Finite Group Databasebeta

← #32053293

Subgroups — #32053293

Order 18176.

982 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986405C211yesmaximal, characteristic, complemented
2C2×#17986398C211yesmaximal, characteristic, complemented
3#22594170C211yesmaximal, characteristic
4#22589675C211yesmaximal, characteristic, complemented
5#22589675C211yesmaximal, characteristic, complemented
6#22589670C211yesmaximal, characteristic, complemented
7#22589665C211yesmaximal, characteristic, complemented
8#22589310C211yesmaximal, characteristic, complemented
9#22589302C211yesmaximal, characteristic, complemented
10#22588539C211yesmaximal, characteristic
11#22588539C211yesmaximal, characteristic
12#22588533C211yesmaximal, characteristic
13#22588528C211yesmaximal, characteristic
14#22588106C211yesmaximal, characteristic
15C2×#544×C71C211yesmaximal, characteristic
16C2×#14556120C2211yescharacteristic
17#17986405—24no—
18#17986398—24no—
19C2×#14556061C2211yescharacteristic
20C2×#14556061C2211yescharacteristic
21C2×#14556061C2211yescharacteristic
22C2×#14556061C2211yescharacteristic
23C2×#14556059C2211yescharacteristic
24C2×#14556059C2211yescharacteristic
25C2×#14556059C2211yescharacteristic
26C2×#14556059C2211yescharacteristic
27C2×C22⋊(C71⋊C8)C2211yescharacteristic
28C2×C22⋊(C71⋊C8)C2211yescharacteristic
29C2×#14556039C2211yescharacteristic
30C2×C4×C71⋊C8C2211yescharacteristic
31#17985932—21no—
32#17985932—21no—
33#17985922—21no—
34#17985922—21no—
35#17985897C2211yescharacteristic
36#17985897C2211yescharacteristic
37#17985869C2211yescharacteristic
38#17985859C2211yescharacteristic
39#17985859C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985761C2211yescharacteristic
48#17985754C2211yescharacteristic
49#17985754C2211yescharacteristic
50#544×C71—24no—
51#544×C71—24no—
52C2×C4⋊Q8×C71—21no—
53C2×C4⋊Q8×C71C2211yescharacteristic
54C2×C4⋊Q8×C71C2211yescharacteristic
55C2×#125×C71—21no—
56C2×#125×C71C2211yescharacteristic
57C2×#125×C71C2211yescharacteristic
58C2×C22⋊Q8×C71—21no—
59C2×C22⋊Q8×C71—21no—
60C2×#117×C71C2211yescharacteristic
61C2×#117×C71C2211yescharacteristic
62C22×C4⋊C4×C71C2211yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67#14556120—22no—
68#14556120—22no—
69C2×C4⋊Dic71—21no—
70C2×C4⋊Dic71—21no—
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×C4⋊Dic71C2311yescharacteristic
75C2×Dic71⋊C4—21no—
76C2×Dic71⋊C4—21no—
77C2×Dic71⋊C4—21no—
78C2×Dic71⋊C4—21no—
79C2×C22⋊Dic71—21no—
80C2×C22⋊Dic71—21no—
81#14556061—22no—
82#14556061—22no—
83#14556061—22no—
84#14556061—22no—
85#14556059—22no—
86#14556059—22no—
87#14556059—22no—
88#14556059—22no—
89C22⋊(C71⋊C8)Q814yescomplemented
90C22⋊(C71⋊C8)Q814yescomplemented
91#14556039—22no—
92C4×C71⋊C8—22no—
93C71⋊#95—21no—
94C71⋊#95—21no—
95C71⋊#95—21no—
96C71⋊#95—21no—
97C22×Q8×C71—21no—
98C22×Q8×C71—21no—
99C23×C4×C71—21no—
100C23×C4×C71C2311yescharacteristic
101C4⋊Q8×C71—28no—
102C4⋊Q8×C71—22no—
103C4⋊Q8×C71—22no—
104C4⋊Q8×C71—22no—
105C4⋊Q8×C71D414yes—
106#125×C71—28no—
107#125×C71—22no—
108#125×C71—22no—
109#125×C71—22no—
110#125×C71D414yes—
111C22⋊Q8×C71—28no—
112C22⋊Q8×C71—28no—
113#117×C71—22no—
114#117×C71—22no—
115#117×C71—22no—
116#117×C71—22no—
117C2×C4⋊C4×C71—21no—
118C2×C4⋊C4×C71—21no—
119C2×C4⋊C4×C71—21no—
120C2×C4⋊C4×C71—21no—
121C2×C4⋊C4×C71—21no—
122C2×C4⋊C4×C71—21no—
123C2×C4⋊C4×C71—21no—
124C2×C4⋊C4×C71—22no—
125C2×C4⋊C4×C71—22no—
126C2×C4⋊C4×C71D411yescharacteristic
127C2×C4⋊C4×C71D411yescharacteristic
128C2×C4⋊C4×C71C2311yescharacteristic
129C2×C4⋊C4×C71C2311yescharacteristic
130C2×C4⋊C4×C71C2311yescharacteristic
131C2×C4⋊C4×C71C2311yescharacteristic
132C2×C22⋊C4×C71—21no—
133C2×C22⋊C4×C71D411yescharacteristic
134C2×C22⋊C4×C71D411yescharacteristic
135C2×C42×C71—21no—
136C2×C42×C71C2311yescharacteristic
137C2×C42×C71C2311yescharacteristic
138C22×Dic71—21no—
139C22×Dic71—21no—
140C22×Dic71—21no—
141C22×Dic71—21no—
142C2×C71⋊C8—21no—
143C2×C71⋊C8—21no—
144C2×C71⋊C8—21no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8—21no—
148C2×C71⋊C8—21no—
149C2×C71⋊C8—21no—
150C2×C71⋊C8C2×Q812yes—
151C2×C71⋊C8C2×Q812yes—
152C2×C71⋊C8C2×Q812yes—
153C2×C71⋊C8C2×Q812yes—
154C4⋊Dic71—21no—
155C4⋊Dic71—21no—
156C4⋊Dic71—21no—
157C4⋊Dic71—21no—
158C4⋊Dic71—21no—
159C4⋊Dic71—21no—
160C4⋊Dic71—21no—
161C4⋊Dic71—21no—
162C4⋊Dic71—81no—
163C4⋊Dic71—81no—
164Dic71⋊C4—42no—
165Dic71⋊C4—42no—
166Dic71⋊C4—81no—
167Dic71⋊C4—81no—
168C22⋊Dic71—81no—
169C22⋊Dic71—81no—
170C24×C71C2×D411yescharacteristic
171C2×Q8×C71—24no—
172C2×Q8×C71—24no—
173C2×Q8×C71—81no—
174C2×Q8×C71—81no—
175C2×Q8×C71—81no—
176C2×Q8×C71—81no—
177C22×C4×C71—21no—
178C22×C4×C71—21no—
179C22×C4×C71—21no—
180C22×C4×C71—21no—
181C22×C4×C71—21no—
182C22×C4×C71—21no—
183C22×C4×C71—21no—
184C22×C4×C71—21no—
185C22×C4×C71—24no—
186C22×C4×C71—81no—
187C22×C4×C71—81no—
188C22×C4×C71C2×D411yescharacteristic
189C22×C4×C71C2×D411yescharacteristic
190C22×C4×C71C2×D411yescharacteristic
191C22×C4×C71C2×D411yescharacteristic
192C22×C4×C71C2×D411yescharacteristic
193C22×C4×C71C2×Q812yes—
194C22×C4×C71C2×Q812yes—
195C22×C4×C71C2411yescharacteristic
196C4⋊C4×C71—24no—
197C4⋊C4×C71—24no—
198C4⋊C4×C71—24no—
199C4⋊C4×C71—24no—
200C4⋊C4×C71—24no—
201C4⋊C4×C71—24no—
202C4⋊C4×C71—24no—
203C4⋊C4×C71—22no—
204C4⋊C4×C71—22no—
205C4⋊C4×C71—21no—
206C4⋊C4×C71—21no—
207C4⋊C4×C71—21no—
208C4⋊C4×C71—21no—
209C4⋊C4×C71C2×D412yes—
210C4⋊C4×C71C2×D412yes—
211C4⋊C4×C71C2×D412yes—
212C4⋊C4×C71C2×D412yes—
213C22⋊C4×C71—24no—
214C22⋊C4×C71—22no—
215C22⋊C4×C71—22no—
216C42×C71—24no—
217C42×C71—21no—
218C42×C71—21no—
219C42×C71C2×D412yes—
220C42×C71C2×D412yes—
221C23×C71—21no—
222C23×C71—21no—
223C23×C71—21no—
224C23×C71—21no—
225C23×C71—21no—
226C23×C71—21no—
227C23×C71C4⋊Q812yes—
228C23×C71C22×D411yescharacteristic
229Q8×C71—84no—
230Q8×C71—84no—
231C2×C4×C71—21no—
232C2×C4×C71—21no—
233C2×C4×C71—21no—
234C2×C4×C71—21no—
235C2×C4×C71—21no—
236C2×C4×C71—21no—
237C2×C4×C71—22no—
238C2×C4×C71—22no—
239C2×C4×C71—22no—
240C2×C4×C71—22no—
241C2×C4×C71—21no—
242C2×C4×C71—22no—
243C2×C4×C71—21no—
244C2×C4×C71—22no—
245C2×C4×C71—22no—
246C2×C4×C71—22no—
247C2×C4×C71—21no—
248C2×C4×C71—21no—
249C2×C4×C71—21no—
250C2×C4×C71—21no—
251C2×C4×C71—41no—
252C2×C4×C71—41no—
253C2×C4×C71—41no—
254C2×C4×C71—41no—
255C2×C4×C71—41no—
256C2×C4×C71—41no—
257C2×C4×C71—41no—
258C2×C4×C71—41no—
259C2×C4×C71—84no—
260C2×C4×C71#12012yes—
261C2×C4×C71#12012yes—
262C2×C4×C71C4⋊Q812yes—
263C2×C4×C71C22×D411yescharacteristic
264C2×C4×C71C22×D411yescharacteristic
265C2×C4×C71C22×Q811yescharacteristic
266C2×C4×C71C22×Q811yescharacteristic
267C2×C4×C712−1+411yescharacteristic
268C2×C4×C712−1+411yescharacteristic
269C2×Dic71—41no—
270C2×Dic71—41no—
271C2×Dic71—41no—
272C2×Dic71—41no—
273C2×Dic71—41no—
274C2×Dic71—41no—
275C2×Dic71—41no—
276C2×Dic71—41no—
277C2×Dic71—41no—
278C2×Dic71—41no—
279C2×Dic71—41no—
280C2×Dic71—41no—
281C71⋊C8—22no—
282C71⋊C8—22no—
283C71⋊C8—22no—
284C71⋊C8—22no—
285C22×C71—22no—
286C22×C71—22no—
287C22×C71—22no—
288C22×C71—41no—
289C22×C71—41no—
290C22×C71—41no—
291C22×C71—41no—
292C22×C71C2×#12011yescharacteristic
293C22×C71C2×C4⋊Q811yescharacteristic
294C22×C71#53611yescharacteristic
295C22×C71#57611yescharacteristic
296C22×C71#57711yescharacteristic
297C22×C71#57711yescharacteristic
298C22×C71#57811yescharacteristic
299C4×C71—22no—
300C4×C71—22no—
301C4×C71—22no—
302C4×C71—41no—
303C4×C71—42no—
304C4×C71—42no—
305C4×C71—42no—
306C4×C71—41no—
307C4×C71—42no—
308C4×C71D4×Q811yescharacteristic
309C4×C71D4×Q811yescharacteristic
310C4×C71D4×Q811yescharacteristic
311C4×C71D4×Q811yescharacteristic
312Dic71—81no—
313Dic71—81no—
314Dic71—81no—
315Dic71—81no—
316#25822—711nomaximal, Sylow
317C2×C71—42no—
318C2×C71#243411yescharacteristic
319C2×C71#301211yescharacteristic
320C2×C71#301311yescharacteristic
321C2×C71#316311yescharacteristic
322C2×C71#316411yescharacteristic
323C2×C71#324111yescharacteristic
324C2×C71#324211yescharacteristic
325C2×#544D7111yescharacteristic
326C2×#501—711no—
327C2×#499—711no—
328#2629—711no—
329#2059—711no—
330#2059—711no—
331#2058—711no—
332#2057—711no—
333#1992—711no—
334#1990—711no—
335#1854—711no—
336#1854—711no—
337#1853—711no—
338#1852—711no—
339#1750—711no—
340C71#2582211yescharacteristic, complemented, Sylow
341#544—24no—
342#544—24no—
343C2×C4⋊Q8—21no—
344C2×C4⋊Q8C2×D7111yescharacteristic
345C2×C4⋊Q8C2×D7111yescharacteristic
346C2×#125—21no—
347C2×#125—711no—
348C2×#125C2×D7111yescharacteristic
349C2×#125C2×D7111yescharacteristic
350C2×C22⋊Q8—21no—
351C2×C22⋊Q8—21no—
352C2×#117C2×D7111yescharacteristic
353C2×#117C2×D7111yescharacteristic
354C22×C4⋊C4C2×D7111yescharacteristic
355#501—1424no—
356#499—1424no—
357C2×#107—711no—
358C2×#107—711no—
359C2×#107—711no—
360C2×#107—711no—
361C2×#106—711no—
362C2×#106—711no—
363C2×#106—711no—
364C2×#106—711no—
365C2×C22⋊C8—711no—
366C2×C22⋊C8—711no—
367C2×#97—711no—
368C2×C4×C8—711no—
369#398—1421no—
370#398—1421no—
371#396—1421no—
372#396—1421no—
373#389—711no—
374#389—711no—
375#384—711no—
376#380—711no—
377#380—711no—
378#340—711no—
379#340—711no—
380#340—711no—
381#340—711no—
382#340—711no—
383#340—711no—
384#340—711no—
385#340—711no—
386#336—711no—
387#336—711no—
388C22×Q8—21no—
389C22×Q8—21no—
390C23×C4—21no—
391C23×C4C22×D7111yescharacteristic
392C22×C8—711no—
393C22×C8—711no—
394C22×C8—711no—
395C22×C8—711no—
396C4⋊Q8—28no—
397C4⋊Q8—22no—
398C4⋊Q8—22no—
399C4⋊Q8—22no—
400C4⋊Q8C22⋊D7114yes—
401#125—28no—
402#125—22no—
403#125—22no—
404#125—22no—
405#125—1422no—
406#125—1422no—
407#125C22⋊D7114yes—
408C22⋊Q8—28no—
409C22⋊Q8—28no—
410#117—22no—
411#117—22no—
412#117—22no—
413#117—22no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—711no—
424C2×C4⋊C4—711no—
425C2×C4⋊C4—711no—
426C2×C4⋊C4—711no—
427C2×C4⋊C4—1421no—
428C2×C4⋊C4—1421no—
429C2×C4⋊C4—1421no—
430C2×C4⋊C4—1421no—
431C2×C4⋊C4—1421no—
432C2×C4⋊C4—1421no—
433C2×C4⋊C4C22⋊D7111yescharacteristic
434C2×C4⋊C4C22⋊D7111yescharacteristic
435C2×C4⋊C4C22×D7111yescharacteristic
436C2×C4⋊C4C22×D7111yescharacteristic
437C2×C4⋊C4C22×D7111yescharacteristic
438C2×C4⋊C4C22×D7111yescharacteristic
439C2×C22⋊C4—21no—
440C2×C22⋊C4—1421no—
441C2×C22⋊C4—1421no—
442C2×C22⋊C4C22⋊D7111yescharacteristic
443C2×C22⋊C4C22⋊D7111yescharacteristic
444C2×C42—21no—
445C2×C42C22×D7111yescharacteristic
446C2×C42C22×D7111yescharacteristic
447#107—1422no—
448#107—1422no—
449#107—1422no—
450#107—1422no—
451#106—1422no—
452#106—1422no—
453#106—1422no—
454#106—1422no—
455C22⋊C8—714no—
456C22⋊C8—714no—
457#97—1422no—
458C4×C8—1422no—
459#95—1421no—
460#95—1421no—
461#95—1421no—
462#95—1421no—
463C24C2×C22⋊D7111yescharacteristic
464C2×Q8—24no—
465C2×Q8—24no—
466C2×Q8—81no—
467C2×Q8—81no—
468C2×Q8—81no—
469C2×Q8—81no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—24no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—1421no—
482C22×C4—1421no—
483C22×C4—1421no—
484C22×C4—1421no—
485C22×C4C2×C22⋊D7111yescharacteristic
486C22×C4C2×C22⋊D7111yescharacteristic
487C22×C4C2×C22⋊D7111yescharacteristic
488C22×C4C2×C22⋊D7111yescharacteristic
489C22×C4C2×C22⋊D7111yescharacteristic
490C22×C4Q8×D7112yes—
491C22×C4Q8×D7112yes—
492C22×C4C23×D7111yescharacteristic
493C2×C8—712no—
494C2×C8—712no—
495C2×C8—712no—
496C2×C8—712no—
497C2×C8—1421no—
498C2×C8—1421no—
499C2×C8—1421no—
500C2×C8—1421no—
501C2×C8—1421no—
502C2×C8—1421no—
503C2×C8—1421no—
504C2×C8—1421no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—2842no—
519C4⋊C4—2842no—
520C4⋊C4—5681no—
521C4⋊C4—5681no—
522C4⋊C4—5681no—
523C4⋊C4—5681no—
524C4⋊C4—1421no—
525C4⋊C4—1421no—
526C4⋊C4—1421no—
527C4⋊C4—1421no—
528C4⋊C4—1421no—
529C4⋊C4—1421no—
530C4⋊C4—1421no—
531C4⋊C4—1421no—
532C4⋊C4C2×C22⋊D7112yes—
533C4⋊C4C2×C22⋊D7112yes—
534C4⋊C4C2×C22⋊D7112yes—
535C4⋊C4C2×C22⋊D7112yes—
536C22⋊C4—24no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—5681no—
540C22⋊C4—5681no—
541C42—24no—
542C42—21no—
543C42—21no—
544C42C2×C22⋊D7112yes—
545C42C2×C22⋊D7112yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23#1455612612yes—
553C23C22×C22⋊D7111yescharacteristic
554Q8—84no—
555Q8—84no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—2841no—
586C2×C4—2841no—
587C2×C4—2841no—
588C2×C4—2841no—
589C2×C4—2841no—
590C2×C4—2841no—
591C2×C4—2841no—
592C2×C4—2841no—
593C2×C4—2841no—
594C2×C4—2841no—
595C2×C4—2841no—
596C2×C4—2841no—
597C2×C4#1455609712yes—
598C2×C4#1455609712yes—
599C2×C4#1455612612yes—
600C2×C4C22×C22⋊D7111yescharacteristic
601C2×C4C22×C22⋊D7111yescharacteristic
602C2×C4C2×Q8×D7111yescharacteristic
603C2×C4C2×Q8×D7111yescharacteristic
604C2×C4#1455617611yescharacteristic
605C2×C4#1455617611yescharacteristic
606C8—1422no—
607C8—1422no—
608C8—1422no—
609C8—1422no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22C2×#1455609711yescharacteristic
618C22C2×#1455612611yescharacteristic
619C22#1798657311yescharacteristic
620C22#1798680211yescharacteristic
621C22#1798680711yescharacteristic
622C22#1798681011yescharacteristic
623C22#1798681411yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—5681no—
634C4—5681no—
635C4—5681no—
636C4—5681no—
637C4Q8×C22⋊D7111yescharacteristic
638C4Q8×C22⋊D7111yescharacteristic
639C4Q8×C22⋊D7111yescharacteristic
640C4Q8×C22⋊D7111yescharacteristic
641C2—42no—
642C2#2259198011yescharacteristic
643C2#2259764411yescharacteristic
644C2#2259765511yescharacteristic
645C2#2259886611yescharacteristic
646C2#2259887411yescharacteristic
647C2#2259959311yescharacteristic
648C2#2259960811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×#544×C71C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenterC2×#544D71
hypocenterC71#25822
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. C2×#17986405C2semidirect
2. C2×#17986398C2semidirect
4. #22589675C2semidirect
5. #22589675C2semidirect
6. #22589670C2semidirect
7. #22589665C2semidirect
8. #22589310C2semidirect
9. #22589302C2semidirect
89. C22⋊(C71⋊C8)Q8semidirect
90. C22⋊(C71⋊C8)Q8semidirect
340. C71#25822semidirect

Maximal subgroups

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

subgroupindex
C2×#179864052
C2×#179863982
#225941702
#225896752
#225896702
#225896652
#225893102
#225893022
#225885392
#225885332
#225885282
#225881062
C2×#544×C712
#2582271