Finite Group Databasebeta

← C2×#22598861 #32867065

Subgroups — C2×#22598861 #32867065

Order 18176.

1902 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986555C211yesmaximal, characteristic
2C2×#17986554C211yesmaximal, characteristic
3#22598861C2116yesmaximal, direct product factor
4C2×#17986409C211yesmaximal, characteristic, complemented
5C2×#17986408C211yesmaximal, characteristic, complemented
6C2×#17986407C212yesmaximal, complemented
7C2×#17986406C212yesmaximal, complemented
8C2×#17986405C211yesmaximal, characteristic, complemented
9C2×#17986036C212yesmaximal
10C2×#17986034C212yesmaximal
11C2×#97×D71C211yesmaximal, characteristic
12C2×#501×C71C211yesmaximal, characteristic, complemented
13C22×C8×D71C2212yes—
14#17986555—24no—
15#17986555C2218yes—
16#17986554—24no—
17#17986554C2218yes—
18C2×#14556128C2211yescharacteristic
19C2×#14556127C2211yescharacteristic
20C2×#14556127C2211yescharacteristic
21C2×#14556120C2211yescharacteristic
22C2×#14556119C2212yes—
23C2×#14556108—21no—
24C2×#14556106—21no—
25C2×#14556106—21no—
26C2×#14556104—21no—
27C2×Q8×Dic71—21no—
28C2×#14556093—21no—
29C2×#14556093—21no—
30C2×#14556093—21no—
31C2×#14556086C2211yescharacteristic
32C2×#14556086C2212yes—
33C2×#14556086C2211yescharacteristic
34C2×C42×D71C2211yescharacteristic
35#17986409C2218yescomplemented
36#17986408C2218yescomplemented
37#17986407C22116yescomplemented
38#17986406C22116yescomplemented
39#17986405C2218yescomplemented
40#17986036C22116yes—
41#17986034C22116yes—
42C2×#14556062C2212yes—
43C2×#14556061C2212yes—
44C2×#14556061C2212yes—
45C2×#14556060C2212yes—
46C2×#14556059C2212yes—
47C2×#14556059C2212yes—
48C2×#14556039C2211yescharacteristic
49C2×#14556038C2212yes—
50#97×D71C2218yes—
51C2×C4⋊Q8×C71C2211yescharacteristic
52C2×#125×C71C2211yescharacteristic
53#501×C71C2218yescomplemented
54C2×#107×C71C2212yes—
55C2×#106×C71C2212yes—
56C2×#97×C71C2211yescharacteristic
57C22×Dic142—21no—
58C22×Dic142—21no—
59C22×Dic142—21no—
60C22×C4×D71D411yescharacteristic
61C22×C4×D71D411yescharacteristic
62C22×C4×D71C2311yescharacteristic
63C22×C71⋊C8C2312yes—
64C2×C8×D71Q8116yescomplemented
65C2×C8×D71C2318yes—
66#14556128—22no—
67#14556128C2314yes—
68#14556127—22no—
69#14556127—22no—
70#14556127C2314yes—
71#14556127C2314yes—
72#14556120—22no—
73#14556120C2314yes—
74#14556119—24no—
75#14556119C2318yes—
76#14556108—28no—
77#14556106—28no—
78#14556106—28no—
79#14556104—28no—
80Q8×Dic71—28no—
81#14556093—28no—
82#14556093—28no—
83#14556093—28no—
84#14556086—24no—
85#14556086—22no—
86#14556086—22no—
87#14556086C2314yes—
88#14556086C2318yes—
89#14556086C2314yes—
90C2×C4⋊Dic71—22no—
91C2×C4⋊Dic71—21no—
92C2×C4⋊Dic71C2312yes—
93C2×C4⋊Dic71C2311yescharacteristic
94C2×C4⋊Dic71C2311yescharacteristic
95C2×Dic71⋊C4—22no—
96C2×Dic71⋊C4—21no—
97C2×Dic71⋊C4—21no—
98C2×Dic71⋊C4—21no—
99C2×Dic71⋊C4—21no—
100C2×(C2×D71)⋊C4—22no—
101C2×(C2×D71)⋊C4—21no—
102C2×(C2×D71)⋊C4—21no—
103C2×C4×Dic71—21no—
104C2×C4×Dic71—21no—
105C2×C4×Dic71—22no—
106C2×C4×Dic71D411yescharacteristic
107C2×C4×Dic71D411yescharacteristic
108C2×C4×Dic71C2311yescharacteristic
109C42×D71D414yes—
110C42×D71C2314yes—
111#14556062C2318yes—
112#14556061C2318yes—
113#14556061C2318yes—
114#14556060C2318yes—
115#14556059C2318yes—
116#14556059C2318yes—
117#14556039C2314yes—
118#14556038C2318yes—
119C22×Q8×C71—21no—
120C22×C8×C71C2312yes—
121C4⋊Q8×C71—22no—
122C4⋊Q8×C71C2314yes—
123#125×C71—22no—
124#125×C71C2314yes—
125C2×C4⋊C4×C71—21no—
126C2×C4⋊C4×C71—21no—
127C2×C4⋊C4×C71—21no—
128C2×C4⋊C4×C71C2311yescharacteristic
129C2×C4⋊C4×C71C2312yes—
130C2×C4⋊C4×C71C2311yescharacteristic
131C2×C42×C71C2311yescharacteristic
132#107×C71C2318yes—
133#106×C71C2318yes—
134#97×C71C2314yes—
135C23×D71C2×D411yescharacteristic
136C2×Dic142—24no—
137C2×Dic142—24no—
138C2×Dic142—24no—
139C2×Dic142—44no—
140C2×Dic142—44no—
141C2×Dic142—44no—
142C22×Dic71—21no—
143C22×Dic71—21no—
144C22×Dic71—22no—
145C22×Dic71C2×D411yescharacteristic
146C22×Dic71C2×D411yescharacteristic
147C22×Dic71C2×D411yescharacteristic
148C2×C4×D71—42no—
149C2×C4×D71—42no—
150C2×C4×D71C2×D414yes—
151C2×C4×D71C2×D414yes—
152C2×C4×D71C2×D412yes—
153C2×C4×D71C2×Q818yes—
154C2×C4×D71C2412yes—
155C2×C71⋊C8C2×Q818yes—
156C2×C71⋊C8C2414yes—
157C8×D71C2×Q8132yescomplemented
158C4⋊Dic71—24no—
159C4⋊Dic71—22no—
160C4⋊Dic71—21no—
161C4⋊Dic71—21no—
162C4⋊Dic71—28no—
163C4⋊Dic71C2412yes—
164C4⋊Dic71C2414yes—
165C4⋊Dic71C2412yes—
166Dic71⋊C4—24no—
167Dic71⋊C4—24no—
168Dic71⋊C4—28no—
169Dic71⋊C4—24no—
170Dic71⋊C4—24no—
171(C2×D71)⋊C4—28no—
172(C2×D71)⋊C4—24no—
173(C2×D71)⋊C4—24no—
174C4×Dic71—24no—
175C4×Dic71—24no—
176C4×Dic71—28no—
177C4×Dic71C2×D412yes—
178C4×Dic71C2×D414yes—
179C4×Dic71C2×D414yes—
180C4×Dic71C2412yes—
181C2×Q8×C71—24no—
182C2×Q8×C71—44no—
183C22×C4×C71—21no—
184C22×C4×C71—21no—
185C22×C4×C71—22no—
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×D411yescharacteristic
188C22×C4×C71C2411yescharacteristic
189C2×C8×C71C2×Q818yes—
190C2×C8×C71C2414yes—
191C4⋊C4×C71—24no—
192C4⋊C4×C71—24no—
193C4⋊C4×C71—24no—
194C4⋊C4×C71—21no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—22no—
197C4⋊C4×C71C2412yes—
198C4⋊C4×C71C2414yes—
199C4⋊C4×C71C2412yes—
200C42×C71C2×D412yes—
201C42×C71C2412yes—
202C23×C71C22×D411yescharacteristic
203C22×D71—22no—
204C22×D71—22no—
205C22×D71C4⋊Q814yes—
206C22×D71C22×D412yes—
207Q8×C71—48no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—24no—
213C2×C4×C71—44no—
214C2×C4×C71—42no—
215C2×C4×C71—42no—
216C2×C4×C71C22×D412yes—
217C2×C4×C71C22×D412yes—
218C2×C4×C71C22×D411yescharacteristic
219C2×C4×C71C22×Q814yes—
220C2×C4×C71C2511yescharacteristic
221C2×Dic71—22no—
222C2×Dic71—22no—
223C2×Dic71—22no—
224C2×Dic71—22no—
225C2×Dic71—24no—
226C2×Dic71—42no—
227C2×Dic71—42no—
228C2×Dic71—44no—
229C2×Dic71C4⋊Q814yes—
230C2×Dic71C22×D412yes—
231C2×Dic71C22×D412yes—
232C2×Dic71C22×D412yes—
233C4×D71—44no—
234C4×D71—44no—
235C4×D71C4⋊Q818yes—
236C4×D71C22×Q818yes—
237Dic142—48no—
238Dic142—48no—
239Dic142—48no—
240C8×C71C22×Q818yes—
241C71⋊C8C22×Q818yes—
242C22×C71C2×C4⋊Q812yes—
243C22×C71#57512yes—
244C22×C71#57512yes—
245C22×C71C23×D411yescharacteristic
246C2×D71—28no—
247C2×D71—22no—
248C2×D71—22no—
249C2×D71C2×C4⋊Q814yes—
250C4×C71—22no—
251C4×C71—22no—
252C4×C71—42no—
253C4×C71—42no—
254C4×C71—44no—
255C4×C71C2×C4⋊Q812yes—
256C4×C71C23×Q812yes—
257Dic71—22no—
258Dic71—22no—
259Dic71—42no—
260Dic71—42no—
261Dic71—44no—
262Dic71C2×C4⋊Q814yes—
263C2×#3162—711nomaximal, Sylow
264C2×C71#316214yescomplemented
265C2×C71C22×C4⋊Q811yescharacteristic
266C2×C71C2×#57511yescharacteristic
267C2×C71C2×#57511yescharacteristic
268D71—24no—
269C2×#533—711no—
270C2×#533—711no—
271#3162—7116no—
272C2×#501—712no—
273C2×#501—712no—
274C2×#501—711no—
275C2×#501—711no—
276C2×#501—711no—
277C2×#501D7111yescharacteristic, complemented
278C2×#427—712no—
279C2×#427—712no—
280C22×#97—711no—
281C71C2×#316211yescharacteristic, complemented, Sylow
282C23×C8—712no—
283#533—718no—
284#533—718no—
285#533—1424no—
286#533—1424no—
287C2×C4⋊Q8—711no—
288C2×C4⋊Q8—711no—
289C2×C4⋊Q8—711no—
290C2×C4⋊Q8C2×D7111yescharacteristic
291C2×#125—712no—
292C2×#125—711no—
293C2×#125C2×D7111yescharacteristic
294C2×C22⋊Q8—1421no—
295C2×C22⋊Q8—1421no—
296C2×C22⋊Q8—1421no—
297C2×C22⋊Q8—1421no—
298C2×C4×Q8—1421no—
299C2×C4×Q8—1421no—
300C2×C4×Q8—1421no—
301C2×C4×Q8—1421no—
302C2×#117—712no—
303C2×#117—711no—
304C2×#117—711no—
305C22×C42—711no—
306#501—718no—
307#501—718no—
308#501—7116no—
309#501—7116no—
310#501—718no—
311#501C2×D7118yescomplemented
312#427—7116no—
313#427—7116no—
314C2×#107—712no—
315C2×#107—712no—
316C2×#107—712no—
317C2×#107C2×D7112yes—
318C2×#106—712no—
319C2×#106—712no—
320C2×#106—712no—
321C2×#106C2×D7112yes—
322C2×#97—712no—
323C2×#97—711no—
324C2×#97—718no—
325C2×#97C2×D7111yescharacteristic
326C22×Q8—21no—
327C22×Q8—1421no—
328C22×Q8—1421no—
329C22×Q8—1421no—
330C23×C4—711no—
331C23×C4—711no—
332C23×C4—711no—
333C22×C8—712no—
334C22×C8—7116no—
335C22×C8—718no—
336C22×C8C22×D7112yes—
337C4⋊Q8—22no—
338C4⋊Q8—714no—
339C4⋊Q8—714no—
340C4⋊Q8—714no—
341C4⋊Q8—1422no—
342C4⋊Q8—1422no—
343C4⋊Q8—1422no—
344C4⋊Q8C22×D7114yes—
345#125—22no—
346#125—718no—
347#125—714no—
348#125—1424no—
349#125—1422no—
350#125C22×D7114yes—
351C22⋊Q8—1428no—
352C22⋊Q8—1428no—
353C22⋊Q8—1428no—
354C22⋊Q8—1428no—
355C4×Q8—1428no—
356C4×Q8—1428no—
357C4×Q8—1428no—
358C4×Q8—1428no—
359#117—714no—
360#117—718no—
361#117—714no—
362#117—1422no—
363#117—1422no—
364#117—1424no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—712no—
369C2×C4⋊C4—711no—
370C2×C4⋊C4—711no—
371C2×C4⋊C4—1422no—
372C2×C4⋊C4—1422no—
373C2×C4⋊C4—1421no—
374C2×C4⋊C4—1421no—
375C2×C4⋊C4—1421no—
376C2×C4⋊C4—1421no—
377C2×C4⋊C4—1421no—
378C2×C4⋊C4C22×D7112yes—
379C2×C4⋊C4C22×D7111yescharacteristic
380C2×C4⋊C4C22×D7111yescharacteristic
381C2×C22⋊C4—1421no—
382C2×C22⋊C4—1421no—
383C2×C22⋊C4—1422no—
384C2×C42—711no—
385C2×C42—711no—
386C2×C42—711no—
387C2×C42—714no—
388C2×C42—714no—
389C2×C42—1421no—
390C2×C42—1421no—
391C2×C42—1422no—
392C2×C42C22×D7111yescharacteristic
393#107—718no—
394#107—718no—
395#107—718no—
396#107C22×D7118yes—
397#106—718no—
398#106—718no—
399#106—718no—
400#106C22×D7118yes—
401#97—718no—
402#97—714no—
403#97C22×D7114yes—
404C24—711no—
405C2×Q8—24no—
406C2×Q8—44no—
407C2×Q8—2844no—
408C2×Q8—2844no—
409C2×Q8—2844no—
410C2×Q8—1424no—
411C2×Q8—1424no—
412C2×Q8—1424no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—2842no—
417C22×C4—2842no—
418C22×C4—714no—
419C22×C4—714no—
420C22×C4—711no—
421C22×C4—712no—
422C22×C4—718no—
423C22×C4—712no—
424C22×C4—711no—
425C22×C4—711no—
426C22×C4—1421no—
427C22×C4—1421no—
428C22×C4—1422no—
429C22×C4D4×D7111yescharacteristic
430C22×C4D4×D7111yescharacteristic
431C22×C4C23×D7111yescharacteristic
432C2×C8—718no—
433C2×C8—714no—
434C2×C8—7132no—
435C2×C8Q8×D7118yes—
436C2×C8C23×D7114yes—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—712no—
444C4⋊C4—714no—
445C4⋊C4—712no—
446C4⋊C4—1428no—
447C4⋊C4—1424no—
448C4⋊C4—1424no—
449C4⋊C4—1428no—
450C4⋊C4—1424no—
451C4⋊C4—1424no—
452C4⋊C4—1422no—
453C4⋊C4—1421no—
454C4⋊C4—1421no—
455C4⋊C4—1424no—
456C4⋊C4C23×D7112yes—
457C4⋊C4C23×D7114yes—
458C4⋊C4C23×D7112yes—
459C22⋊C4—1424no—
460C22⋊C4—1424no—
461C22⋊C4—1428no—
462C42—714no—
463C42—714no—
464C42—712no—
465C42—712no—
466C42—1424no—
467C42—1424no—
468C42—1428no—
469C42D4×D7112yes—
470C42C23×D7112yes—
471C23—712no—
472C23—714no—
473C23—1422no—
474C23—1422no—
475C23C2×D4×D7111yescharacteristic
476Q8—48no—
477Q8—2848no—
478Q8—2848no—
479Q8—2848no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—2844no—
489C2×C4—2844no—
490C2×C4—2842no—
491C2×C4—2844no—
492C2×C4—2842no—
493C2×C4—712no—
494C2×C4—714no—
495C2×C4—718no—
496C2×C4—718no—
497C2×C4—712no—
498C2×C4—712no—
499C2×C4—1422no—
500C2×C4—1424no—
501C2×C4—1422no—
502C2×C4—1422no—
503C2×C4—1422no—
504C2×C4C2×D4×D7112yes—
505C2×C4C2×D4×D7112yes—
506C2×C4C2×D4×D7111yescharacteristic
507C2×C4C2×Q8×D7114yes—
508C2×C4C24×D7111yescharacteristic
509C8—718no—
510C8C2×Q8×D7118yes—
511C22—714no—
512C22—1422no—
513C22—1422no—
514C22—1428no—
515C22C4⋊Q8×D7112yes—
516C22#1798679512yes—
517C22#1798679812yes—
518C22C22×D4×D7111yescharacteristic
519C4—22no—
520C4—22no—
521C4—44no—
522C4—42no—
523C4—42no—
524C4—2842no—
525C4—2844no—
526C4—2842no—
527C4—714no—
528C4—1422no—
529C4—1422no—
530C4C4⋊Q8×D7112yes—
531C4C22×Q8×D7112yes—
532C2—1424no—
533C2#2259886114yesdirect product factor
534C2C2×C4⋊Q8×D7111yescharacteristic
535C2C2×#1798679511yescharacteristic
536C2C2×#1798679811yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutator, cosocleC2×C4×C71C25
FrattiniC2×C4C24×D71
FittingC2×#501×C71C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
hypercenterC2×#501D71
hypocenterC71C2×#3162
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
3. #22598861C2direct (N × H)
4. C2×#17986409C2semidirect
5. C2×#17986408C2semidirect
6. C2×#17986407C2semidirect
7. C2×#17986406C2semidirect
8. C2×#17986405C2semidirect
12. C2×#501×C71C2semidirect
35. #17986409C22semidirect
36. #17986408C22semidirect
37. #17986407C22semidirect
38. #17986406C22semidirect
39. #17986405C22semidirect
53. #501×C71C22semidirect
64. C2×C8×D71Q8semidirect
157. C8×D71C2×Q8semidirect
264. C2×C71#3162semidirect
277. C2×#501D71semidirect
281. C71C2×#3162semidirect
311. #501C2×D71semidirect
533. C2#22598861direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179865552
C2×#179865542
#225988612
C2×#179864092
C2×#179864082
C2×#179864072
C2×#179864062
C2×#179864052
C2×#179860362
C2×#179860342
C2×#97×D712
C2×#501×C712
C2×#316271