Finite Group Databasebeta

← C2×#22589994 #32150189

Subgroups — C2×#22589994 #32150189

Order 18176.

1214 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985983C211yesmaximal, characteristic
2C2×#17985973C211yesmaximal, characteristic
3#22589994C218yesmaximal, direct product factor
4C2×#17985789C214yesmaximal, complemented
5C2×#536×C71C211yesmaximal, characteristic
6C22×#11765849—21no—
7C2×#14556087—21no—
8#17985983—22no—
9#17985983C2214yes—
10C2×#14556047—21no—
11C2×#14556045C2212yes—
12#17985973—22no—
13#17985973C2214yes—
14C2×#14556043—21no—
15C2×#14556043C2212yes—
16#17985789C22116yescomplemented
17C22×#41×C71C2211yescharacteristic
18C23×Q8×C71C411yescharacteristic, complemented
19#536×C71—22no—
20#536×C71—22no—
21#536×C71C414yescomplemented
22#536×C71C2214yes—
23C2×C4⋊Q8×C71—21no—
24C2×#124×C71—21no—
25C2×#123×C71C412yescomplemented
26C2×#123×C71C2212yes—
27C2×C22⋊Q8×C71—21no—
28C2×C22⋊Q8×C71—21no—
29C2×#117×C71C411yescharacteristic, complemented
30C2×#11765849—22no—
31C2×#11765849—24no—
32C2×#11765849—41no—
33C2×#11765849—42no—
34C22×C71⋊C8—41no—
35#14556087—24no—
36#14556087—42no—
37C2×C4⋊Dic71—41no—
38C2×C22⋊Dic71—22no—
39C2×C4×Dic71—41no—
40#14556047—24no—
41#14556045—22no—
42#14556045C2314yes—
43#14556043—24no—
44#14556043—22no—
45#14556043C2314yes—
46C2×#41×C71—41no—
47C2×#41×C71—41no—
48C2×#41×C71—42no—
49C2×#41×C71C2×C412yes—
50C2×#41×C71D412yes—
51C2×#41×C71D412yes—
52C2×#41×C71C2312yes—
53C22×Q8×C71—21no—
54C22×Q8×C71—21no—
55C22×Q8×C71—22no—
56C22×Q8×C71—42no—
57C22×Q8×C71—41no—
58C22×Q8×C71—41no—
59C22×Q8×C71C2×C414yescomplemented
60C22×Q8×C71D411yescharacteristic
61C22×D4×C71C2×C411yescharacteristic
62C22×D4×C71D411yescharacteristic
63C22×D4×C71C2311yescharacteristic
64C23×C4×C71C2×C411yescharacteristic
65C23×C4×C71C2×C412yes—
66C23×C4×C71D411yescharacteristic
67C23×C4×C71D411yescharacteristic
68C4⋊Q8×C71—24no—
69C4⋊Q8×C71—24no—
70#124×C71—24no—
71#124×C71—24no—
72#123×C71—24no—
73#123×C71—22no—
74#123×C71—22no—
75#123×C71C2×C414yes—
76#123×C71C2×C418yescomplemented
77#123×C71C2314yes—
78C22⋊Q8×C71—24no—
79C22⋊Q8×C71—24no—
80C22⋊Q8×C71—24no—
81C22⋊Q8×C71—24no—
82#117×C71—22no—
83#117×C71C2×C414yescomplemented
84C2×C4⋊C4×C71—21no—
85C2×C4⋊C4×C71—22no—
86C2×C4⋊C4×C71—22no—
87C2×C22⋊C4×C71—22no—
88C2×C22⋊C4×C71—22no—
89C2×C22⋊C4×C71C2×C412yes—
90C2×C42×C71—21no—
91C22×Dic71—42no—
92#11765849—24no—
93#11765849—42no—
94#11765849—44no—
95C2×C71⋊C8—42no—
96C2×C71⋊C8—44no—
97C4⋊Dic71—44no—
98C22⋊Dic71—24no—
99C22⋊Dic71—42no—
100C4×Dic71—44no—
101C24×C71C22⋊C411yescharacteristic
102C24×C71C22×C411yescharacteristic
103C24×C71C2×D411yescharacteristic
104#41×C71—42no—
105#41×C71—42no—
106#41×C71—42no—
107#41×C71—42no—
108#41×C71—42no—
109#41×C71—42no—
110#41×C71—42no—
111#41×C71—42no—
112C2×Q8×C71—24no—
113C2×Q8×C71—24no—
114C2×Q8×C71—41no—
115C2×Q8×C71—41no—
116C2×Q8×C71—44no—
117C2×Q8×C71—44no—
118C2×Q8×C71—42no—
119C2×Q8×C71—42no—
120C2×Q8×C71—44no—
121C2×Q8×C71—44no—
122C2×Q8×C71—42no—
123C2×Q8×C71—41no—
124C2×Q8×C71—41no—
125C2×Q8×C71C22⋊C412yes—
126C2×Q8×C71C2×D412yes—
127C2×D4×C71—41no—
128C2×D4×C71—41no—
129C2×D4×C71—41no—
130C2×D4×C71—41no—
131C2×D4×C71—41no—
132C2×D4×C71—41no—
133C2×D4×C71C22⋊C412yes—
134C2×D4×C71C22⋊C412yes—
135C2×D4×C71C22×C411yescharacteristic
136C2×D4×C71C22×C412yes—
137C2×D4×C71C2×D411yescharacteristic
138C2×D4×C71C2×D412yes—
139C2×D4×C71C2×D411yescharacteristic
140C2×D4×C71C2411yescharacteristic
141C22×C4×C71—22no—
142C22×C4×C71—21no—
143C22×C4×C71—21no—
144C22×C4×C71—21no—
145C22×C4×C71—21no—
146C22×C4×C71—22no—
147C22×C4×C71—21no—
148C22×C4×C71—21no—
149C22×C4×C71—22no—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—42no—
154C22×C4×C71—42no—
155C22×C4×C71C22⋊C411yescharacteristic
156C22×C4×C71C22⋊C411yescharacteristic
157C22×C4×C71C22⋊C412yes—
158C22×C4×C71C22⋊C412yes—
159C22×C4×C71C22⋊C412yes—
160C22×C4×C71C22⋊C411yescharacteristic
161C22×C4×C71C22×C412yes—
162C22×C4×C71C22×C414yes—
163C22×C4×C71C2×D412yes—
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2×D412yes—
166C4⋊C4×C71—24no—
167C4⋊C4×C71—24no—
168C4⋊C4×C71—28no—
169C4⋊C4×C71—24no—
170C22⋊C4×C71—24no—
171C22⋊C4×C71—28no—
172C22⋊C4×C71—22no—
173C22⋊C4×C71—24no—
174C22⋊C4×C71C22×C414yes—
175C42×C71—24no—
176C23×C71—21no—
177C23×C71—21no—
178C23×C71—21no—
179C23×C71—21no—
180C23×C71—21no—
181C23×C71—21no—
182C23×C71—42no—
183C23×C71—42no—
184C23×C71—42no—
185C23×C71C2×C22⋊C411yescharacteristic
186C23×C71C2×C22⋊C411yescharacteristic
187C23×C71C2×C22⋊C412yes—
188C23×C71C2×C22⋊C411yescharacteristic
189C23×C71C23×C411yescharacteristic
190C23×C71C22×D411yescharacteristic
191Q8×C71—42no—
192Q8×C71—42no—
193Q8×C71—48no—
194Q8×C71—44no—
195Q8×C71—44no—
196D4×C71—42no—
197D4×C71—42no—
198D4×C71—42no—
199D4×C71—41no—
200D4×C71—41no—
201D4×C71—41no—
202D4×C71—42no—
203D4×C71—41no—
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—21no—
207C2×C4×C71—21no—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—22no—
216C2×C4×C71—22no—
217C2×C4×C71—22no—
218C2×C4×C71—22no—
219C2×C4×C71—22no—
220C2×C4×C71—22no—
221C2×C4×C71—22no—
222C2×C4×C71—24no—
223C2×C4×C71—24no—
224C2×C4×C71—44no—
225C2×C4×C71—44no—
226C2×C4×C71—42no—
227C2×C4×C71—42no—
228C2×C4×C71—44no—
229C2×C4×C71#9912yes—
230C2×C4×C71#9911yescharacteristic
231C2×C4×C71#9911yescharacteristic
232C2×C4×C71C2×C22⋊C412yes—
233C2×C4×C71C2×C22⋊C412yes—
234C2×C4×C71C2×C22⋊C411yescharacteristic
235C2×C4×C71C2×C22⋊C412yes—
236C2×C4×C71C22×D411yescharacteristic
237C2×Dic71—44no—
238C2×Dic71—84no—
239C71⋊C8—44no—
240C22×C71—22no—
241C22×C71—21no—
242C22×C71—21no—
243C22×C71—21no—
244C22×C71—21no—
245C22×C71—22no—
246C22×C71—41no—
247C22×C71—42no—
248C22×C71—41no—
249C22×C71—41no—
250C22×C71—41no—
251C22×C71—42no—
252C22×C71—42no—
253C22×C71—42no—
254C22×C71—42no—
255C22×C71—81no—
256C22×C71—81no—
257C22×C71C2×#9911yescharacteristic
258C22×C71C2×#9911yescharacteristic
259C22×C71C22×C22⋊C411yescharacteristic
260C4×C71—21no—
261C4×C71—22no—
262C4×C71—21no—
263C4×C71—22no—
264C4×C71—44no—
265C4×C71—44no—
266C4×C71C2×#9912yes—
267C4×C71C2×#9912yes—
268Dic71—84no—
269C2×#2123—711nomaximal, Sylow
270C2×C71—21no—
271C2×C71—21no—
272C2×C71—41no—
273C2×C71—41no—
274C2×C71—41no—
275C2×C71—41no—
276C2×C71—42no—
277C2×C71#212312yescomplemented
278C2×C71C22×#9911yescharacteristic
279C2×#536D7111yescharacteristic
280C2×#413—711no—
281C2×#410—711no—
282#2123—718no—
283C2×#352—714no—
284C71C2×#212311yescharacteristic, complemented, Sylow
285C22×#41C2×D7111yescharacteristic
286C23×Q8Dic7111yescharacteristic, complemented
287C22×M16—1421no—
288#536—22no—
289#536—22no—
290#536Dic7114yescomplemented
291#536C2×D7114yes—
292C2×C4⋊Q8—21no—
293C2×#124—21no—
294C2×#123Dic7112yescomplemented
295C2×#123C2×D7112yes—
296C2×C22⋊Q8—21no—
297C2×C22⋊Q8—21no—
298C2×#117—1421no—
299C2×#117Dic7111yescharacteristic, complemented
300#413—714no—
301#413—1422no—
302C2×#101—1421no—
303C2×#100—712no—
304#410—714no—
305#410—1422no—
306C2×#99—712no—
307C2×#99—1421no—
308#352—7116no—
309C2×#41—41no—
310C2×#41—41no—
311C2×#41—42no—
312C2×#41C2×Dic7112yes—
313C2×#41C22⋊D7112yes—
314C2×#41C22⋊D7112yes—
315C2×#41C22×D7112yes—
316C22×Q8—21no—
317C22×Q8—22no—
318C22×Q8—21no—
319C22×Q8—42no—
320C22×Q8—41no—
321C22×Q8—41no—
322C22×Q8C2×Dic7114yescomplemented
323C22×Q8C22⋊D7111yescharacteristic
324C22×D4C2×Dic7111yescharacteristic
325C22×D4C22⋊D7111yescharacteristic
326C22×D4C22×D7111yescharacteristic
327C23×C4C2×Dic7111yescharacteristic
328C23×C4C2×Dic7112yes—
329C23×C4C22⋊D7111yescharacteristic
330C23×C4C22⋊D7111yescharacteristic
331C2×M16—2841no—
332C2×M16—2842no—
333C2×M16—1422no—
334C2×M16—1424no—
335C22×C8—2841no—
336C4⋊Q8—24no—
337C4⋊Q8—24no—
338#124—24no—
339#124—24no—
340#123—24no—
341#123—22no—
342#123—22no—
343#123C2×Dic7114yes—
344#123C2×Dic7118yescomplemented
345#123C22×D7114yes—
346C22⋊Q8—24no—
347C22⋊Q8—24no—
348C22⋊Q8—24no—
349C22⋊Q8—24no—
350#117—22no—
351#117—2842no—
352#117—1424no—
353#117C2×Dic7114yescomplemented
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—2841no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—1422no—
361C2×C22⋊C4C2×Dic7112yes—
362C2×C42—21no—
363C2×C42—2841no—
364#101—1424no—
365#100—714no—
366#100—1422no—
367#99—714no—
368#99—1422no—
369#99—1424no—
370C24C22⋊Dic7111yescharacteristic
371C24C22×Dic7111yescharacteristic
372C24C2×C22⋊D7111yescharacteristic
373#41—42no—
374#41—42no—
375#41—42no—
376#41—42no—
377#41—42no—
378#41—42no—
379#41—42no—
380#41—42no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—41no—
393C2×Q8—41no—
394C2×Q8C22⋊Dic7112yes—
395C2×Q8C2×C22⋊D7112yes—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4C22⋊Dic7112yes—
403C2×D4C22⋊Dic7112yes—
404C2×D4C22×Dic7111yescharacteristic
405C2×D4C22×Dic7112yes—
406C2×D4C2×C22⋊D7111yescharacteristic
407C2×D4C2×C22⋊D7111yescharacteristic
408C2×D4C2×C22⋊D7112yes—
409C2×D4C23×D7111yescharacteristic
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—2842no—
425C22×C4C22⋊Dic7111yescharacteristic
426C22×C4C22⋊Dic7111yescharacteristic
427C22×C4C22⋊Dic7112yes—
428C22×C4C22⋊Dic7112yes—
429C22×C4C22⋊Dic7112yes—
430C22×C4C22⋊Dic7111yescharacteristic
431C22×C4C22×Dic7112yes—
432C22×C4C22×Dic7114yes—
433C22×C4C2×C22⋊D7111yescharacteristic
434C22×C4C2×C22⋊D7112yes—
435C22×C4C2×C22⋊D7112yes—
436M16—2842no—
437M16—2844no—
438M16—1424no—
439C2×C8—2842no—
440C2×C8—2844no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—28no—
445C4⋊C4—2844no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—28no—
449C22⋊C4—22no—
450C22⋊C4—2842no—
451C22⋊C4—1424no—
452C22⋊C4C22×Dic7114yes—
453C42—24no—
454C42—2844no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23C2×C22⋊Dic7111yescharacteristic
465C23C2×C22⋊Dic7111yescharacteristic
466C23C2×C22⋊Dic7112yes—
467C23C2×C22⋊Dic7111yescharacteristic
468C23C23×Dic7111yescharacteristic
469C23C22×C22⋊D7111yescharacteristic
470Q8—42no—
471Q8—42no—
472Q8—48no—
473Q8—44no—
474Q8—44no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—41no—
479D4—41no—
480D4—42no—
481D4—41no—
482D4—41no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—2844no—
509C2×C4—5684no—
510C2×C4#1455604312yes—
511C2×C4#1455604311yescharacteristic
512C2×C4#1455604311yescharacteristic
513C2×C4C2×C22⋊Dic7112yes—
514C2×C4C2×C22⋊Dic7112yes—
515C2×C4C2×C22⋊Dic7111yescharacteristic
516C2×C4C2×C22⋊Dic7112yes—
517C2×C4C22×C22⋊D7111yescharacteristic
518C8—2844no—
519C22—22no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—22no—
524C22—21no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—42no—
529C22—41no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22C2×#1455604311yescharacteristic
537C22C2×#1455604311yescharacteristic
538C22C22×C22⋊Dic7111yescharacteristic
539C4—21no—
540C4—22no—
541C4—21no—
542C4—22no—
543C4—44no—
544C4—44no—
545C4—5684no—
546C4C2×#1455604312yes—
547C4C2×#1455604312yes—
548C2—21no—
549C2—21no—
550C2—41no—
551C2—41no—
552C2—41no—
553C2—41no—
554C2—42no—
555C2#2258999412yesdirect product factor
556C2C22×#1455604311yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC23×C71C23×C4
FrattiniC2×D4C23×D71
FittingC2×#536×C71C2
socleC22×C71C2×#99
prime radicalC22×#41×C71C22
cosocleC2×D4×C71C24
hypercenterC2×#536D71
hypocenterC71C2×#2123
LCSG ▷ C23×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
3. #22589994C2direct (N × H)
4. C2×#17985789C2semidirect
16. #17985789C22semidirect
18. C23×Q8×C71C4semidirect
21. #536×C71C4semidirect
25. C2×#123×C71C4semidirect
29. C2×#117×C71C4semidirect
59. C22×Q8×C71C2×C4semidirect
76. #123×C71C2×C4semidirect
83. #117×C71C2×C4semidirect
277. C2×C71#2123semidirect
284. C71C2×#2123semidirect
286. C23×Q8Dic71semidirect
290. #536Dic71semidirect
294. C2×#123Dic71semidirect
299. C2×#117Dic71semidirect
322. C22×Q8C2×Dic71semidirect
344. #123C2×Dic71semidirect
353. #117C2×Dic71semidirect
555. C2#22589994direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179859832
C2×#179859732
#225899942
C2×#179857892
C2×#536×C712
C2×#212371