Finite Group Databasebeta

← #31918839

Subgroups — #31918839

Order 18176.

2782 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556086C211yesmaximal, characteristic
2C22×#14556085C211yesmaximal, characteristic
3C22×C4⋊C4×D71C211yesmaximal, characteristic
4#22586570C211yesmaximal, characteristic, complemented
5#22586569C211yesmaximal, characteristic, complemented
6#22586569C211yesmaximal, characteristic, complemented
7Dic71⋊#95C214yesmaximal, complemented
8(C2×D71)⋊#95C214yesmaximal
9C71×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D71C2212yes—
11C23×C4×D71C2211yescharacteristic
12C2×#14556086—24no—
13C2×#14556086—24no—
14C2×#14556086C418yescomplemented
15C2×#14556085—24no—
16C2×#14556085—24no—
17C2×#14556085C418yescomplemented
18C22×C4⋊Dic71C2211yescharacteristic
19C22×C4⋊Dic71C2211yescharacteristic
20C22×Dic71⋊C4C2212yes—
21C22×Dic71⋊C4C2212yes—
22C2×C4⋊C4×D71—24no—
23C2×C4⋊C4×D71—24no—
24C2×C4⋊C4×D71C418yescomplemented
25C22×(C2×D71)⋊C4C2212yes—
26C22×(C2×D71)⋊C4C2212yes—
27C22×C4×Dic71C2211yescharacteristic
28C22×C4×Dic71C2212yes—
29C2×C71⋊#95C2214yes—
30C2×C71⋊#95C2214yes—
31C2×C71⋊#95C2214yes—
32C22×C4⋊C4×C71C2211yescharacteristic
33C22×C4⋊C4×C71C2211yescharacteristic
34C22×C42×C71C2211yescharacteristic
35C2×#95×C71C2214yes—
36C24×D71C2311yescharacteristic
37C23×Dic71C2312yes—
38C23×Dic71C2312yes—
39C23×Dic71C2312yes—
40C23×Dic71C2311yescharacteristic
41C22×C4×D71—24no—
42C22×C4×D71—24no—
43C22×C4×D71—44no—
44C22×C4×D71—44no—
45C22×C4×D71C2×C418yes—
46C22×C4×D71C2×C414yes—
47C22×C4×D71C2×C414yes—
48C22×C4×D71C2×C414yes—
49C22×C4×D71D414yes—
50C22×C4×D71D414yes—
51C22×C4×D71D414yes—
52C22×C4×D71Q814yes—
53#14556086—232no—
54#14556085—232no—
55C2×C4⋊Dic71—22no—
56C2×C4⋊Dic71—22no—
57C2×C4⋊Dic71—22no—
58C2×C4⋊Dic71—22no—
59C2×C4⋊Dic71C2×C414yes—
60C2×C4⋊Dic71C2×C414yescomplemented
61C2×Dic71⋊C4—24no—
62C2×Dic71⋊C4—24no—
63C2×Dic71⋊C4—24no—
64C2×Dic71⋊C4—24no—
65C2×Dic71⋊C4C2×C418yes—
66C2×Dic71⋊C4C2×C418yescomplemented
67C4⋊C4×D71—232no—
68C2×(C2×D71)⋊C4—24no—
69C2×(C2×D71)⋊C4—24no—
70C2×(C2×D71)⋊C4—24no—
71C2×(C2×D71)⋊C4—24no—
72C2×(C2×D71)⋊C4C2×C418yes—
73C2×(C2×D71)⋊C4C2×C418yes—
74C2×C4×Dic71—24no—
75C2×C4×Dic71—24no—
76C2×C4×Dic71—22no—
77C2×C4×Dic71—22no—
78C2×C4×Dic71C2×C414yescomplemented
79C2×C4×Dic71C2×C418yescomplemented
80C71⋊#95—44no—
81C71⋊#95—44no—
82C71⋊#95—44no—
83C23×C4×C71C2312yes—
84C23×C4×C71C2312yes—
85C23×C4×C71C2312yes—
86C23×C4×C71C2311yescharacteristic
87C2×C4⋊C4×C71—22no—
88C2×C4⋊C4×C71—22no—
89C2×C4⋊C4×C71—22no—
90C2×C4⋊C4×C71—22no—
91C2×C4⋊C4×C71C2×C414yes—
92C2×C4⋊C4×C71C2×C414yescomplemented
93C2×C42×C71—22no—
94C2×C42×C71—22no—
95C2×C42×C71C2×C414yescomplemented
96#95×C71—44no—
97C23×D71—22no—
98C23×D71—22no—
99C23×D71—22no—
100C23×D71—22no—
101C23×D71C22×C412yes—
102C23×D71C22×C412yes—
103C23×D71C22×C412yes—
104C23×D71C2×D412yes—
105C23×D71C2×D412yes—
106C23×D71C2×D412yes—
107C23×D71C2×Q812yes—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—22no—
111C22×Dic71—22no—
112C22×Dic71—22no—
113C22×Dic71—22no—
114C22×Dic71—24no—
115C22×Dic71—24no—
116C22×Dic71—22no—
117C22×Dic71—22no—
118C22×Dic71—24no—
119C22×Dic71—22no—
120C22×Dic71—22no—
121C22×Dic71—24no—
122C22×Dic71C22×C414yes—
123C22×Dic71C22×C414yes—
124C22×Dic71C22×C414yes—
125C22×Dic71C22×C412yes—
126C22×Dic71C22×C412yes—
127C22×Dic71C22×C412yes—
128C22×Dic71C2×D412yes—
129C22×Dic71C2×D412yes—
130C22×Dic71C2×D412yes—
131C22×Dic71C2×Q812yes—
132C2×C4×D71—216no—
133C2×C4×D71—416no—
134C2×C4×D71—48no—
135C2×C4×D71—416no—
136C2×C4×D71—48no—
137C2×C4×D71C42116yes—
138C2×C4×D71C22⋊C4116yes—
139C2×C4×D71C22⋊C4116yes—
140C2×C4×D71C22⋊C4116yes—
141C2×C4×D71C4⋊C4116yes—
142C2×C4×D71C4⋊C4116yes—
143C2×C4×D71C4⋊C4116yes—
144C4⋊Dic71—28no—
145C4⋊Dic71—28no—
146Dic71⋊C4—216no—
147Dic71⋊C4—216no—
148(C2×D71)⋊C4—216no—
149(C2×D71)⋊C4—216no—
150C4×Dic71—28no—
151C4×Dic71—216no—
152C24×C71C2411yescharacteristic
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—22no—
157C22×C4×C71—22no—
158C22×C4×C71—22no—
159C22×C4×C71—24no—
160C22×C4×C71—24no—
161C22×C4×C71—22no—
162C22×C4×C71—24no—
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—24no—
166C22×C4×C71—22no—
167C22×C4×C71C22×C414yes—
168C22×C4×C71C22×C414yes—
169C22×C4×C71C22×C414yes—
170C22×C4×C71C22×C412yes—
171C22×C4×C71C22×C412yes—
172C22×C4×C71C22×C412yes—
173C22×C4×C71C2×D412yes—
174C22×C4×C71C2×D412yes—
175C22×C4×C71C2×D412yes—
176C22×C4×C71C2×Q812yes—
177C4⋊C4×C71—28no—
178C4⋊C4×C71—28no—
179C42×C71—28no—
180C23×C71C23×C411yescharacteristic
181C23×C71C23×C411yescharacteristic
182C23×C71C23×C411yescharacteristic
183C23×C71C22×D411yescharacteristic
184C23×C71C22×D411yescharacteristic
185C23×C71C22×D411yescharacteristic
186C23×C71C22×Q811yescharacteristic
187C23×C71H(5,𝔽2)12yes—
188C23×C71H(5,𝔽2)12yes—
189C23×C712−1+412yes—
190C23×C712−1+412yes—
191C22×D71—24no—
192C22×D71—24no—
193C22×D71—24no—
194C22×D71—24no—
195C22×D71—24no—
196C22×D71—24no—
197C22×D71—22no—
198C22×D71—24no—
199C22×D71—22no—
200C22×D71—22no—
201C22×D71—24no—
202C22×D71—22no—
203C22×D71—24no—
204C22×D71—24no—
205C22×D71—24no—
206C22×D71—24no—
207C22×D71C2×C4214yes—
208C22×D71C2×C22⋊C414yes—
209C22×D71C2×C22⋊C414yes—
210C22×D71C2×C22⋊C414yes—
211C22×D71C2×C4⋊C414yes—
212C22×D71C2×C4⋊C414yes—
213C22×D71C2×C4⋊C414yes—
214C2×C4×C71—24no—
215C2×C4×C71—24no—
216C2×C4×C71—24no—
217C2×C4×C71—28no—
218C2×C4×C71—28no—
219C2×C4×C71—24no—
220C2×C4×C71—24no—
221C2×C4×C71—42no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—44no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
227C2×C4×C71—42no—
228C2×C4×C71—42no—
229C2×C4×C71—44no—
230C2×C4×C71—42no—
231C2×C4×C71C2×C4214yes—
232C2×C4×C71C2×C22⋊C414yes—
233C2×C4×C71C2×C22⋊C414yes—
234C2×C4×C71C2×C22⋊C414yes—
235C2×C4×C71C2×C4⋊C414yes—
236C2×C4×C71C2×C4⋊C414yes—
237C2×C4×C71C2×C4⋊C414yes—
238C2×Dic71—24no—
239C2×Dic71—24no—
240C2×Dic71—24no—
241C2×Dic71—28no—
242C2×Dic71—24no—
243C2×Dic71—24no—
244C2×Dic71—28no—
245C2×Dic71—42no—
246C2×Dic71—44no—
247C2×Dic71—42no—
248C2×Dic71—42no—
249C2×Dic71—42no—
250C2×Dic71—42no—
251C2×Dic71—42no—
252C2×Dic71—42no—
253C2×Dic71—42no—
254C2×Dic71—44no—
255C2×Dic71C2×C4214yes—
256C2×Dic71C2×C22⋊C414yes—
257C2×Dic71C2×C22⋊C414yes—
258C2×Dic71C2×C22⋊C414yes—
259C2×Dic71C2×C4⋊C414yes—
260C2×Dic71C2×C4⋊C414yes—
261C2×Dic71C2×C4⋊C414yes—
262C4×D71—432no—
263C4×D71#95164yescomplemented
264C22×C71C22×C4211yescharacteristic
265C22×C71C22×C22⋊C411yescharacteristic
266C22×C71C22×C22⋊C411yescharacteristic
267C22×C71C22×C22⋊C411yescharacteristic
268C22×C71C22×C4⋊C411yescharacteristic
269C22×C71C22×C4⋊C411yescharacteristic
270C22×C71C22×C4⋊C411yescharacteristic
271C22×C71#51812yes—
272C22×C71#51912yes—
273C22×C71#52012yes—
274C22×C71#52012yes—
275C22×C71#52012yes—
276C22×C71#52012yes—
277C22×C71#53511yescharacteristic
278C22×C71#53512yes—
279C22×C71#53511yescharacteristic
280C22×C71#53611yescharacteristic
281C22×C71#53612yes—
282C22×C71#53611yescharacteristic
283C22×C71#53712yes—
284C22×C71#53712yes—
285C22×C71#54412yes—
286C22×C71#54412yes—
287C2×D71—24no—
288C2×D71—24no—
289C2×D71—28no—
290C2×D71—24no—
291C2×D71—24no—
292C2×D71—28no—
293C2×D71—24no—
294C2×D71—28no—
295C2×D71—24no—
296C2×D71—28no—
297C2×D71C2×#9518yes—
298C4×C71—44no—
299C4×C71—44no—
300C4×C71—44no—
301C4×C71C2×#9518yes—
302Dic71—44no—
303Dic71—44no—
304Dic71—44no—
305Dic71C2×#9518yes—
306#13795—711nomaximal, Sylow
307C2×C71C22×#9511yescharacteristic
308C2×C71#227412yes—
309C2×C71#230911yescharacteristic
310C2×C71#231011yescharacteristic
311C2×C71#231112yes—
312C2×C71#231111yescharacteristic
313C2×C71#231111yescharacteristic
314C2×C71#231612yes—
315C2×C71#231711yescharacteristic
316C2×C71#231712yes—
317C2×C71#231711yescharacteristic
318D71—28no—
319C22×#117—711no—
320C22×#117—711no—
321C23×C4⋊C4—711no—
322C4⋊#95—711no—
323C4⋊#95—714no—
324C4⋊#95—711no—
325C4⋊#95—711no—
326C4⋊#95D7111yescharacteristic, complemented
327C22⋊#95—714no—
328C71#1379511yescharacteristic, complemented, Sylow
329C24×C4—712no—
330C24×C4—711no—
331C2×#117—718no—
332C2×#117—718no—
333C2×#117—1424no—
334C2×#117—1424no—
335C2×#117—1424no—
336C2×#117—1424no—
337C22×C4⋊C4—712no—
338C22×C4⋊C4—711no—
339C22×C4⋊C4—712no—
340C22×C4⋊C4—711no—
341C22×C4⋊C4—718no—
342C22×C4⋊C4—1424no—
343C22×C4⋊C4—1424no—
344C22×C4⋊C4C2×D7111yescharacteristic
345C22×C4⋊C4C2×D7111yescharacteristic
346C22×C22⋊C4—712no—
347C22×C22⋊C4—712no—
348C22×C42—712no—
349C22×C42—711no—
350C22×C42C2×D7111yescharacteristic
351C2×#95—714no—
352C2×#95—714no—
353C2×#95—714no—
354C2×#95C2×D7114yes—
355C25—711no—
356C23×C4—2844no—
357C23×C4—2844no—
358C23×C4—718no—
359C23×C4—711no—
360C23×C4—714no—
361C23×C4—714no—
362C23×C4—714no—
363C23×C4—714no—
364C23×C4—714no—
365C23×C4—714no—
366C23×C4—714no—
367C23×C4—712no—
368C23×C4—712no—
369C23×C4—712no—
370C23×C4—1424no—
371C23×C4—1424no—
372C23×C4C22×D7112yes—
373C23×C4C22×D7112yes—
374C23×C4C22×D7112yes—
375C23×C4C22×D7111yescharacteristic
376#117—14232no—
377#117—14232no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—718no—
383C2×C4⋊C4—714no—
384C2×C4⋊C4—718no—
385C2×C4⋊C4—714no—
386C2×C4⋊C4—14232no—
387C2×C4⋊C4—1424no—
388C2×C4⋊C4—1424no—
389C2×C4⋊C4—1424no—
390C2×C4⋊C4—1424no—
391C2×C4⋊C4—1422no—
392C2×C4⋊C4—1422no—
393C2×C4⋊C4—1422no—
394C2×C4⋊C4—1422no—
395C2×C4⋊C4C4×D7114yes—
396C2×C4⋊C4C4×D7114yescomplemented
397C2×C22⋊C4—718no—
398C2×C22⋊C4—718no—
399C2×C22⋊C4—1424no—
400C2×C22⋊C4—1424no—
401C2×C22⋊C4—1424no—
402C2×C22⋊C4—1424no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42—718no—
406C2×C42—714no—
407C2×C42—1424no—
408C2×C42—1424no—
409C2×C42—1422no—
410C2×C42—1422no—
411C2×C42C4×D7114yescomplemented
412#95—44no—
413#95—2844no—
414#95—2844no—
415#95—2844no—
416C24—712no—
417C24—712no—
418C24—712no—
419C24—712no—
420C24—712no—
421C24—712no—
422C24—712no—
423C24—1422no—
424C24—1422no—
425C24—1422no—
426C24—1422no—
427C24C23×D7111yescharacteristic
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—28416no—
443C22×C4—2848no—
444C22×C4—28416no—
445C22×C4—2848no—
446C22×C4—712no—
447C22×C4—712no—
448C22×C4—712no—
449C22×C4—712no—
450C22×C4—712no—
451C22×C4—712no—
452C22×C4—712no—
453C22×C4—7116no—
454C22×C4—7116no—
455C22×C4—7116no—
456C22×C4—7116no—
457C22×C4—7116no—
458C22×C4—7116no—
459C22×C4—7116no—
460C22×C4—714no—
461C22×C4—714no—
462C22×C4—714no—
463C22×C4—14216no—
464C22×C4—1422no—
465C22×C4—1422no—
466C22×C4—1422no—
467C22×C4—1422no—
468C22×C4—1422no—
469C22×C4—1422no—
470C22×C4—1422no—
471C22×C4—1424no—
472C22×C4—1422no—
473C22×C4—1424no—
474C22×C4—1424no—
475C22×C4—1422no—
476C22×C4—1422no—
477C22×C4—1424no—
478C22×C4C2×C4×D7114yes—
479C22×C4C2×C4×D7114yes—
480C22×C4C2×C4×D7114yes—
481C22×C4C2×C4×D7112yes—
482C22×C4C2×C4×D7112yes—
483C22×C4C2×C4×D7112yes—
484C22×C4D4×D7112yes—
485C22×C4D4×D7112yes—
486C22×C4D4×D7112yes—
487C22×C4Q8×D7112yes—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—14216no—
491C4⋊C4—14216no—
492C4⋊C4—1428no—
493C4⋊C4—1428no—
494C22⋊C4—14216no—
495C22⋊C4—14216no—
496C42—28no—
497C42—14216no—
498C42—1428no—
499C23—714no—
500C23—714no—
501C23—714no—
502C23—714no—
503C23—714no—
504C23—714no—
505C23—714no—
506C23—1422no—
507C23—1424no—
508C23—1422no—
509C23—1422no—
510C23—1424no—
511C23—1422no—
512C23—1424no—
513C23—1424no—
514C23—1424no—
515C23—1424no—
516C23—1424no—
517C23—1424no—
518C23—1424no—
519C23—1424no—
520C23—1424no—
521C23—1424no—
522C23C22×C4×D7111yescharacteristic
523C23C22×C4×D7111yescharacteristic
524C23C22×C4×D7111yescharacteristic
525C23C2×D4×D7111yescharacteristic
526C23C2×D4×D7111yescharacteristic
527C23C2×D4×D7111yescharacteristic
528C23C2×Q8×D7111yescharacteristic
529C23#1455617312yes—
530C23#1455617412yes—
531C23#1455617512yes—
532C23#1455617612yes—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—28432no—
551C2×C4—2842no—
552C2×C4—2844no—
553C2×C4—2842no—
554C2×C4—2842no—
555C2×C4—2842no—
556C2×C4—2842no—
557C2×C4—2842no—
558C2×C4—2842no—
559C2×C4—2842no—
560C2×C4—2844no—
561C2×C4—714no—
562C2×C4—714no—
563C2×C4—714no—
564C2×C4—714no—
565C2×C4—714no—
566C2×C4—714no—
567C2×C4—714no—
568C2×C4—7164no—
569C2×C4—1424no—
570C2×C4—1424no—
571C2×C4—1424no—
572C2×C4—1428no—
573C2×C4—1424no—
574C2×C4—1424no—
575C2×C4—1428no—
576C2×C4C42×D7114yes—
577C2×C4C22⋊C4×D7114yes—
578C2×C4C22⋊C4×D7114yes—
579C2×C4C22⋊C4×D7114yes—
580C2×C4C4⋊C4×D7114yes—
581C2×C4C4⋊C4×D7114yes—
582C2×C4C4⋊C4×D7114yes—
583C22—718no—
584C22—1424no—
585C22—1424no—
586C22—1428no—
587C22—1424no—
588C22—1424no—
589C22—1428no—
590C22—1428no—
591C22—1424no—
592C22—1428no—
593C22—1424no—
594C22C2×C42×D7111yescharacteristic
595C22C2×C22⋊C4×D7111yescharacteristic
596C22C2×C22⋊C4×D7111yescharacteristic
597C22C2×C22⋊C4×D7111yescharacteristic
598C22C2×C4⋊C4×D7111yescharacteristic
599C22C2×C4⋊C4×D7111yescharacteristic
600C22C2×C4⋊C4×D7111yescharacteristic
601C22#1798647112yes—
602C22#1798647512yes—
603C22#1798647812yes—
604C22#1798647912yes—
605C22#1798648012yes—
606C22#1798648112yes—
607C22#1798656211yescharacteristic
608C22#1798656512yes—
609C22#1798656711yescharacteristic
610C22#1798656911yescharacteristic
611C22#1798657212yes—
612C22#1798657411yescharacteristic
613C22#1798657812yes—
614C22#1798657912yes—
615C22#1798662612yes—
616C22#1798662912yes—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—2844no—
621C4—2844no—
622C4—2844no—
623C4—718no—
624C4#95×D7118yes—
625C2—1428no—
626C2C2×#95×D7111yescharacteristic
627C2#2259069412yes—
628C2#2259091911yescharacteristic
629C2#2259092311yescharacteristic
630C2#2259092912yes—
631C2#2259093011yescharacteristic
632C2#2259093111yescharacteristic
633C2#2259095512yes—
634C2#2259096111yescharacteristic
635C2#2259096212yes—
636C2#2259096511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC22×C71C22×C42
FittingC71×C4⋊#95C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC4⋊#95D71
hypocenterC71#13795
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
4. #22586570C2semidirect
5. #22586569C2semidirect
6. #22586569C2semidirect
7. Dic71⋊#95C2semidirect
9. C71×C4⋊#95C2semidirect
14. C2×#14556086C4semidirect
17. C2×#14556085C4semidirect
24. C2×C4⋊C4×D71C4semidirect
60. C2×C4⋊Dic71C2×C4semidirect
66. C2×Dic71⋊C4C2×C4semidirect
78. C2×C4×Dic71C2×C4semidirect
79. C2×C4×Dic71C2×C4semidirect
92. C2×C4⋊C4×C71C2×C4semidirect
95. C2×C42×C71C2×C4semidirect
263. C4×D71#95semidirect
326. C4⋊#95D71semidirect
328. C71#13795semidirect
396. C2×C4⋊C4C4×D71semidirect
411. C2×C42C4×D71semidirect

Maximal subgroups

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

subgroupindex
C22×#145560862
C22×#145560852
C22×C4⋊C4×D712
#225865702
#225865692
Dic71⋊#952
(C2×D71)⋊#952
C71×C4⋊#952
#1379571