Finite Group Databasebeta

← #31912409

Subgroups — #31912409

Order 18176.

768 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985973C211yesmaximal, characteristic
2#22596785C211yesmaximal, characteristic
3#22586460C214yesmaximal, complemented
4C2×#410×C71C211yesmaximal, characteristic
5C22×#11765849C411yescharacteristic, complemented
6C22×#11765849C411yescharacteristic, complemented
7C2×#14556087—21no—
8#17985973—21no—
9#17985973—21no—
10#17985973C412yescomplemented
11#17985973C2212yes—
12C2×#14556043—21no—
13C2×#14556043C412yescomplemented
14#17985966—21no—
15#17985966—21no—
16#17985966C412yescomplemented
17#17985966C2212yes—
18#17985963—21no—
19#17985963—21no—
20C22×#41×C71C2211yescharacteristic
21C2×#117×C71—21no—
22#410×C71—21no—
23#410×C71—21no—
24#410×C71C412yescomplemented
25#410×C71C2212yes—
26C2×#99×C71—21no—
27C2×#99×C71C412yescomplemented
28C2×#11765849—21no—
29C2×#11765849—21no—
30C2×#11765849—21no—
31C2×#11765849—41no—
32C2×#11765849—41no—
33C2×#11765849—41no—
34C2×#11765849—41no—
35C2×#11765849C2×C412yescomplemented
36C22×C71⋊C8—21no—
37C22×C71⋊C8C2×C412yescomplemented
38#14556087—22no—
39#14556087—22no—
40#14556087—41no—
41#14556087—41no—
42C2×C4⋊Dic71—41no—
43C2×C22⋊Dic71—22no—
44C2×C4×Dic71—41no—
45#14556043—22no—
46#14556043—22no—
47#14556043—22no—
48#14556043C2×C414yescomplemented
49C22⋊(C71⋊C8)—22no—
50C22⋊(C71⋊C8)—22no—
51C22⋊(C71⋊C8)—22no—
52C22⋊(C71⋊C8)—22no—
53C2×#41×C71—41no—
54C2×#41×C71—41no—
55C2×#41×C71—41no—
56C2×#41×C71—41no—
57C2×#41×C71C2×C411yescharacteristic
58C2×#41×C71C2×C411yescharacteristic
59C2×#41×C71C2×C411yescharacteristic
60C2×#41×C71D411yescharacteristic
61C2×#41×C71D411yescharacteristic
62C2×#41×C71D411yescharacteristic
63C2×#41×C71Q811yescharacteristic
64C2×#41×C71C2311yescharacteristic
65C22×Q8×C71Q811yescharacteristic
66C22×D4×C71C2×C411yescharacteristic
67C22×D4×C71C2×C411yescharacteristic
68C22×D4×C71D411yescharacteristic
69C23×C4×C71C2×C411yescharacteristic
70C23×C4×C71D411yescharacteristic
71C23×C4×C71D411yescharacteristic
72#117×C71—22no—
73#117×C71—22no—
74#117×C71—41no—
75#117×C71—41no—
76C2×C4⋊C4×C71—41no—
77C2×C22⋊C4×C71—22no—
78C2×C42×C71—41no—
79#99×C71—22no—
80#99×C71—22no—
81#99×C71—22no—
82#99×C71C2×C414yescomplemented
83C22×Dic71—42no—
84#11765849—42no—
85#11765849—41no—
86#11765849—41no—
87#11765849—41no—
88#11765849—41no—
89#11765849—42no—
90C2×C71⋊C8—22no—
91C2×C71⋊C8—22no—
92C2×C71⋊C8—42no—
93C2×C71⋊C8—42no—
94C4⋊Dic71—42no—
95C4⋊Dic71—42no—
96C22⋊Dic71—24no—
97C22⋊Dic71—42no—
98C4×Dic71—42no—
99C4×Dic71—42no—
100C24×C71C4211yescharacteristic
101C24×C71C22⋊C411yescharacteristic
102C24×C71C22⋊C411yescharacteristic
103#41×C71—41no—
104#41×C71—41no—
105#41×C71—41no—
106#41×C71—41no—
107#41×C71—41no—
108#41×C71—41no—
109#41×C71—41no—
110#41×C71—41no—
111#41×C71—41no—
112#41×C71—41no—
113#41×C71—41no—
114#41×C71—41no—
115#41×C71—41no—
116#41×C71—41no—
117#41×C71—41no—
118#41×C71—41no—
119C2×Q8×C71—41no—
120C2×Q8×C71—41no—
121C2×Q8×C71C4⋊C411yescharacteristic
122C2×Q8×C71C4⋊C411yescharacteristic
123C2×Q8×C71C4⋊C411yescharacteristic
124C2×Q8×C71C2×Q811yescharacteristic
125C2×D4×C71—41no—
126C2×D4×C71—41no—
127C2×D4×C71—41no—
128C2×D4×C71—41no—
129C2×D4×C71—41no—
130C2×D4×C71—41no—
131C2×D4×C71C4211yescharacteristic
132C2×D4×C71C4211yescharacteristic
133C2×D4×C71C22⋊C411yescharacteristic
134C2×D4×C71C22⋊C411yescharacteristic
135C2×D4×C71C22⋊C411yescharacteristic
136C2×D4×C71C22⋊C411yescharacteristic
137C2×D4×C71C4⋊C411yescharacteristic
138C2×D4×C71C4⋊C411yescharacteristic
139C2×D4×C71C4⋊C411yescharacteristic
140C2×D4×C71C22×C411yescharacteristic
141C2×D4×C71C22×C411yescharacteristic
142C2×D4×C71C2×D411yescharacteristic
143C22×C4×C71—21no—
144C22×C4×C71—21no—
145C22×C4×C71—21no—
146C22×C4×C71—21no—
147C22×C4×C71—21no—
148C22×C4×C71—21no—
149C22×C4×C71—21no—
150C22×C4×C71—21no—
151C22×C4×C71—21no—
152C22×C4×C71—21no—
153C22×C4×C71—21no—
154C22×C4×C71—21no—
155C22×C4×C71—42no—
156C22×C4×C71C4211yescharacteristic
157C22×C4×C71C22⋊C411yescharacteristic
158C22×C4×C71C22⋊C411yescharacteristic
159C22×C4×C71C22⋊C411yescharacteristic
160C22×C4×C71C22⋊C411yescharacteristic
161C22×C4×C71C22⋊C411yescharacteristic
162C22×C4×C71C22⋊C411yescharacteristic
163C22×C4×C71C4⋊C411yescharacteristic
164C22×C4×C71C4⋊C411yescharacteristic
165C22×C4×C71C4⋊C411yescharacteristic
166C22×C4×C71C4⋊C411yescharacteristic
167C22×C4×C71C4⋊C411yescharacteristic
168C22×C4×C71C4⋊C411yescharacteristic
169C22×C4×C71C22×C411yescharacteristic
170C22×C4×C71C2×D411yescharacteristic
171C22×C4×C71C2×D411yescharacteristic
172C4⋊C4×C71—42no—
173C4⋊C4×C71—42no—
174C22⋊C4×C71—24no—
175C22⋊C4×C71—42no—
176C42×C71—42no—
177C42×C71—42no—
178C23×C71—21no—
179C23×C71—21no—
180C23×C71—21no—
181C23×C71—21no—
182C23×C71—21no—
183C23×C71—21no—
184C23×C71—81no—
185C23×C71—81no—
186C23×C71—81no—
187C23×C71#9511yescharacteristic
188C23×C71#9511yescharacteristic
189C23×C71#9511yescharacteristic
190C23×C71#9511yescharacteristic
191C23×C71C2×C4211yescharacteristic
192C23×C71C2×C22⋊C411yescharacteristic
193C23×C71C2×C22⋊C411yescharacteristic
194Q8×C71—41no—
195Q8×C71—41no—
196Q8×C71—41no—
197Q8×C71—41no—
198D4×C71—41no—
199D4×C71—41no—
200D4×C71—41no—
201D4×C71—41no—
202D4×C71—41no—
203D4×C71—41no—
204D4×C71—41no—
205D4×C71—41no—
206D4×C71—41no—
207D4×C71—41no—
208D4×C71—41no—
209D4×C71—41no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—21no—
230C2×C4×C71—21no—
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—21no—
238C2×C4×C71—21no—
239C2×C4×C71—21no—
240C2×C4×C71—44no—
241C2×C4×C71—84no—
242C2×C4×C71#9511yescharacteristic
243C2×C4×C71#9511yescharacteristic
244C2×C4×C71#9511yescharacteristic
245C2×C4×C71#9511yescharacteristic
246C2×C4×C71#9911yescharacteristic
247C2×C4×C71#9911yescharacteristic
248C2×C4×C71#9911yescharacteristic
249C2×C4×C71#9911yescharacteristic
250C2×C4×C71C2×C22⋊C411yescharacteristic
251C2×C4×C71C2×C4⋊C411yescharacteristic
252C2×C4×C71C2×C4⋊C411yescharacteristic
253C2×C4×C71C2×C4⋊C411yescharacteristic
254C2×Dic71—44no—
255C2×Dic71—84no—
256C71⋊C8—42no—
257C71⋊C8—42no—
258C22×C71—21no—
259C22×C71—21no—
260C22×C71—21no—
261C22×C71—21no—
262C22×C71—21no—
263C22×C71—21no—
264C22×C71—41no—
265C22×C71—41no—
266C22×C71—41no—
267C22×C71—41no—
268C22×C71—41no—
269C22×C71—41no—
270C22×C71—41no—
271C22×C71—42no—
272C22×C71—42no—
273C22×C71—81no—
274C22×C71—81no—
275C22×C71—81no—
276C22×C71—81no—
277C22×C71#34211yescharacteristic
278C22×C71#34211yescharacteristic
279C22×C71C2×#9511yescharacteristic
280C4×C71—21no—
281C4×C71—21no—
282C4×C71—21no—
283C4×C71—21no—
284C4×C71—21no—
285C4×C71—21no—
286C4×C71—84no—
287C4×C71#34211yescharacteristic
288C4×C71#34211yescharacteristic
289C4×C71C2×#9911yescharacteristic
290C4×C71C2×#9911yescharacteristic
291Dic71—84no—
292#12577—711nomaximal, Sylow
293C2×C71—21no—
294C2×C71—21no—
295C2×C71—21no—
296C2×C71—41no—
297C2×C71—41no—
298C2×C71—41no—
299C2×C71—41no—
300C2×C71—41no—
301C2×C71—41no—
302C2×C71C2×#34211yescharacteristic
303C2×#410—711no—
304C2×#410D7111yescharacteristic
305#2879—711no—
306#1391—714no—
307C71#1257711yescharacteristic, complemented, Sylow
308C22×#41C2×D7111yescharacteristic
309C22×M16—711no—
310C22×M16—711no—
311C2×#117—21no—
312C2×#117—1421no—
313#410—21no—
314#410—21no—
315#410—712no—
316#410—712no—
317#410—1421no—
318#410—1421no—
319#410Dic7112yescomplemented
320#410C2×D7112yes—
321C2×#99—21no—
322C2×#99—712no—
323C2×#99—1421no—
324C2×#99Dic7112yescomplemented
325#408—712no—
326#408—712no—
327#408—1421no—
328#408—1421no—
329#407—1421no—
330#407—1421no—
331C2×#41—41no—
332C2×#41—41no—
333C2×#41—41no—
334C2×#41—41no—
335C2×#41Dic14211yescharacteristic
336C2×#41C4×D7111yescharacteristic
337C2×#41C4×D7111yescharacteristic
338C2×#41D28411yescharacteristic
339C2×#41C2×Dic7111yescharacteristic
340C2×#41C22⋊D7111yescharacteristic
341C2×#41C22⋊D7111yescharacteristic
342C2×#41C22×D7111yescharacteristic
343C22×Q8Dic14211yescharacteristic
344C22×D4C4×D7111yescharacteristic
345C22×D4C2×Dic7111yescharacteristic
346C22×D4C22⋊D7111yescharacteristic
347C23×C4C4×D7111yescharacteristic
348C23×C4D28411yescharacteristic
349C23×C4C22⋊D7111yescharacteristic
350C2×M16—2841no—
351C2×M16—2841no—
352C2×M16—2841no—
353C2×M16—2841no—
354C2×M16—712no—
355C2×M16—1421no—
356C2×M16—1421no—
357C2×M16—1421no—
358C22×C8—712no—
359C22×C8—1421no—
360#117—22no—
361#117—22no—
362#117—41no—
363#117—41no—
364#117—2841no—
365#117—2841no—
366#117—1422no—
367#117—1422no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4—2841no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—1422no—
372C2×C42—41no—
373C2×C42—2841no—
374#99—22no—
375#99—22no—
376#99—22no—
377#99—714no—
378#99—1422no—
379#99—1422no—
380#99—1422no—
381#99C2×Dic7114yescomplemented
382C22⋊C8—1422no—
383C22⋊C8—1422no—
384C22⋊C8—1422no—
385C22⋊C8—1422no—
386C24C4×Dic7111yescharacteristic
387C24(C2×D71)⋊C411yescharacteristic
388C24C22⋊Dic7111yescharacteristic
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—41no—
395#41—41no—
396#41—41no—
397#41—41no—
398#41—41no—
399#41—41no—
400#41—41no—
401#41—41no—
402#41—41no—
403#41—41no—
404#41—41no—
405C2×Q8—41no—
406C2×Q8—41no—
407C2×Q8Dic71⋊C411yescharacteristic
408C2×Q8Dic71⋊C411yescharacteristic
409C2×Q8C4⋊Dic7111yescharacteristic
410C2×Q8C2×Dic14211yescharacteristic
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4C4×Dic7111yescharacteristic
418C2×D4C4×Dic7111yescharacteristic
419C2×D4(C2×D71)⋊C411yescharacteristic
420C2×D4(C2×D71)⋊C411yescharacteristic
421C2×D4C22⋊Dic7111yescharacteristic
422C2×D4C22⋊Dic7111yescharacteristic
423C2×D4Dic71⋊C411yescharacteristic
424C2×D4Dic71⋊C411yescharacteristic
425C2×D4C4⋊Dic7111yescharacteristic
426C2×D4C2×C4×D7111yescharacteristic
427C2×D4C22×Dic7111yescharacteristic
428C2×D4C2×C22⋊D7111yescharacteristic
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—2842no—
443C22×C4C4×Dic7111yescharacteristic
444C22×C4(C2×D71)⋊C411yescharacteristic
445C22×C4(C2×D71)⋊C411yescharacteristic
446C22×C4(C2×D71)⋊C411yescharacteristic
447C22×C4(C2×D71)⋊C411yescharacteristic
448C22×C4(C2×D71)⋊C411yescharacteristic
449C22×C4C22⋊Dic7111yescharacteristic
450C22×C4Dic71⋊C411yescharacteristic
451C22×C4Dic71⋊C411yescharacteristic
452C22×C4Dic71⋊C411yescharacteristic
453C22×C4Dic71⋊C411yescharacteristic
454C22×C4C4⋊Dic7111yescharacteristic
455C22×C4C4⋊Dic7111yescharacteristic
456C22×C4C2×C4×D7111yescharacteristic
457C22×C4C2×C22⋊D7111yescharacteristic
458C22×C4C2×D28411yescharacteristic
459M16—2842no—
460M16—2841no—
461M16—2841no—
462M16—2841no—
463M16—2841no—
464M16—2842no—
465C2×C8—2842no—
466C2×C8—2842no—
467C2×C8—1422no—
468C2×C8—1422no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—2842no—
472C4⋊C4—2842no—
473C22⋊C4—24no—
474C22⋊C4—42no—
475C22⋊C4—2842no—
476C22⋊C4—1424no—
477C42—42no—
478C42—42no—
479C42—2842no—
480C42—2842no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23C71⋊#9511yescharacteristic
491C23C71⋊#9511yescharacteristic
492C23C71⋊#9511yescharacteristic
493C23C71⋊#9511yescharacteristic
494C23C2×C4×Dic7111yescharacteristic
495C23C2×(C2×D71)⋊C411yescharacteristic
496C23C2×C22⋊Dic7111yescharacteristic
497Q8—41no—
498Q8—41no—
499Q8—41no—
500Q8—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—41no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—44no—
544C2×C4—84no—
545C2×C4—2844no—
546C2×C4—5684no—
547C2×C4C71⋊#9511yescharacteristic
548C2×C4C71⋊#9511yescharacteristic
549C2×C4C71⋊#9511yescharacteristic
550C2×C4C71⋊#9511yescharacteristic
551C2×C4#1455604211yescharacteristic
552C2×C4#1455604211yescharacteristic
553C2×C4#1455604211yescharacteristic
554C2×C4#1455604211yescharacteristic
555C2×C4C2×(C2×D71)⋊C411yescharacteristic
556C2×C4C2×Dic71⋊C411yescharacteristic
557C2×C4C2×Dic71⋊C411yescharacteristic
558C2×C4C2×C4⋊Dic7111yescharacteristic
559C8—2842no—
560C8—2842no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—42no—
575C22—42no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22#1798576511yescharacteristic
581C22#1798576511yescharacteristic
582C22C2×C71⋊#9511yescharacteristic
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—84no—
590C4—5684no—
591C4#1798576511yescharacteristic
592C4#1798576511yescharacteristic
593C4C2×#1455604211yescharacteristic
594C4C2×#1455604211yescharacteristic
595C2—21no—
596C2—21no—
597C2—21no—
598C2—41no—
599C2—41no—
600C2—41no—
601C2—41no—
602C2—41no—
603C2—41no—
604C2C2×#1798576511yescharacteristic
subgroupG/H
centerC4C2×#14556042
commutatorC23×C71C2×C42
FrattiniC2×#41C22×D71
FittingC2×#410×C71C2
socleC2×C71C2×#342
prime radicalC22×#41×C71C22
cosocleC2×#41×C71C23
hypercenterC2×#410D71
hypocenterC71#12577
LCSG ▷ C23×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
3. #22586460C2semidirect
5. C22×#11765849C4semidirect
6. C22×#11765849C4semidirect
10. #17985973C4semidirect
13. C2×#14556043C4semidirect
16. #17985966C4semidirect
24. #410×C71C4semidirect
27. C2×#99×C71C4semidirect
35. C2×#11765849C2×C4semidirect
37. C22×C71⋊C8C2×C4semidirect
48. #14556043C2×C4semidirect
82. #99×C71C2×C4semidirect
307. C71#12577semidirect
319. #410Dic71semidirect
324. C2×#99Dic71semidirect
381. #99C2×Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179859732
#225967852
#225864602
C2×#410×C712
#1257771