Finite Group Databasebeta

← #32157781

Subgroups — #32157781

Order 18176.

3038 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556104C211yesmaximal, characteristic
2C2×#17986477C211yesmaximal, characteristic, complemented
3C2×#117×D71C211yesmaximal, characteristic
4#22591372C214yesmaximal, complemented
5#22591350C214yesmaximal, complemented
6#22591204C214yesmaximal, complemented
7#22591182C214yesmaximal
8#22591025C214yesmaximal, complemented
9#22590956C211yesmaximal, characteristic, complemented
10#22590878C214yesmaximal, complemented
11#22590719C211yesmaximal, characteristic
12#22590696C211yesmaximal, characteristic, complemented
13C71×#2310C211yesmaximal, characteristic, complemented
14C23×C4×D71C2211yescharacteristic
15C2×#14556104C4116yescomplemented
16C2×#14556104C2212yes—
17C2×#14556104C2212yes—
18C2×#14556104C2212yes—
19C2×#14556104C2212yes—
20#17986477—28no—
21C2×Q8×Dic71C2214yescomplemented
22C2×Q8×Dic71C2214yescomplemented
23C2×#14556087C2214yes—
24C2×#14556087C2211yescharacteristic
25C2×#14556087C2211yescharacteristic
26C2×#14556086C2212yes—
27C2×#14556085C2212yes—
28C2×#14556084C2212yes—
29#117×D71—28no—
30C22×C4⋊Dic71C2211yescharacteristic
31C22×Dic71⋊C4C2212yes—
32C2×C4⋊C4×D71C2212yes—
33C22×(C2×D71)⋊C4C2212yes—
34C2×C22⋊C4×D71C2212yes—
35C2×C42×D71C2212yes—
36#17985897C2214yescomplemented
37#17985896C2218yescomplemented
38#17985895C2214yescomplemented
39#17985891C2218yes—
40#17985888C2218yes—
41#17985886C2218yescomplemented
42#17985881C2218yescomplemented
43#17985873C2218yes—
44#17985860C2212yes—
45#17985857C2214yes—
46#17985856C2212yes—
47#17985856C2212yes—
48#17985854C2212yes—
49C4⋊C4×Dic71C2214yescomplemented
50C4×C4⋊Dic71C2214yescomplemented
51#17985850C2218yescomplemented
52C22⋊C4×Dic71C2214yes—
53C4×(C2×D71)⋊C4C2212yes—
54C4×(C2×D71)⋊C4C2212yes—
55#17985846C2218yes—
56C23×Q8×C71C2211yescharacteristic
57C2×#117×C71C2211yescharacteristic
58C2×#117×C71C2211yescharacteristic
59#391×C71C2214yescomplemented
60#391×C71C2214yescomplemented
61#381×C71C2214yes—
62C24×D71C2311yescharacteristic
63C23×Dic71C2311yescharacteristic
64C23×Dic71C2312yes—
65C22×C4×D71—24no—
66C22×C4×D71—24no—
67C22×C4×D71C2×C418yes—
68C22×C4×D71C2314yes—
69C22×C4×D71C2311yescharacteristic
70C22×C4×D71C2311yescharacteristic
71C22×C4×D71C2311yescharacteristic
72C22×C4×D71C2311yescharacteristic
73#14556104C2×C4116yes—
74#14556104C2×C4116yes—
75#14556104C2×C4116yescomplemented
76#14556104C2×C4116yescomplemented
77Q8×Dic71—216no—
78Q8×Dic71—216no—
79#14556087—216no—
80#14556087—24no—
81#14556087D418yes—
82#14556086—28no—
83#14556085—28no—
84#14556084—28no—
85C2×C4⋊Dic71C2×C418yescomplemented
86C2×C4⋊Dic71C2312yes—
87C2×C4⋊Dic71C2318yes—
88C2×C4⋊Dic71C2312yes—
89C2×C4⋊Dic71C2312yes—
90C2×C4⋊Dic71C2312yes—
91C2×Dic71⋊C4C2×C4116yescomplemented
92C2×Dic71⋊C4C2314yes—
93C2×Dic71⋊C4C2318yes—
94C2×Dic71⋊C4C2314yes—
95C2×Dic71⋊C4C2314yes—
96C4⋊C4×D71—28no—
97C2×C22⋊Dic71C2312yes—
98C2×C22⋊Dic71C2312yes—
99C2×(C2×D71)⋊C4C2×C4116yes—
100C2×(C2×D71)⋊C4C2314yes—
101C2×(C2×D71)⋊C4C2314yes—
102C2×(C2×D71)⋊C4C2314yes—
103C2×(C2×D71)⋊C4C2314yes—
104C2×(C2×D71)⋊C4C2314yes—
105C22⋊C4×D71—28no—
106C2×C4×Dic71C2318yes—
107C2×C4×Dic71C2312yes—
108C2×C4×Dic71C2314yes—
109C2×C4×Dic71C2318yes—
110C2×C4×Dic71C2312yes—
111C42×D71—28no—
112C71⋊#95C2318yes—
113C71⋊#95C2314yes—
114C71⋊#95C2314yes—
115C71⋊#95C2318yes—
116C22×Q8×C71—24no—
117C22×Q8×C71—24no—
118C22×Q8×C71C2×C418yescomplemented
119C22×Q8×C71C2312yes—
120C22×Q8×C71C2312yes—
121C23×C4×C71C2312yes—
122C23×C4×C71C2311yescharacteristic
123#117×C71—24no—
124#117×C71—24no—
125C2×C4⋊C4×C71C2312yes—
126C2×C4⋊C4×C71C2312yes—
127C2×C22⋊C4×C71C2312yes—
128C2×C22⋊C4×C71C2312yes—
129C2×C42×C71C2312yes—
130C2×C42×C71C2312yes—
131#95×C71C2318yes—
132#95×C71C2318yes—
133C23×D71—22no—
134C23×D71—22no—
135C23×D71—24no—
136C23×D71—42no—
137C23×D71C22×C414yes—
138C23×D71C2411yescharacteristic
139C23×D71C2411yescharacteristic
140C22×Dic71—22no—
141C22×Dic71—22no—
142C22×Dic71—44no—
143C22×Dic71C22×C414yes—
144C22×Dic71C22×C418yes—
145C22×Dic71C2414yes—
146C22×Dic71C2411yescharacteristic
147C22×Dic71C2411yescharacteristic
148C22×Dic71C2414yes—
149C22×Dic71C2414yes—
150C2×C4×D71—232no—
151C2×C4×D71—22no—
152C2×C4×D71—22no—
153C2×C4×D71—22no—
154C2×C4×D71—22no—
155C2×C4×D71—22no—
156C2×C4×D71—22no—
157C2×C4×D71—22no—
158C2×C4×D71—22no—
159C2×C4×D71—28no—
160C2×C4×D71—28no—
161C2×C4×D71—44no—
162C2×C4×D71C22×C414yes—
163C2×C4×D71C22×C414yes—
164C2×C4×D71C22×C414yes—
165C2×C4×D71C22×C414yes—
166C4⋊Dic71—24no—
167C4⋊Dic71—216no—
168C4⋊Dic71C22×C418yes—
169C4⋊Dic71C22×C418yes—
170C4⋊Dic71C2×D418yes—
171Dic71⋊C4—28no—
172Dic71⋊C4—48no—
173Dic71⋊C4C22×C4116yes—
174Dic71⋊C4C22×C4116yes—
175C22⋊Dic71—24no—
176C22⋊Dic71C2×D418yes—
177(C2×D71)⋊C4—28no—
178(C2×D71)⋊C4—28no—
179(C2×D71)⋊C4—44no—
180(C2×D71)⋊C4C22×C4116yes—
181(C2×D71)⋊C4C22×C4116yes—
182C4×Dic71—24no—
183C4×Dic71—28no—
184C4×Dic71—216no—
185C4×Dic71—48no—
186C4×Dic71C2×D418yes—
187C24×C71C2411yescharacteristic
188C2×Q8×C71—232no—
189C2×Q8×C71—24no—
190C2×Q8×C71—24no—
191C2×Q8×C71—24no—
192C2×Q8×C71—24no—
193C2×Q8×C71C22×C418yes—
194C2×Q8×C71C22×C418yes—
195C22×C4×C71—24no—
196C22×C4×C71—24no—
197C22×C4×C71C22×C418yes—
198C22×C4×C71C22×C414yes—
199C22×C4×C71C2×D414yes—
200C22×C4×C71#4114yes—
201C22×C4×C71C2414yes—
202C22×C4×C71C2414yes—
203C22×C4×C71C2414yes—
204C22×C4×C71C2411yescharacteristic
205C22×C4×C71C2411yescharacteristic
206C4⋊C4×C71—24no—
207C4⋊C4×C71—24no—
208C22⋊C4×C71—24no—
209C22⋊C4×C71—24no—
210C42×C71—24no—
211C42×C71—24no—
212C23×C71—22no—
213C23×C71—22no—
214C23×C71C23×C412yes—
215C23×C71C22×D412yes—
216C23×C71C2×#4112yes—
217C23×C71C2511yescharacteristic
218C22×D71—21no—
219C22×D71—21no—
220C22×D71—22no—
221C22×D71—28no—
222C22×D71—21no—
223C22×D71—21no—
224C22×D71—22no—
225C22×D71—22no—
226C22×D71—28no—
227C22×D71—44no—
228C22×D71—48no—
229C22×D71—48no—
230C22×D71C23×C412yes—
231C22×D71C23×C412yes—
232C22×D712−1+414yes—
233Q8×C71—216no—
234Q8×C71—216no—
235C2×C4×C71—216no—
236C2×C4×C71—24no—
237C2×C4×C71—24no—
238C2×C4×C71—24no—
239C2×C4×C71—24no—
240C2×C4×C71—24no—
241C2×C4×C71—24no—
242C2×C4×C71—24no—
243C2×C4×C71—24no—
244C2×C4×C71C4×D4116yes—
245C2×C4×C71C23×C418yes—
246C2×C4×C71C23×C412yes—
247C2×C4×C71C23×C412yes—
248C2×C4×C71C22×D412yes—
249C2×C4×C71C22×D412yes—
250C2×C4×C71C2×#4112yes—
251C2×C4×C71C2×#4112yes—
252C2×Dic71—28no—
253C2×Dic71—21no—
254C2×Dic71—21no—
255C2×Dic71—21no—
256C2×Dic71—21no—
257C2×Dic71—24no—
258C2×Dic71—24no—
259C2×Dic71—28no—
260C2×Dic71—24no—
261C2×Dic71—24no—
262C2×Dic71—24no—
263C2×Dic71—48no—
264C2×Dic71C23×C418yes—
265C2×Dic71C23×C412yes—
266C2×Dic71C23×C412yes—
267C2×Dic71C22×D418yes—
268C4×D71—28no—
269C4×D71—28no—
270C4×D71—28no—
271C4×D71—28no—
272C4×D71—416no—
273C22×C71—22no—
274C22×C71—22no—
275C22×C71—24no—
276C22×C71—24no—
277C22×C71C2×C4×D414yes—
278C22×C71C24×C411yescharacteristic
279C22×C71C23×D411yescharacteristic
280C22×C71C22×#4111yescharacteristic
281C22×C71C2×2−1+412yes—
282C22×C71#58512yes—
283C2×D71—22no—
284C2×D71—22no—
285C2×D71—24no—
286C2×D71—24no—
287C2×D71—22no—
288C2×D71—22no—
289C2×D71—22no—
290C2×D71—416no—
291C2×D71—42no—
292C2×D71—42no—
293C2×D71#51914yescomplemented
294C4×C71—28no—
295C4×C71—44no—
296C4×C71—44no—
297C4×C71C2×C4×D414yes—
298C4×C71C2×C4×D414yes—
299Dic71—22no—
300Dic71—22no—
301Dic71—28no—
302Dic71—28no—
303Dic71—44no—
304#34203—711nomaximal, Sylow
305C2×C71—24no—
306C2×C71C22×C4×D411yescharacteristic
307C2×C71C2×#51911yescharacteristic
308C2×C71#342811yescharacteristic
309C2×C71#348912yes—
310C2×C71#349512yes—
311D71—24no—
312D71—42no—
313C22×C22⋊Q8—711no—
314C2×#519—711no—
315C22×#117—711no—
316#2362—714no—
317#2360—714no—
318#2344—714no—
319#2342—714no—
320#2323—714no—
321#2316—711no—
322#2310D7111yescharacteristic, complemented
323C4×C22⋊Q8—714no—
324#2280—711no—
325#2274—711no—
326C71#3420311yescharacteristic, complemented, Sylow
327C23×Q8C2×D7111yescharacteristic
328C24×C4—711no—
329C2×C22⋊Q8—7116no—
330C2×C22⋊Q8—712no—
331C2×C22⋊Q8—712no—
332C2×C22⋊Q8—712no—
333C2×C22⋊Q8—712no—
334#519—1428no—
335C2×C4×Q8—714no—
336C2×C4×Q8—714no—
337C2×#117—711no—
338C2×#117—711no—
339C2×#117—714no—
340C2×#117—712no—
341C2×#117—712no—
342C2×#117—712no—
343C2×#117—1428no—
344C2×#117C2×D7111yescharacteristic
345C2×#117C2×D7111yescharacteristic
346C22×C4⋊C4—711no—
347C22×C4⋊C4—712no—
348C22×C4⋊C4—712no—
349C22×C22⋊C4—712no—
350C22×C22⋊C4—712no—
351C22×C42—712no—
352#391C2×D7114yescomplemented
353#391C2×D7114yescomplemented
354#389—714no—
355#389—714no—
356#389—718no—
357#388—718no—
358#388—718no—
359#387—718no—
360#387—718no—
361#385—718no—
362#381—712no—
363#381C2×D7114yes—
364#380—712no—
365#380—712no—
366#380—714no—
367#379—712no—
368C4×C4⋊C4—714no—
369C4×C4⋊C4—714no—
370C4×C4⋊C4—718no—
371C4×C22⋊C4—714no—
372C4×C22⋊C4—712no—
373C4×C22⋊C4—712no—
374C4×C22⋊C4—718no—
375C25—711no—
376C22×Q8—24no—
377C22×Q8—24no—
378C22×Q8C4×D7118yescomplemented
379C22×Q8C22×D7112yes—
380C22×Q8C22×D7112yes—
381C23×C4—712no—
382C23×C4—714no—
383C23×C4—711no—
384C23×C4—718no—
385C23×C4—711no—
386C23×C4—711no—
387C23×C4—711no—
388C23×C4—711no—
389C23×C4—1424no—
390C23×C4—1424no—
391C23×C4C22×D7112yes—
392C23×C4C22×D7111yescharacteristic
393C22⋊Q8—7116no—
394C22⋊Q8—7116no—
395C22⋊Q8—7116no—
396C22⋊Q8—7116no—
397C4×Q8—14216no—
398C4×Q8—14216no—
399#117—24no—
400#117—24no—
401#117—718no—
402#117—1428no—
403#117—1428no—
404#117—14216no—
405#117—1428no—
406#117—1424no—
407C2×C4⋊C4—718no—
408C2×C4⋊C4—7116no—
409C2×C4⋊C4—718no—
410C2×C4⋊C4—712no—
411C2×C4⋊C4—714no—
412C2×C4⋊C4—712no—
413C2×C4⋊C4—712no—
414C2×C4⋊C4—714no—
415C2×C4⋊C4—714no—
416C2×C4⋊C4—712no—
417C2×C4⋊C4—718no—
418C2×C4⋊C4—1428no—
419C2×C4⋊C4C22×D7112yes—
420C2×C4⋊C4C22×D7112yes—
421C2×C22⋊C4—714no—
422C2×C22⋊C4—712no—
423C2×C22⋊C4—7116no—
424C2×C22⋊C4—714no—
425C2×C22⋊C4—714no—
426C2×C22⋊C4—712no—
427C2×C22⋊C4—714no—
428C2×C22⋊C4—714no—
429C2×C22⋊C4—1428no—
430C2×C22⋊C4C22×D7112yes—
431C2×C22⋊C4C22×D7112yes—
432C2×C42—718no—
433C2×C42—712no—
434C2×C42—714no—
435C2×C42—712no—
436C2×C42—718no—
437C2×C42—1428no—
438C2×C42C22×D7112yes—
439C2×C42C22×D7112yes—
440#95—714no—
441#95—718no—
442#95—714no—
443#95—718no—
444#95C22×D7118yes—
445#95C22×D7118yes—
446C24—2842no—
447C24—714no—
448C24—711no—
449C24—711no—
450C24—1424no—
451C24—1422no—
452C24—1422no—
453C24C23×D7111yescharacteristic
454C2×Q8—24no—
455C2×Q8—24no—
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—232no—
459C2×Q8C2×C4×D7118yes—
460C2×Q8C2×C4×D7118yes—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—2844no—
464C22×C4—2844no—
465C22×C4—718no—
466C22×C4—714no—
467C22×C4—711no—
468C22×C4—714no—
469C22×C4—711no—
470C22×C4—714no—
471C22×C4—714no—
472C22×C4—714no—
473C22×C4—714no—
474C22×C4—714no—
475C22×C4—714no—
476C22×C4—1428no—
477C22×C4—1428no—
478C22×C4—1422no—
479C22×C4—1422no—
480C22×C4—1422no—
481C22×C4—1422no—
482C22×C4—1422no—
483C22×C4—1422no—
484C22×C4—1422no—
485C22×C4—1422no—
486C22×C4—1422no—
487C22×C4—1422no—
488C22×C4—14232no—
489C22×C4C2×C4×D7118yes—
490C22×C4C2×C4×D7114yes—
491C22×C4D4×D7114yes—
492C22×C4#1176586414yes—
493C22×C4C23×D7114yes—
494C22×C4C23×D7114yes—
495C22×C4C23×D7114yes—
496C22×C4C23×D7111yescharacteristic
497C22×C4C23×D7111yescharacteristic
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—2848no—
501C4⋊C4—7116no—
502C4⋊C4—7116no—
503C4⋊C4—718no—
504C4⋊C4—718no—
505C4⋊C4—718no—
506C4⋊C4—1428no—
507C4⋊C4—14216no—
508C4⋊C4—1424no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—2844no—
512C22⋊C4—718no—
513C22⋊C4—7116no—
514C22⋊C4—7116no—
515C22⋊C4—1428no—
516C22⋊C4—1428no—
517C22⋊C4—1424no—
518C42—24no—
519C42—24no—
520C42—2848no—
521C42—718no—
522C42—1428no—
523C42—14216no—
524C42—1424no—
525C23—22no—
526C23—22no—
527C23—2848no—
528C23—2848no—
529C23—2844no—
530C23—712no—
531C23—714no—
532C23—712no—
533C23—1428no—
534C23—1421no—
535C23—1421no—
536C23—1422no—
537C23—1422no—
538C23—1428no—
539C23—1421no—
540C23—1421no—
541C23—1422no—
542C23C22×C4×D7112yes—
543C23C2×D4×D7112yes—
544C23C2×#1176586412yes—
545C23C24×D7111yescharacteristic
546Q8—216no—
547Q8—216no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—216no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28416no—
558C2×C4—2848no—
559C2×C4—718no—
560C2×C4—712no—
561C2×C4—712no—
562C2×C4—718no—
563C2×C4—1424no—
564C2×C4—1424no—
565C2×C4—1424no—
566C2×C4—1421no—
567C2×C4—1421no—
568C2×C4—1421no—
569C2×C4—1421no—
570C2×C4—1424no—
571C2×C4—1424no—
572C2×C4—1428no—
573C2×C4—1428no—
574C2×C4—1428no—
575C2×C4—1428no—
576C2×C4—1428no—
577C2×C4—1428no—
578C2×C4(C22⋊D71)⋊C4116yes—
579C2×C4C22×C4×D7118yes—
580C2×C4C22×C4×D7112yes—
581C2×C4C22×C4×D7112yes—
582C2×C4C2×D4×D7112yes—
583C2×C4C2×D4×D7112yes—
584C2×C4C2×#1176586412yes—
585C2×C4C2×#1176586412yes—
586C22—24no—
587C22—22no—
588C22—22no—
589C22—24no—
590C22—28416no—
591C22—2842no—
592C22—2842no—
593C22—714no—
594C22—1422no—
595C22—1422no—
596C22—1422no—
597C22—1424no—
598C22—1422no—
599C22—1422no—
600C22—1424no—
601C22C2×(C22⋊D71)⋊C414yes—
602C22C23×C4×D7111yescharacteristic
603C22C22×D4×D7111yescharacteristic
604C22C22×#1176586411yescharacteristic
605C222−1+4×D7112yes—
606C22#1798683612yes—
607C4—28no—
608C4—44no—
609C4—44no—
610C4—2844no—
611C4—1428no—
612C4—1422no—
613C4—1422no—
614C4—1428no—
615C4C2×(C22⋊D71)⋊C414yes—
616C4C2×(C22⋊D71)⋊C414yes—
617C2—24no—
618C2—2842no—
619C2—1424no—
620C2C22×(C22⋊D71)⋊C411yescharacteristic
621C2#519×D7111yescharacteristic
622C2#2260154211yescharacteristic
623C2#2260201812yes—
624C2#2260208912yes—
subgroupG/H
center, FrattiniC23C24×D71
commutatorC22×C71C24×C4
FittingC71×#2310C2
socle, cosocleC23×C71C25
prime radicalC24×D71C23
hypercenter#2310D71
hypocenterC71#34203
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. C2×#17986477C2semidirect
4. #22591372C2semidirect
5. #22591350C2semidirect
6. #22591204C2semidirect
8. #22591025C2semidirect
9. #22590956C2semidirect
10. #22590878C2semidirect
12. #22590696C2semidirect
13. C71×#2310C2semidirect
15. C2×#14556104C4semidirect
21. C2×Q8×Dic71C22semidirect
22. C2×Q8×Dic71C22semidirect
36. #17985897C22semidirect
37. #17985896C22semidirect
38. #17985895C22semidirect
41. #17985886C22semidirect
42. #17985881C22semidirect
49. C4⋊C4×Dic71C22semidirect
50. C4×C4⋊Dic71C22semidirect
51. #17985850C22semidirect
59. #391×C71C22semidirect
60. #391×C71C22semidirect
75. #14556104C2×C4semidirect
76. #14556104C2×C4semidirect
85. C2×C4⋊Dic71C2×C4semidirect
91. C2×Dic71⋊C4C2×C4semidirect
118. C22×Q8×C71C2×C4semidirect
293. C2×D71#519semidirect
322. #2310D71semidirect
326. C71#34203semidirect
352. #391C2×D71semidirect
353. #391C2×D71semidirect
378. C22×Q8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C22×#145561042
C2×#179864772
C2×#117×D712
#225913722
#225913502
#225912042
#225911822
#225910252
#225909562
#225908782
#225907192
#225906962
C71×#23102
#3420371