic
34#19098461C2211yescharacteristic
35#19098461C2211yescharacteristic
36#19098461C2211yescharacteristic
37#19098458—21no—
38#19098458—21no—
39C4×C4⋊Dic79C2211yescharacteristic
40#19098325C2211yescharacteristic
41#19098325C2211yescharacteristic
42#19098325C2211yescharacteristic
43#19098325C2211yescharacteristic
44#19098325C2211yescharacteristic
45#19098325C2211yescharacteristic
46#19098318C2211yescharacteristic
47#19098318C2211yescharacteristic
48#19098318C2211yescharacteristic
49#19098318C2211yescharacteristic
50#554×C79—24no—
51#554×C79—24no—
52C2×C4⋊Q8×C79C2211yescharacteristic
53C2×#125×C79C2211yescharacteristic
54C2×#125×C79C2211yescharacteristic
55C2×C22⋊Q8×C79—21no—
56C2×C22⋊Q8×C79—21no—
57C2×C22⋊Q8×C79—21no—
58C2×C4×Q8×C79C2211yescharacteristic
59C2×C4×D4×C79C2211yescharacteristic
60C2×C4×D4×C79C2211yescharacteristic
61C2×C4×D4×C79C2211yescharacteristic
62C22×C4⋊C4×C79—21no—
63C22×C79⋊C8C2311yescharacteristic
64C22×C79⋊C8C2311yescharacteristic
65C22×C79⋊C8C2311yescharacteristic
66C22×C79⋊C8C2311yescharacteristic
67#14579153—22no—
68#14579153—22no—
69C2×C4⋊Dic79C2311yescharacteristic
70C2×C4⋊Dic79C2311yescharacteristic
71C2×C4⋊Dic79C2311yescharacteristic
72C2×C4⋊Dic79C2311yescharacteristic
73C2×Dic79⋊C4—21no—
74C2×Dic79⋊C4—21no—
75C2×Dic79⋊C4—21no—
76C2×Dic79⋊C4—21no—
77C2×Dic79⋊C4—21no—
78C2×Dic79⋊C4—21no—
79C2×C4×Dic79—21no—
80#14579094—22no—
81#14579094—22no—
82#14579094—22no—
83#14579092—22no—
84#14579092—22no—
85#14579092—22no—
86C4⋊(C79⋊C8)—22no—
87#14579082—22no—
88#14579082—22no—
89#14579082Q814yescomplemented
90#14579082Q814yescomplemented
91C4×C79⋊C8—22no—
92C79⋊#95—21no—
93C79⋊#95—21no—
94C79⋊#95—21no—
95C79⋊#95—21no—
96C79⋊#95—21no—
97C22×Q8×C79—21no—
98C22×Q8×C79D411yescharacteristic
99C22×D4×C79C2311yescharacteristic
100C23×C4×C79—21no—
101C23×C4×C79—21no—
102C23×C4×C79—21no—
103C4⋊Q8×C79—22no—
104C4⋊Q8×C79—22no—
105#125×C79—22no—
106#125×C79—22no—
107#125×C79—22no—
108#125×C79D414yes—
109C22⋊Q8×C79—28no—
110C22⋊Q8×C79—28no—
111C22⋊Q8×C79—28no—
112C4×Q8×C79—22no—
113C4×Q8×C79—22no—
114C4×D4×C79—22no—
115C4×D4×C79—22no—
116C4×D4×C79—22no—
117C4×D4×C79—22no—
118C4×D4×C79—22no—
119C4×D4×C79D414yes—
120C2×C4⋊C4×C79—28no—
121C2×C4⋊C4×C79—21no—
122C2×C4⋊C4×C79—21no—
123C2×C4⋊C4×C79—21no—
124C2×C4⋊C4×C79—21no—
125C2×C4⋊C4×C79—21no—
126C2×C4⋊C4×C79D411yescharacteristic
127C2×C4⋊C4×C79D411yescharacteristic
128C2×C4⋊C4×C79D411yescharacteristic
129C2×C4⋊C4×C79C2311yescharacteristic
130C2×C4⋊C4×C79C2311yescharacteristic
131C2×C4⋊C4×C79C2311yescharacteristic
132C2×C22⋊C4×C79—21no—
133C2×C22⋊C4×C79—21no—
134C2×C22⋊C4×C79—21no—
135C2×C42×C79C2311yescharacteristic
136C2×C42×C79C2311yescharacteristic
137C2×C42×C79C2311yescharacteristic
138C22×Dic79—21no—
139C22×Dic79—21no—
140C22×Dic79—21no—
141C22×Dic79—21no—
142C2×C79⋊C8—21no—
143C2×C79⋊C8—21no—
144C2×C79⋊C8—21no—
145C2×C79⋊C8—21no—
146C2×C79⋊C8—21no—
147C2×C79⋊C8—21no—
148C2×C79⋊C8—21no—
149C2×C79⋊C8—21no—
150C2×C79⋊C8—21no—
151C2×C79⋊C8—21no—
152C2×C79⋊C8C2×Q812yes—
153C2×C79⋊C8C2×Q812yes—
154C4⋊Dic79—21no—
155C4⋊Dic79—21no—
156C4⋊Dic79—21no—
157C4⋊Dic79—21no—
158C4⋊Dic79—21no—
159C4⋊Dic79—21no—
160C4⋊Dic79C2×Q812yes—
161C4⋊Dic79C2×Q812yes—
162Dic79⋊C4—42no—
163Dic79⋊C4—42no—
164Dic79⋊C4—42no—
165Dic79⋊C4—42no—
166Dic79⋊C4—81no—
167Dic79⋊C4—81no—
168C4×Dic79—81no—
169C24×C79—21no—
170C2×Q8×C79—24no—
171C2×Q8×C79—22no—
172C2×Q8×C79—22no—
173C2×Q8×C79—41no—
174C2×Q8×C79—81no—
175C2×Q8×C79—81no—
176C2×D4×C79—21no—
177C2×D4×C79—21no—
178C2×D4×C79—21no—
179C2×D4×C79C2×D412yes—
180C2×D4×C79C2×Q812yes—
181C2×D4×C79C2×Q812yes—
182C22×C4×C79—24no—
183C22×C4×C79—21no—
184C22×C4×C79—21no—
185C22×C4×C79—21no—
186C22×C4×C79—24no—
187C22×C4×C79—24no—
188C22×C4×C79—42no—
189C22×C4×C79—42no—
190C22×C4×C79—81no—
191C22×C4×C79—81no—
192C22×C4×C79—81no—
193C22×C4×C79—81no—
194C22×C4×C79C2×D411yescharacteristic
195C22×C4×C79C2×D411yescharacteristic
196C22×C4×C79C2×D411yescharacteristic
197C22×C4×C79C2×D411yescharacteristic
198C22×C4×C79C2×D411yescharacteristic
199C22×C4×C79C2×D411yescharacteristic
200C22×C4×C79C2411yescharacteristic
201C4⋊C4×C79—24no—
202C4⋊C4×C79—24no—
203C4⋊C4×C79—24no—
204C4⋊C4×C79—22no—
205C4⋊C4×C79—24no—
206C4⋊C4×C79—24no—
207C4⋊C4×C79—22no—
208C4⋊C4×C79—22no—
209C4⋊C4×C79—21no—
210C4⋊C4×C79—21no—
211C4⋊C4×C79—21no—
212C4⋊C4×C79C2×D412yes—
213C4⋊C4×C79C2×D412yes—
214C4⋊C4×C79C2×D412yes—
215C22⋊C4×C79—24no—
216C22⋊C4×C79—24no—
217C22⋊C4×C79—24no—
218C42×C79—21no—
219C42×C79—21no—
220C42×C79—21no—
221C42×C79—21no—
222C42×C79C2×D412yes—
223C42×C79C2×D412yes—
224C23×C79—22no—
225C23×C79—22no—
226C23×C79—22no—
227C23×C79—41no—
228C23×C79—41no—
229C23×C79—41no—
230C23×C79—41no—
231C23×C79C22×D411yescharacteristic
232Q8×C79—44no—
233Q8×C79—84no—
234D4×C79—22no—
235D4×C79—22no—
236D4×C79—22no—
237D4×C79C4⋊Q814yes—
238C2×C4×C79—21no—
239C2×C4×C79—21no—
240C2×C4×C79—21no—
241C2×C4×C79—21no—
242C2×C4×C79—21no—
243C2×C4×C79—21no—
244C2×C4×C79—21no—
245C2×C4×C79—21no—
246C2×C4×C79—21no—
247C2×C4×C79—22no—
248C2×C4×C79—22no—
249C2×C4×C79—22no—
250C2×C4×C79—21no—
251C2×C4×C79—21no—
252C2×C4×C79—21no—
253C2×C4×C79—21no—
254C2×C4×C79—21no—
255C2×C4×C79—21no—
256C2×C4×C79—48no—
257C2×C4×C79—41no—
258C2×C4×C79—41no—
259C2×C4×C79—41no—
260C2×C4×C79—41no—
261C2×C4×C79—41no—
262C2×C4×C79—41no—
263C2×C4×C79—84no—
264C2×C4×C79—84no—
265C2×C4×C79#12012yes—
266C2×C4×C79#12012yes—
267C2×C4×C79#13512yes—
268C2×C4×C79C22×D411yescharacteristic
269C2×C4×C79C22×D411yescharacteristic
270C2×C4×C79C22×Q811yescharacteristic
271C2×C4×C79C22×Q811yescharacteristic
272C2×C4×C792−1+411yescharacteristic
273C2×C4×C792−1+411yescharacteristic
274C2×Dic79—22no—
275C2×Dic79—22no—
276C2×Dic79—41no—
277C2×Dic79—41no—
278C2×Dic79—41no—
279C2×Dic79—41no—
280C2×Dic79—41no—
281C2×Dic79—41no—
282C2×Dic79—41no—
283C2×Dic79—41no—
284C2×Dic79—41no—
285C2×Dic79—41no—
286C79⋊C8—22no—
287C79⋊C8—22no—
288C79⋊C8—41no—
289C79⋊C8—41no—
290C22×C79—24no—
291C22×C79—42no—
292C22×C79—42no—
293C22×C79—42no—
294C22×C79—42no—
295C22×C79—42no—
296C22×C79—42no—
297C22×C79C2×#12011yescharacteristic
298C22×C79D4×Q811yescharacteristic
299C22×C79D4×Q811yescharacteristic
300C22×C79C2×#13511yescharacteristic
301C22×C79#57511yescharacteristic
302C22×C79#57611yescharacteristic
303C22×C79#57711yescharacteristic
304C4×C79—22no—
305C4×C79—22no—
306C4×C79—22no—
307C4×C79—41no—
308C4×C79—42no—
309C4×C79—41no—
310C4×C79—41no—
311C4×C79—42no—
312C4×C79—42no—
313C4×C79C2×C4⋊Q811yescharacteristic
314C4×C79#53611yescharacteristic
315C4×C79D4×Q811yescharacteristic
316C4×C79D4×Q811yescharacteristic
317Dic79—42no—
318Dic79—42no—
319Dic79—81no—
320Dic79—81no—
321#25843—791nomaximal, Sylow
322C2×C79—44no—
323C2×C79#243411yescharacteristic
324C2×C79#300211yescharacteristic
325C2×C79#301211yescharacteristic
326C2×C79#338011yescharacteristic
327C2×C79#338111yescharacteristic
328C2×C79#338811yescharacteristic
329C2×C79#338911yescharacteristic
330C2×#554D7911yescharacteristic
331C2×#499—791no—
332C2×#478—791no—
333#2584—791no—
334#2059—791no—
335#2059—791no—
336#2058—791no—
337#2057—791no—
338#1989—791no—
339#1984—791no—
340#1864—791no—
341#1864—791no—
342#1862—791no—
343#1862—791no—
344C4×#102—791no—
345C79#2584311yescharacteristic, complemented, Sylow
346#554—24no—
347#554—24no—
348C2×C4⋊Q8C2×D7911yescharacteristic
349C2×#125—791no—
350C2×#125C2×D7911yescharacteristic
351C2×#125C2×D7911yescharacteristic
352C2×C22⋊Q8—21no—
353C2×C22⋊Q8—21no—
354C2×C22⋊Q8—21no—
355C2×C4×Q8C2×D7911yescharacteristic
356C2×C4×D4C2×D7911yescharacteristic
357C2×C4×D4C2×D7911yescharacteristic
358C2×C4×D4C2×D7911yescharacteristic
359C22×C4⋊C4—21no—
360#499—1584no—
361#478—1584no—
362C2×#107—791no—
363C2×#107—791no—
364C2×#107—791no—
365C2×#106—791no—
366C2×#106—791no—
367C2×#106—791no—
368C2×C4⋊C8—791no—
369C2×#102—791no—
370C2×#102—791no—
371C2×#102—791no—
372C2×#102—791no—
373C2×C4×C8—791no—
374#398—1581no—
375#395—1581no—
376#389—791no—
377#389—791no—
378#389—791no—
379#389—791no—
380#389—1581no—
381#389—1581no—
382C4×C4⋊C4—791no—
383#340—791no—
384#340—791no—
385#340—791no—
386#340—791no—
387#340—791no—
388#340—791no—
389#336—791no—
390#336—791no—
391#336—791no—
392#336—791no—
393C22×Q8—21no—
394C22×Q8C22⋊D7911yescharacteristic
395C22×D4C22×D7911yescharacteristic
396C23×C4—21no—
397C23×C4—21no—
398C23×C4—21no—
399C22×C8—791no—
400C22×C8—791no—
401C22×C8—791no—
402C22×C8—791no—
403C4⋊Q8—22no—
404C4⋊Q8—22no—
405#125—22no—
406#125—22no—
407#125—22no—
408#125—1582no—
409#125—1582no—
410#125C22⋊D7914yes—
411C22⋊Q8—28no—
412C22⋊Q8—28no—
413C22⋊Q8—28no—
414C4×Q8—22no—
415C4×Q8—22no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4C22⋊D7914yes—
422C2×C4⋊C4—28no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—791no—
429C2×C4⋊C4—791no—
430C2×C4⋊C4—791no—
431C2×C4⋊C4—791no—
432C2×C4⋊C4—1581no—
433C2×C4⋊C4—1581no—
434C2×C4⋊C4—1581no—
435C2×C4⋊C4—1581no—
436C2×C4⋊C4—1581no—
437C2×C4⋊C4—1581no—
438C2×C4⋊C4C22⋊D7911yescharacteristic
439C2×C4⋊C4C22⋊D7911yescharacteristic
440C2×C4⋊C4C22⋊D7911yescharacteristic
441C2×C4⋊C4C22×D7911yescharacteristic
442C2×C4⋊C4C22×D7911yescharacteristic
443C2×C4⋊C4C22×D7911yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C42—1581no—
448C2×C42C22×D7911yescharacteristic
449C2×C42C22×D7911yescharacteristic
450C2×C42C22×D7911yescharacteristic
451#107—1582no—
452#107—1582no—
453#107—1582no—
454#106—1582no—
455#106—1582no—
456#106—1582no—
457C4⋊C8—1582no—
458#102—794no—
459#102—794no—
460#102—1582no—
461#102—1582no—
462C4×C8—1582no—
463#95—1581no—
464#95—1581no—
465#95—1581no—
466#95—1581no—
467#95—1581no—
468C24—21no—
469C2×Q8—24no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8—41no—
473C2×Q8—81no—
474C2×Q8—81no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4C2×C22⋊D7912yes—
479C2×D4Q8×D7912yes—
480C2×D4Q8×D7912yes—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—81no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—1581no—
494C22×C4—1581no—
495C22×C4—1581no—
496C22×C4—1581no—
497C22×C4C2×C22⋊D7911yescharacteristic
498C22×C4C2×C22⋊D7911yescharacteristic
499C22×C4C2×C22⋊D7911yescharacteristic
500C22×C4C2×C22⋊D7911yescharacteristic
501C22×C4C2×C22⋊D7911yescharacteristic
502C22×C4C2×C22⋊D7911yescharacteristic
503C22×C4C23×D7911yescharacteristic
504C2×C8—792no—
505C2×C8—792no—
506C2×C8—1581no—
507C2×C8—1581no—
508C2×C8—1581no—
509C2×C8—1581no—
510C2×C8—1581no—
511C2×C8—1581no—
512C2×C8—1581no—
513C2×C8—1581no—
514C2×C8—1581no—
515C2×C8—1581no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—3162no—
528C4⋊C4—3162no—
529C4⋊C4—3162no—
530C4⋊C4—3162no—
531C4⋊C4—792no—
532C4⋊C4—792no—
533C4⋊C4—6321no—
534C4⋊C4—6321no—
535C4⋊C4—1581no—
536C4⋊C4—1581no—
537C4⋊C4—1581no—
538C4⋊C4—1581no—
539C4⋊C4—1581no—
540C4⋊C4—1581no—
541C4⋊C4C2×C22⋊D7912yes—
542C4⋊C4C2×C22⋊D7912yes—
543C4⋊C4C2×C22⋊D7912yes—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—6321no—
552C42C2×C22⋊D7912yes—
553C42C2×C22⋊D7912yes—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23C22×C22⋊D7911yescharacteristic
562Q8—44no—
563Q8—84no—
564D4—22no—
565D4—22no—
566D4—22no—
567D4#1457915914yes—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—48no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—3161no—
596C2×C4—3161no—
597C2×C4—3161no—
598C2×C4—3161no—
599C2×C4—3161no—
600C2×C4—3161no—
601C2×C4—3161no—
602C2×C4—3161no—
603C2×C4—3161no—
604C2×C4—3161no—
605C2×C4—1582no—
606C2×C4—1582no—
607C2×C4#1457913012yes—
608C2×C4#1457913012yes—
609C2×C4#1457918112yes—
610C2×C4C22×C22⋊D7911yescharacteristic
611C2×C4C22×C22⋊D7911yescharacteristic
612C2×C4C2×Q8×D7911yescharacteristic
613C2×C4C2×Q8×D7911yescharacteristic
614C2×C4#1457920911yescharacteristic
615C2×C4#1457920911yescharacteristic
616C8—3161no—
617C8—3161no—
618C8—1582no—
619C8—1582no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22C2×#1457913011yescharacteristic
628C22Q8×C22⋊D7911yescharacteristic
629C22Q8×C22⋊D7911yescharacteristic
630C22C2×#1457918111yescharacteristic
631C22#1909936111yescharacteristic
632C22#1909936611yescharacteristic
633C22#1909937411yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—42no—
640C4—42no—
641C4—41no—
642C4—42no—
643C4—3162no—
644C4—3162no—
645C4—6321no—
646C4—6321no—
647C4C2×#1457915911yescharacteristic
648C4#1909913711yescharacteristic
649C4Q8×C22⋊D7911yescharacteristic
650C4Q8×C22⋊D7911yescharacteristic
651C2—44no—
652C2#2376704011yescharacteristic
653C2#2377261811yescharacteristic
654C2#2377270411yescharacteristic
655C2#2377626311yescharacteristic
656C2#2377627511yescharacteristic
657C2#2377634511yescharacteristic
658C2#2377635411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D79
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC2×#554×C79C2
socleC23×C79C22×D4
prime radicalC22×D4×C79C23
hypercenterC2×#554D79
hypocenterC79#25843
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#19098962C2semidirect
3. #23768788C2semidirect
4. #23764735C2semidirect
5. #23764735C2semidirect
6. #23764730C2semidirect
7. #23764725C2semidirect
8. #23764358C2semidirect
9. #23764337C2semidirect
89. #14579082Q8semidirect
90. #14579082Q8semidirect
345. C79#25843semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#190989622
C2×#190988632
#237687882
#237647352
#237647302
#237647252
#237643582
#237643372
#237636612
#237636472
C4×#145790822
C2×#554×C792
#2584379