Finite Group Databasebeta

← C2×#70059 #10947979

Subgroups — C2×#70059 #10947979

Order 768.

1230 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5825C211yesmaximal, characteristic
2C2×C3×#410C211yesmaximal, characteristic
3C2×#5240C211yesmaximal, characteristic
4#70059C218yesmaximal, direct product factor
5C2×#4473C214yesmaximal, complemented
6C2×#2129—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C23×Dic6C411yescharacteristic, complemented
9C2×C3×#117—21no—
10#5825—22no—
11#5825—22no—
12#5825C414yescomplemented
13#5825C2214yes—
14C2×#944C412yescomplemented
15C2×#936C2212yes—
16C2×#935—21no—
17C2×#935—21no—
18C2×#926—21no—
19C2×#926C411yescharacteristic, complemented
20C2×#924—21no—
21C2×#924—21no—
22C3×#410—22no—
23C3×#410C2214yes—
24C2×C3×#99—21no—
25C2×C3×#99C2212yes—
26#5240—22no—
27#5240C2214yes—
28C2×#834—21no—
29C2×#834C2212yes—
30#4473C22116yescomplemented
31C2×#536—31no—
32C2×#410—31no—
33C2×#410D311yescharacteristic
34#2129—38no—
35C2×#354—34no—
36C2×C3×#41—41no—
37C2×C3×#41—41no—
38C2×C3×#41—42no—
39C2×C3×#41C2×C412yes—
40C2×C3×#41D412yes—
41C2×C3×#41D412yes—
42C2×C3×#41C2312yes—
43C22×C3×Q8C2×C411yescharacteristic
44C22×C3×D4D411yescharacteristic
45C22×C3×D4D411yescharacteristic
46C22×C3×D4C2311yescharacteristic
47C23×C3×C4C2×C411yescharacteristic
48C23×C3×C4D411yescharacteristic
49C23×C3×C4D411yescharacteristic
50C23×Dic3—21no—
51C22×Dic6—21no—
52C22×Dic6—22no—
53C22×Dic6—42no—
54C22×Dic6—41no—
55C22×Dic6—41no—
56C22×Dic6C2×C412yes—
57C22×Dic6C2×C414yescomplemented
58C3×#117—24no—
59C3×#117—42no—
60C2×C3×C4⋊C4—41no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C42—41no—
63C2×#217—21no—
64C2×#217—22no—
65C2×#217—22no—
66C2×#217—22no—
67#944—24no—
68#944C2×C418yescomplemented
69#936—22no—
70#936—22no—
71#936C2×C414yes—
72#936C2314yes—
73#935—24no—
74#935—24no—
75#935—24no—
76#935—24no—
77#926—22no—
78#926—24no—
79#926—42no—
80#926C2×C414yescomplemented
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—41no—
83#924—24no—
84#924—24no—
85#924—24no—
86#924—24no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×C4×Dic3—41no—
90C2×C4×Dic3C2×C412yes—
91C3×#99—24no—
92C3×#99—22no—
93C3×#99C2314yes—
94#834—24no—
95#834—22no—
96#834C2314yes—
97C22×#41C2×D311yescharacteristic
98C23×Q8—31no—
99#536—34no—
100#536—34no—
101#536—62no—
102#536—62no—
103C2×C4⋊Q8—32no—
104C2×#124—32no—
105C2×#123—61no—
106C2×#123—61no—
107C2×C22⋊Q8—61no—
108C2×C22⋊Q8—61no—
109C2×#117—21no—
110C2×#117—31no—
111C2×#117—61no—
112#410—22no—
113#410—34no—
114#410—62no—
115#410C2×D314yes—
116C2×#99—21no—
117C2×#99—32no—
118C2×#99—61no—
119C2×#99C2×D312yes—
120#354—316no—
121C24×C3C22⋊C411yescharacteristic
122C24×C3C2×D411yescharacteristic
123C24×C3C2×D411yescharacteristic
124C3×#41—42no—
125C3×#41—42no—
126C3×#41—42no—
127C3×#41—42no—
128C3×#41—42no—
129C3×#41—42no—
130C3×#41—42no—
131C3×#41—42no—
132C2×C3×Q8—41no—
133C2×C3×Q8—41no—
134C2×C3×Q8C22⋊C412yes—
135C2×C3×Q8C22×C412yes—
136C2×C3×D4—41no—
137C2×C3×D4—41no—
138C2×C3×D4—41no—
139C2×C3×D4—41no—
140C2×C3×D4—41no—
141C2×C3×D4—41no—
142C2×C3×D4C22⋊C412yes—
143C2×C3×D4C22⋊C412yes—
144C2×C3×D4C22×C411yescharacteristic
145C2×C3×D4C2×D412yes—
146C2×C3×D4C2×D411yescharacteristic
147C2×C3×D4C2×D412yes—
148C2×C3×D4C2×D411yescharacteristic
149C2×C3×D4C2411yescharacteristic
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—42no—
160C22×C3×C4C22⋊C411yescharacteristic
161C22×C3×C4C22⋊C411yescharacteristic
162C22×C3×C4C22⋊C412yes—
163C22×C3×C4C22⋊C412yes—
164C22×C3×C4C22⋊C412yes—
165C22×C3×C4C22⋊C411yescharacteristic
166C22×C3×C4C22×C411yescharacteristic
167C22×C3×C4C22×C412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—24no—
173C22×Dic3—42no—
174C22×Dic3—42no—
175C22×Dic3—42no—
176C2×Dic6—24no—
177C2×Dic6—22no—
178C2×Dic6—44no—
179C2×Dic6—44no—
180C2×Dic6—42no—
181C2×Dic6—41no—
182C2×Dic6—41no—
183C2×Dic6—42no—
184C2×Dic6—42no—
185C2×Dic6—44no—
186C2×Dic6—44no—
187C2×Dic6C22×C414yes—
188C3×C4⋊C4—44no—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—42no—
191C3×C42—44no—
192#217—24no—
193#217—24no—
194#217—24no—
195#217—24no—
196#217—24no—
197#217—24no—
198#217—42no—
199C4⋊Dic3—24no—
200C4⋊Dic3—44no—
201Dic3⋊C4—28no—
202Dic3⋊C4—28no—
203C4×Dic3—22no—
204C4×Dic3—44no—
205C4×Dic3C22×C414yes—
206C2×#41—41no—
207C2×#41—41no—
208C2×#41—42no—
209C2×#41C4×D312yes—
210C2×#41D1212yes—
211C2×#41#6712yes—
212C2×#41C22×D312yes—
213C22×Q8—32no—
214C22×Q8—34no—
215C22×Q8—62no—
216C22×Q8—61no—
217C22×Q8—122no—
218C22×Q8—121no—
219C22×Q8—121no—
220C22×Q8C4×D311yescharacteristic
221C22×D4D1211yescharacteristic
222C22×D4#6711yescharacteristic
223C22×D4C22×D311yescharacteristic
224C23×C4—61no—
225C23×C4C4×D311yescharacteristic
226C23×C4D1211yescharacteristic
227C23×C4#6711yescharacteristic
228C4⋊Q8—38no—
229C4⋊Q8—64no—
230#124—34no—
231#124—34no—
232#124—62no—
233#124—62no—
234#123—64no—
235#123—64no—
236#123—64no—
237#123—64no—
238C22⋊Q8—64no—
239C22⋊Q8—64no—
240C22⋊Q8—64no—
241C22⋊Q8—64no—
242#117—24no—
243#117—34no—
244#117—42no—
245#117—62no—
246#117—64no—
247#117—122no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—121no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—61no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—62no—
258C2×C42—32no—
259C2×C42—41no—
260C2×C42—121no—
261#99—24no—
262#99—22no—
263#99—34no—
264#99—62no—
265#99—64no—
266#99C22×D314yes—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3C2×C22⋊C411yescharacteristic
277C23×C3C2×C22⋊C411yescharacteristic
278C23×C3C2×C22⋊C411yescharacteristic
279C23×C3C2×C22⋊C412yes—
280C23×C3C22×D411yescharacteristic
281C23×C3C22×D411yescharacteristic
282C3×Q8—42no—
283C3×Q8—42no—
284C3×D4—42no—
285C3×D4—42no—
286C3×D4—41no—
287C3×D4—41no—
288C3×D4—42no—
289C3×D4—41no—
290C3×D4—42no—
291C3×D4—41no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—44no—
311C2×C3×C4—84no—
312C2×C3×C4#9912yes—
313C2×C3×C4#9911yescharacteristic
314C2×C3×C4#9911yescharacteristic
315C2×C3×C4C2×C22⋊C411yescharacteristic
316C2×C3×C4C2×C22⋊C412yes—
317C2×C3×C4C2×C22⋊C412yes—
318C2×C3×C4C2×C22⋊C412yes—
319C2×C3×C4C23×C411yescharacteristic
320C2×Dic3—24no—
321C2×Dic3—24no—
322C2×Dic3—48no—
323C2×Dic3—44no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—44no—
327C2×Dic3—84no—
328Dic6—44no—
329Dic6—44no—
330Dic6—44no—
331Dic6—44no—
332C24(C2×D3)⋊C411yescharacteristic
333C24C2×D1211yescharacteristic
334C24C2×#6711yescharacteristic
335#41—42no—
336#41—42no—
337#41—42no—
338#41—42no—
339#41—42no—
340#41—42no—
341#41—42no—
342#41—42no—
343C2×Q8—34no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—64no—
347C2×Q8—62no—
348C2×Q8—124no—
349C2×Q8—124no—
350C2×Q8—124no—
351C2×Q8—124no—
352C2×Q8—122no—
353C2×Q8—122no—
354C2×Q8—122no—
355C2×Q8—121no—
356C2×Q8—121no—
357C2×Q8(C2×D3)⋊C412yes—
358C2×Q8C2×C4×D312yes—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4(C2×D3)⋊C412yes—
366C2×D4(C2×D3)⋊C412yes—
367C2×D4C2×C4×D311yescharacteristic
368C2×D4C2×D1212yes—
369C2×D4C2×D1211yescharacteristic
370C2×D4C2×#6712yes—
371C2×D4C2×#6711yescharacteristic
372C2×D4C23×D311yescharacteristic
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—42no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—64no—
386C22×C4—122no—
387C22×C4—122no—
388C22×C4—122no—
389C22×C4(C2×D3)⋊C411yescharacteristic
390C22×C4(C2×D3)⋊C411yescharacteristic
391C22×C4(C2×D3)⋊C412yes—
392C22×C4(C2×D3)⋊C412yes—
393C22×C4(C2×D3)⋊C412yes—
394C22×C4(C2×D3)⋊C411yescharacteristic
395C22×C4C2×C4×D311yescharacteristic
396C22×C4C2×C4×D312yes—
397C22×C4C2×D1212yes—
398C22×C4C2×#6712yes—
399C4⋊C4—44no—
400C4⋊C4—64no—
401C4⋊C4—68no—
402C4⋊C4—68no—
403C4⋊C4—124no—
404C22⋊C4—24no—
405C22⋊C4—42no—
406C22⋊C4—64no—
407C22⋊C4—64no—
408C22⋊C4—64no—
409C22⋊C4—64no—
410C22⋊C4—64no—
411C22⋊C4—64no—
412C22⋊C4—122no—
413C42—34no—
414C42—44no—
415C42—62no—
416C42—124no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—42no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3C2×#9911yescharacteristic
435C22×C3C2×#9911yescharacteristic
436C22×C3C22×C22⋊C411yescharacteristic
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—84no—
442C3×C4C2×#9912yes—
443C3×C4C2×#9912yes—
444Dic3—44no—
445Dic3—44no—
446Dic3—84no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23C2×(C2×D3)⋊C411yescharacteristic
457C23C2×(C2×D3)⋊C411yescharacteristic
458C23C2×(C2×D3)⋊C411yescharacteristic
459C23C2×(C2×D3)⋊C412yes—
460C23C22×D1211yescharacteristic
461C23C22×#6711yescharacteristic
462Q8—42no—
463Q8—42no—
464Q8—124no—
465Q8—124no—
466Q8—124no—
467Q8—124no—
468D4—42no—
469D4—42no—
470D4—41no—
471D4—41no—
472D4—42no—
473D4—41no—
474D4—42no—
475D4—41no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—44no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—84no—
498C2×C4—128no—
499C2×C4—124no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4—124no—
503C2×C4—244no—
504C2×C4#80612yes—
505C2×C4#80611yescharacteristic
506C2×C4#80611yescharacteristic
507C2×C4C2×(C2×D3)⋊C411yescharacteristic
508C2×C4C2×(C2×D3)⋊C412yes—
509C2×C4C2×(C2×D3)⋊C412yes—
510C2×C4C2×(C2×D3)⋊C412yes—
511C2×C4C22×C4×D311yescharacteristic
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—41no—
515C2×C3—41no—
516C2×C3—42no—
517C2×C3—41no—
518C2×C3—41no—
519C2×C3#212912yescomplemented
520C2×C3C22×#9911yescharacteristic
521C22—22no—
522C22—21no—
523C22—22no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—42no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—81no—
537C22—81no—
538C22C2×#80611yescharacteristic
539C22C2×#80611yescharacteristic
540C22C22×(C2×D3)⋊C411yescharacteristic
541C4—21no—
542C4—22no—
543C4—22no—
544C4—21no—
545C4—84no—
546C4—124no—
547C4—124no—
548C4—244no—
549C4C2×#80612yes—
550C4C2×#80612yes—
551C3C2×#212911yescharacteristic, complemented, Sylow
552C2—21no—
553C2—21no—
554C2—41no—
555C2—41no—
556C2—42no—
557C2—41no—
558C2—41no—
559C2#7005912yesdirect product factor
560C2C22×#80611yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC2×C3×C4C23×C4
FrattiniC2×D4C23×D3
FittingC2×C3×#410C2
socleC22×C3C2×#99
prime radicalC22×C3×#41C22
cosocleC2×C3×D4C24
hypercenterC2×#410D3
hypocenterC3C2×#2129
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
4. #70059C2direct (N × H)
5. C2×#4473C2semidirect
8. C23×Dic6C4semidirect
12. #5825C4semidirect
14. C2×#944C4semidirect
19. C2×#926C4semidirect
30. #4473C22semidirect
57. C22×Dic6C2×C4semidirect
68. #944C2×C4semidirect
80. #926C2×C4semidirect
519. C2×C3#2129semidirect
551. C3C2×#2129semidirect
559. C2#70059direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#58252
C2×C3×#4102
C2×#52402
#700592
C2×#44732
C2×#21293