Finite Group Databasebeta

← #33102575

Subgroups — #33102575

Order 18688.

1086 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989296C211yesmaximal, characteristic
2C2×#17988711C211yesmaximal, characteristic, complemented
3#22612883C211yesmaximal, characteristic, complemented
4#22610895C211yesmaximal, characteristic, complemented
5#22610894C211yesmaximal, characteristic, complemented
6#22610890C211yesmaximal, characteristic, complemented
7#22610886C211yesmaximal, characteristic, complemented
8#22610885C211yesmaximal, characteristic, complemented
9#22610877C211yesmaximal, characteristic, complemented
10#22609939C211yesmaximal, characteristic
11#22609929C211yesmaximal, characteristic
12#102×Dic73C211yesmaximal, characteristic
13#102×Dic73C211yesmaximal, characteristic
14C4×#14556512C211yesmaximal, characteristic
15C73×#1804C211yesmaximal, characteristic
16#17989296—24no—
17#17989296C418yescomplemented
18C2×#14556590C2211yescharacteristic
19C2×#14556583C2211yescharacteristic
20C2×#14556582C2211yescharacteristic
21C2×#14556570—21no—
22C2×#14556568—21no—
23C2×#14556568—21no—
24C2×C4×Dic146C2211yescharacteristic
25C2×D4×Dic73C2211yescharacteristic
26C2×D4×Dic73C2211yescharacteristic
27C22×#11924007—21no—
28#17988711—24no—
29C2×#14556520C2211yescharacteristic
30C2×#14556520C2211yescharacteristic
31C2×#14556512C2211yescharacteristic
32C2×#14556512C2211yescharacteristic
33C2×C4×#10596831C2211yescharacteristic
34C2×C8×Dic73C2211yescharacteristic
35C2×C8×Dic73C2211yescharacteristic
36#17988491—21no—
37#17988472—21no—
38#17988469—21no—
39#17988456C2211yescharacteristic
40#17988452C2211yescharacteristic
41C4⋊C4×Dic73C2211yescharacteristic
42C4⋊C4×Dic73C2211yescharacteristic
43C4×C4⋊Dic73C2211yescharacteristic
44#17988438—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988349C2211yescharacteristic
55C2×C4×D4×C73C2211yescharacteristic
56C2×C4⋊C8×C73C2211yescharacteristic
57C2×#102×C73C2211yescharacteristic
58C2×#102×C73C2211yescharacteristic
59#382×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#340×C73C2211yescharacteristic
62C22×Dic146—21no—
63C22×Dic146D411yescharacteristic
64C23×Dic73—21no—
65C23×Dic73—21no—
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556590—22no—
69#14556590C2×C414yes—
70#14556583—22no—
71#14556583C2×C414yes—
72#14556582—22no—
73#14556582C2×C414yes—
74#14556570—28no—
75#14556568—28no—
76#14556568—28no—
77C4×Dic146C2×C414yes—
78C4×Dic146D414yes—
79D4×Dic73—22no—
80D4×Dic73—22no—
81D4×Dic73C2×C414yes—
82D4×Dic73C2×C414yes—
83C2×C4⋊Dic73—21no—
84C2×C4⋊Dic73D411yescharacteristic
85C2×C4⋊Dic73C2311yescharacteristic
86C2×C4⋊Dic73C2311yescharacteristic
87C2×C4⋊Dic73C2311yescharacteristic
88C2×C4⋊Dic73C2311yescharacteristic
89C2×#11924007—21no—
90C2×#11924007—21no—
91C2×#11924007—28no—
92C2×#11924007—21no—
93C2×#11924007—21no—
94C2×#11924007—21no—
95C2×#11924007—21no—
96C2×#11924007D411yescharacteristic
97C2×#11924007D411yescharacteristic
98C2×C22⋊Dic73—21no—
99C2×C22⋊Dic73—21no—
100C2×C4×Dic73—21no—
101C2×C4×Dic73—21no—
102C2×C4×Dic73—21no—
103C2×C4×Dic73C2311yescharacteristic
104C2×C4×Dic73C2311yescharacteristic
105#14556520—22no—
106#14556520—22no—
107#14556512—22no—
108#14556512Q814yescomplemented
109C4×#10596831—22no—
110C8×Dic73—22no—
111C8×Dic73—22no—
112#14556498—21no—
113#14556498—21no—
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117C22×D4×C73C2311yescharacteristic
118C23×C4×C73—21no—
119C22×C8×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C4×D4×C73—22no—
122C4×D4×C73C2×C414yes—
123C2×C4⋊C4×C73C2311yescharacteristic
124C2×C4⋊C4×C73C2311yescharacteristic
125C2×C4⋊C4×C73C2311yescharacteristic
126C2×C22⋊C4×C73—21no—
127C2×C42×C73C2311yescharacteristic
128C4⋊C8×C73—22no—
129#102×C73—22no—
130#102×C73—22no—
131#95×C73—21no—
132#95×C73—21no—
133C2×Dic146—22no—
134C2×Dic146—22no—
135C2×Dic146—24no—
136C2×Dic146—81no—
137C2×Dic146—81no—
138C2×Dic146C22⋊C414yes—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—24no—
143C22×Dic73—24no—
144C22×Dic73—21no—
145C22×Dic73—42no—
146C22×Dic73—42no—
147C22×Dic73—81no—
148C22×Dic73—81no—
149C22×Dic73C2×D411yescharacteristic
150C22×Dic73C2×D411yescharacteristic
151C22×Dic73C2×D411yescharacteristic
152C22×Dic73C2×D411yescharacteristic
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831C2×Q812yes—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—24no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73C22⋊C414yes—
166C4⋊Dic73C22×C412yes—
167C4⋊Dic73C22×C412yes—
168C4⋊Dic73C2×Q812yes—
169#11924007—24no—
170#11924007—24no—
171#11924007—24no—
172#11924007—24no—
173#11924007—42no—
174#11924007—81no—
175#11924007C22⋊C414yes—
176#11924007C22⋊C414yes—
177C22⋊Dic73—24no—
178C22⋊Dic73—24no—
179C4×Dic73—42no—
180C4×Dic73—81no—
181C4×Dic73—81no—
182C4×Dic73C22×C412yes—
183C4×Dic73C22×C412yes—
184C4×Dic73C2×D412yes—
185C4×Dic73C2×D412yes—
186C24×C73—21no—
187C2×D4×C73—21no—
188C2×D4×C73—21no—
189C2×D4×C73—21no—
190C2×D4×C73C22×C412yes—
191C2×D4×C73C2×Q812yes—
192C2×D4×C73#4112yes—
193C22×C4×C73—21no—
194C22×C4×C73—24no—
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73—81no—
198C22×C4×C73—81no—
199C22×C4×C73C2×D411yescharacteristic
200C22×C4×C73C2×D411yescharacteristic
201C22×C4×C73C2411yescharacteristic
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73—21no—
206C2×C8×C73—21no—
207C2×C8×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—21no—
211C4⋊C4×C73—21no—
212C4⋊C4×C73—21no—
213C4⋊C4×C73C22×C412yes—
214C22⋊C4×C73—24no—
215C42×C73C22×C412yes—
216C42×C73C2×D412yes—
217C23×C73—22no—
218C23×C73—22no—
219C23×C73—22no—
220C23×C73—41no—
221C23×C73—41no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73C22×D411yescharacteristic
225D4×C73—22no—
226D4×C73—22no—
227D4×C73—22no—
228D4×C73C4×Q814yes—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—22no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—84no—
243C2×C4×C73C2×C22⋊C412yes—
244C2×C4×C73C2×C22⋊C412yes—
245C2×C4×C73C23×C411yescharacteristic
246C2×C4×C73C22×D411yescharacteristic
247C2×C4×C73C22×Q811yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C732−1+411yescharacteristic
250C2×C4×C732−1+411yescharacteristic
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—22no—
258C2×Dic73—22no—
259C2×Dic73—22no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—48no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—84no—
271C2×Dic73C2×C22⋊C412yes—
272C2×Dic73C2×C22⋊C412yes—
273C2×Dic73C2×C22⋊C412yes—
274C2×Dic73C2×C22⋊C412yes—
275C2×Dic73#13512yes—
276C2×Dic73#13512yes—
277Dic146—28no—
278Dic146—84no—
279C8×C73—41no—
280C8×C73—41no—
281#10596831—22no—
282#10596831—41no—
283C22×C73—24no—
284C22×C73—42no—
285C22×C73—42no—
286C22×C73—42no—
287C22×C73—42no—
288C22×C73—42no—
289C22×C73—42no—
290C22×C73C22×C22⋊C411yescharacteristic
291C22×C73#54811yescharacteristic
292C22×C73D4×Q811yescharacteristic
293C22×C73C2×#13511yescharacteristic
294C22×C73C2×#13511yescharacteristic
295C22×C73#57611yescharacteristic
296C22×C73#57711yescharacteristic
297C4×C73—22no—
298C4×C73—22no—
299C4×C73—42no—
300C4×C73—81no—
301C4×C73—81no—
302C4×C73C2×C4×Q811yescharacteristic
303C4×C73#51911yescharacteristic
304C4×C73#54811yescharacteristic
305C4×C73D4×Q811yescharacteristic
306Dic73—22no—
307Dic73—22no—
308Dic73—22no—
309Dic73—42no—
310Dic73—42no—
311Dic73—42no—
312Dic73—81no—
313Dic73#41614yes—
314#22519—731nomaximal, Sylow
315C2×C73—44no—
316C2×C73Q8×C22⋊C411yescharacteristic
317C2×C73C2×#41611yescharacteristic
318C2×C73#289711yescharacteristic
319C2×C73#336011yescharacteristic
320C2×C73#336111yescharacteristic
321C2×C73#338011yescharacteristic
322C2×C73#338111yescharacteristic
323C2×#554—731no—
324C2×C8×Q8—731no—
325#2355—731no—
326#1991—731no—
327#1991—731no—
328#1990—731no—
329#1989—731no—
330#1989—731no—
331#1987—731no—
332#1804—731no—
333#1804D7311yescharacteristic
334#1802—731no—
335C4×#102—731no—
336C4×#102—731no—
337C4×#102—731no—
338C73#2251911yescharacteristic, complemented, Sylow
339#554—738no—
340#554—1464no—
341C2×C4⋊Q8—731no—
342C2×#125—731no—
343C2×#125—731no—
344C2×C22⋊Q8—1461no—
345C2×C22⋊Q8—1461no—
346C2×C22⋊Q8—1461no—
347C2×C4×Q8—731no—
348C2×C4×D4—731no—
349C2×C4×D4—731no—
350C2×C4×D4C2×D7311yescharacteristic
351C22×C4⋊C4—1461no—
352C8×Q8—1464no—
353C2×C4⋊C8—731no—
354C2×C4⋊C8—731no—
355C2×C4⋊C8C2×D7311yescharacteristic
356C2×#102—731no—
357C2×#102—731no—
358C2×#102C2×D7311yescharacteristic
359C2×#102C2×D7311yescharacteristic
360C2×C4×C8—731no—
361C2×C4×C8—731no—
362C2×C4×C8—731no—
363#391—1461no—
364#387—1461no—
365#387—1461no—
366#384—731no—
367#382—731no—
368#382C2×D7311yescharacteristic
369C4×C4⋊C4—731no—
370C4×C4⋊C4—731no—
371C4×C4⋊C4—731no—
372C4×C4⋊C4—1461no—
373#340—731no—
374#340—731no—
375#340—731no—
376#340—731no—
377#340—731no—
378#340—731no—
379#340—731no—
380#340—731no—
381#340—731no—
382#340—731no—
383#340C2×D7311yescharacteristic
384#340C2×D7311yescharacteristic
385C22×Q8—731no—
386C22×Q8—1461no—
387C22×D4C22×D7311yescharacteristic
388C23×C4—21no—
389C23×C4—1461no—
390C23×C4—1461no—
391C22×C8—731no—
392C22×C8—731no—
393C22×C8C22×D7311yescharacteristic
394C22×C8C22×D7311yescharacteristic
395C4⋊Q8—734no—
396C4⋊Q8—1462no—
397#125—734no—
398#125—734no—
399#125—1462no—
400#125—1462no—
401C22⋊Q8—1468no—
402C22⋊Q8—1468no—
403C22⋊Q8—1468no—
404C4×Q8—734no—
405C4×Q8—734no—
406C4×D4—22no—
407C4×D4—734no—
408C4×D4—734no—
409C4×D4—1462no—
410C4×D4—1462no—
411C4×D4C4×D7314yes—
412C2×C4⋊C4—731no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4—731no—
418C2×C4⋊C4—731no—
419C2×C4⋊C4—1461no—
420C2×C4⋊C4—1461no—
421C2×C4⋊C4—1468no—
422C2×C4⋊C4—1461no—
423C2×C4⋊C4—1461no—
424C2×C4⋊C4—1461no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4C22×D7311yescharacteristic
428C2×C4⋊C4C22×D7311yescharacteristic
429C2×C4⋊C4C22×D7311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1461no—
432C2×C22⋊C4—1461no—
433C2×C42—731no—
434C2×C42—731no—
435C2×C42—1461no—
436C2×C42—1461no—
437C2×C42—1461no—
438C2×C42C22×D7311yescharacteristic
439C4⋊C8—22no—
440C4⋊C8—1462no—
441C4⋊C8—1462no—
442#102—22no—
443#102—22no—
444#102—734no—
445#102—1462no—
446C4×C8—1462no—
447C4×C8—1462no—
448C4×C8—1462no—
449#95—21no—
450#95—21no—
451#95—1461no—
452#95—1461no—
453#95—1461no—
454#95—1461no—
455#95—1461no—
456C24—21no—
457C2×Q8—5841no—
458C2×Q8—5841no—
459C2×Q8—734no—
460C2×Q8—1464no—
461C2×Q8—1462no—
462C2×Q8—1462no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4C2×C4×D7312yes—
467C2×D4Q8×D7312yes—
468C2×D4#1192402712yes—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—2922no—
476C22×C4—2922no—
477C22×C4—5841no—
478C22×C4—5841no—
479C22×C4—731no—
480C22×C4—731no—
481C22×C4—731no—
482C22×C4—731no—
483C22×C4—1461no—
484C22×C4—1464no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4—1464no—
489C22×C4D4×D7311yescharacteristic
490C22×C4D4×D7311yescharacteristic
491C22×C4C23×D7311yescharacteristic
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—732no—
499C2×C8—1461no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—2922no—
510C4⋊C4—5841no—
511C4⋊C4—734no—
512C4⋊C4—734no—
513C4⋊C4—734no—
514C4⋊C4—732no—
515C4⋊C4—732no—
516C4⋊C4—732no—
517C4⋊C4—1464no—
518C4⋊C4—1464no—
519C4⋊C4—1464no—
520C4⋊C4—1461no—
521C4⋊C4—1461no—
522C4⋊C4—1464no—
523C4⋊C4—1464no—
524C4⋊C4—1461no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4C2×C4×D7312yes—
528C22⋊C4—24no—
529C22⋊C4—1464no—
530C22⋊C4—1464no—
531C42—2922no—
532C42—5841no—
533C42—5841no—
534C42—732no—
535C42—732no—
536C42—732no—
537C42—732no—
538C42C2×C4×D7312yes—
539C42D4×D7312yes—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C2×D4×D7311yescharacteristic
548Q8—5844no—
549Q8—1468no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4#1455655614yes—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—84no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2928no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—5844no—
579C2×C4—732no—
580C2×C4—732no—
581C2×C4—732no—
582C2×C4—732no—
583C2×C4—732no—
584C2×C4—732no—
585C2×C4—1462no—
586C2×C4—1462no—
587C2×C4—1462no—
588C2×C4—1461no—
589C2×C4—1461no—
590C2×C4—1461no—
591C2×C4—1461no—
592C2×C4—1461no—
593C2×C4—1461no—
594C2×C4C22⋊C4×D7312yes—
595C2×C4C22⋊C4×D7312yes—
596C2×C4C22×C4×D7311yescharacteristic
597C2×C4C2×D4×D7311yescharacteristic
598C2×C4C2×Q8×D7311yescharacteristic
599C2×C4C2×#1192402711yescharacteristic
600C2×C4#1455663811yescharacteristic
601C2×C4#1455663811yescharacteristic
602C8—41no—
603C8—41no—
604C8—2921no—
605C8—1462no—
606C22—24no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22C2×C22⋊C4×D7311yescharacteristic
614C22#1798924311yescharacteristic
615C22#1798925011yescharacteristic
616C22#135×D7311yescharacteristic
617C22#135×D7311yescharacteristic
618C22#1798939111yescharacteristic
619C22#1798939711yescharacteristic
620C4—22no—
621C4—22no—
622C4—42no—
623C4—81no—
624C4—81no—
625C4—2922no—
626C4—2922no—
627C4—2922no—
628C4—5841no—
629C4—734no—
630C4—1462no—
631C4—1462no—
632C4—1462no—
633C4C2×#1455655611yescharacteristic
634C4#1798906411yescharacteristic
635C4#1798924311yescharacteristic
636C4#1798925011yescharacteristic
637C2—44no—
638C2#2261276311yescharacteristic
639C2#416×D7311yescharacteristic
640C2#2261845611yescharacteristic
641C2#2262257511yescharacteristic
642C2#2262258611yescharacteristic
643C2#2262279311yescharacteristic
644C2#2262280111yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1804C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1804D73
hypocenterC73#22519
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
2. C2×#17988711C2semidirect
3. #22612883C2semidirect
4. #22610895C2semidirect
5. #22610894C2semidirect
6. #22610890C2semidirect
7. #22610886C2semidirect
8. #22610885C2semidirect
9. #22610877C2semidirect
17. #17989296C4semidirect
108. #14556512Q8semidirect
338. C73#22519semidirect

Maximal subgroups

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

subgroupindex
C2×#179892962
C2×#179887112
#226128832
#226108952
#226108942
#226108902
#226108862
#226108852
#226108772
#226099392
#226099292
#102×Dic732
C4×#145565122
C73×#18042
#2251973