Finite Group Databasebeta

← C2×#68427 #10763567

Subgroups — C2×#68427 #10763567

Order 768.

974 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5728C211yesmaximal, characteristic
2C2×#4835C211yesmaximal, characteristic
3C2×#4833C211yesmaximal, characteristic
4C2×C3×#332C211yesmaximal, characteristic, complemented
5C2×#4547C211yesmaximal, characteristic, complemented
6C2×#4481C211yesmaximal, characteristic, complemented
7C2×#4481C211yesmaximal, characteristic, complemented
8#68427C218yesmaximal, direct product factor
9C2×#1554—31nomaximal, Sylow
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#946—21no—
12C2×Q8×Dic3—21no—
13#5728—22no—
14#5728—22no—
15#5728C414yes—
16#5728C2214yes—
17C2×#897C412yes—
18C2×#892—21no—
19C2×#890—21no—
20C2×#887C412yes—
21C2×C42×D3C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×(C2×D3)⋊C8—21no—
26C2×(C2×D3)⋊C8—21no—
27C2×#815—21no—
28C2×C8×Dic3—21no—
29C2×C4⋊(C3⋊C8)C2211yescharacteristic
30C2×C4⋊(C3⋊C8)C2211yescharacteristic
31#4835—22no—
32#4835C2214yes—
33#4833—22no—
34#4833C2214yes—
35C3×#332C2214yescomplemented
36#4547C2214yescomplemented
37#4481C2214yescomplemented
38#4481C2214yescomplemented
39C2×#533—31no—
40C2×#432—31no—
41C2×#431—31no—
42#1554—38no—
43C2×#333—31no—
44C2×#333—31no—
45C2×#332—31no—
46C2×#332D311yescharacteristic, complemented
47C22×C3×Q8—21no—
48C22×C4×D3C2×C411yescharacteristic
49C22×C4×D3D411yescharacteristic
50C22×C4×D3D411yescharacteristic
51C22×Dic6C2×C412yes—
52C22×C3×C8—21no—
53C22×C3×C8—21no—
54C3×C4⋊Q8—21no—
55C3×C4⋊Q8—21no—
56C3×C4⋊Q8C2×C412yes—
57C3×C4⋊Q8C2312yes—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4C2×C412yes—
60C2×C3×C42C2311yescharacteristic
61#946—24no—
62#946—24no—
63Q8×Dic3—24no—
64Q8×Dic3—24no—
65C2×(C2×D3)⋊C4—22no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—22no—
69C2×Dic3⋊C4—22no—
70C2×C4×Dic3—22no—
71C2×C4×Dic3C2×C411yescharacteristic
72C2×C4×Dic3D411yescharacteristic
73C2×C4×Dic3D411yescharacteristic
74C22×C3⋊C8—21no—
75C22×C3⋊C8—21no—
76#897—24no—
77#897C2×C418yes—
78#892—24no—
79#892—24no—
80#890—24no—
81#890—24no—
82#887—24no—
83#887C2×C418yes—
84C42×D3C2×C412yes—
85C42×D3D412yes—
86C42×D3D412yes—
87C42×D3C2312yes—
88#869—21no—
89#869—21no—
90#869C2×C412yes—
91#869C2312yes—
92C3×C4⋊C8—21no—
93C3×C4⋊C8—21no—
94C3×C4⋊C8C2312yes—
95C3×C4⋊C8C2312yes—
96(C2×D3)⋊C8—24no—
97(C2×D3)⋊C8—24no—
98#815—24no—
99C8×Dic3—24no—
100C4⋊(C3⋊C8)—21no—
101C4⋊(C3⋊C8)—21no—
102C4⋊(C3⋊C8)C2312yes—
103C4⋊(C3⋊C8)C2312yes—
104#533—34no—
105#533—34no—
106#533—62no—
107#533—62no—
108C2×C4⋊Q8—31no—
109C2×C4⋊Q8C2×D311yescharacteristic
110C2×#125—61no—
111C2×C22⋊Q8—32no—
112C2×C22⋊Q8—61no—
113C2×C4×Q8—32no—
114C2×C4×Q8—61no—
115C2×#117—61no—
116C22×C42—31no—
117#432—34no—
118#432—62no—
119#431—34no—
120#431—62no—
121C2×C4⋊C8—31no—
122C2×C4⋊C8—31no—
123C2×C4⋊C8C2×D311yescharacteristic
124C2×C4⋊C8C2×D311yescharacteristic
125C2×C22⋊C8—61no—
126C2×C22⋊C8—61no—
127C2×#97—61no—
128C2×C4×C8—61no—
129#333—34no—
130#333—34no—
131#332—34no—
132#332C2×D314yescomplemented
133C23×D3C22⋊C411yescharacteristic
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—42no—
137C2×C3×Q8—42no—
138C22×C3×C4—22no—
139C22×C3×C4C22×C411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×Dic3—22no—
143C22×Dic3C22⋊C411yescharacteristic
144C22×Dic3C22⋊C411yescharacteristic
145C22×Dic3C22⋊C411yescharacteristic
146C2×C4×D3—22no—
147C2×C4×D3—22no—
148C2×C4×D3—22no—
149C2×C4×D3—22no—
150C2×C4×D3—42no—
151C2×C4×D3C22⋊C412yes—
152C2×C4×D3C22⋊C412yes—
153C2×C4×D3C22⋊C412yes—
154C2×C4×D3C22×C412yes—
155C2×C4×D3C2×D412yes—
156C2×C4×D3C2×D412yes—
157C2×Dic6—22no—
158C2×Dic6—44no—
159C2×Dic6C22×C414yes—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—42no—
163C2×C3×C8—42no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4C22×C414yes—
168C3×C42C22×C411yescharacteristic
169C3×C42C2×D411yescharacteristic
170C3×C42C2×D411yescharacteristic
171C3×C42C2411yescharacteristic
172(C2×D3)⋊C4—28no—
173C4⋊Dic3—22no—
174C4⋊Dic3—22no—
175C4⋊Dic3—22no—
176C4⋊Dic3—22no—
177C4⋊Dic3—28no—
178Dic3⋊C4—28no—
179C4×Dic3—28no—
180C4×Dic3C22⋊C412yes—
181C4×Dic3C22⋊C412yes—
182C4×Dic3C22⋊C412yes—
183C4×Dic3C22×C412yes—
184C4×Dic3C2×D412yes—
185C4×Dic3C2×D412yes—
186C2×C3⋊C8—22no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—42no—
189C2×C3⋊C8—42no—
190C22×Q8—21no—
191C22×Q8—32no—
192C23×C4—31no—
193C23×C4—31no—
194C23×C4—31no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—61no—
198C22×C8—61no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—32no—
202C4⋊Q8—32no—
203C4⋊Q8—61no—
204C4⋊Q8—61no—
205C4⋊Q8C4×D312yes—
206C4⋊Q8C22×D312yes—
207#125—64no—
208#125—64no—
209C22⋊Q8—38no—
210C22⋊Q8—64no—
211C22⋊Q8—64no—
212C22⋊Q8—64no—
213C4×Q8—38no—
214C4×Q8—64no—
215C4×Q8—64no—
216C4×Q8—64no—
217#117—64no—
218#117—64no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4C4×D312yes—
225C2×C22⋊C4—62no—
226C2×C42—31no—
227C2×C42—31no—
228C2×C42—31no—
229C2×C42—32no—
230C2×C42—32no—
231C2×C42—32no—
232C2×C42—32no—
233C2×C42—62no—
234C2×C42C22×D311yescharacteristic
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—32no—
238C4⋊C8—32no—
239C4⋊C8—61no—
240C4⋊C8—61no—
241C4⋊C8C22×D312yes—
242C4⋊C8C22×D312yes—
243C22⋊C8—64no—
244C22⋊C8—64no—
245#97—64no—
246C4×C8—64no—
247C23×C3C2×C22⋊C411yescharacteristic
248C22×D3—22no—
249C22×D3—22no—
250C22×D3—22no—
251C22×D3C2×C22⋊C412yes—
252C3×Q8—42no—
253C3×Q8—42no—
254C3×Q8—44no—
255C2×C3×C4—22no—
256C2×C3×C4—24no—
257C2×C3×C4—44no—
258C2×C3×C4C2×C22⋊C411yescharacteristic
259C2×C3×C4C2×C22⋊C411yescharacteristic
260C2×C3×C4C2×C22⋊C411yescharacteristic
261C2×C3×C4#13512yes—
262C2×C3×C4#13512yes—
263C2×C3×C4#13712yes—
264C2×C3×C4#13712yes—
265C2×C3×C4C23×C411yescharacteristic
266C2×C3×C4C22×D411yescharacteristic
267C2×C3×C4C22×D411yescharacteristic
268C2×Dic3—22no—
269C2×Dic3—22no—
270C2×Dic3—22no—
271C2×Dic3—24no—
272C2×Dic3—44no—
273C2×Dic3C2×C22⋊C412yes—
274C2×Dic3C2×C22⋊C412yes—
275C2×Dic3C2×C22⋊C412yes—
276C4×D3—22no—
277C4×D3—22no—
278C4×D3—22no—
279C4×D3—22no—
280C4×D3—24no—
281C4×D3—24no—
282C4×D3—42no—
283C4×D3—42no—
284Dic6—44no—
285Dic6—44no—
286C3×C8—42no—
287C3×C8—42no—
288C3⋊C8—42no—
289C3⋊C8—42no—
290C24—31no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—34no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—62no—
297C2×Q8—124no—
298C22×C4—22no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—31no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—32no—
305C22×C4—32no—
306C22×C4—32no—
307C22×C4—32no—
308C22×C4—62no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—62no—
312C22×C4—62no—
313C22×C4—122no—
314C22×C4C2×C4×D311yescharacteristic
315C22×C4D3×D411yescharacteristic
316C22×C4D3×D411yescharacteristic
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—62no—
322C2×C8—62no—
323C2×C8—122no—
324C2×C8—122no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—68no—
329C4⋊C4—68no—
330C4⋊C4—62no—
331C4⋊C4—62no—
332C4⋊C4—62no—
333C4⋊C4—62no—
334C4⋊C4C2×C4×D314yes—
335C22⋊C4—68no—
336C42—32no—
337C42—32no—
338C42—32no—
339C42—32no—
340C42—32no—
341C42—32no—
342C42—68no—
343C42C2×C4×D311yescharacteristic
344C42D3×D411yescharacteristic
345C42D3×D411yescharacteristic
346C42C23×D311yescharacteristic
347C22×C3#41312yes—
348C22×C3#41612yes—
349C22×C3#41912yes—
350C22×C3C22×C22⋊C411yescharacteristic
351C2×D3—22no—
352C2×D3—22no—
353C2×D3—22no—
354C2×D3—44no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—44no—
358C3×C4#41612yes—
359C3×C4#41912yes—
360C3×C4C2×#13511yescharacteristic
361C3×C4C2×#13511yescharacteristic
362C3×C4C2×#13711yescharacteristic
363C3×C4C2×#13711yescharacteristic
364Dic3—22no—
365Dic3—22no—
366Dic3—22no—
367Dic3—44no—
368C23—32no—
369C23—62no—
370C23—62no—
371C23—62no—
372C23D3×C22⋊C411yescharacteristic
373Q8—42no—
374Q8—42no—
375Q8—44no—
376Q8—124no—
377Q8—124no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—32no—
381C2×C4—32no—
382C2×C4—32no—
383C2×C4—44no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—64no—
392C2×C4—64no—
393C2×C4—64no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—124no—
397C2×C4D3×C22⋊C411yescharacteristic
398C2×C4D3×C22⋊C411yescharacteristic
399C2×C4D3×C22⋊C411yescharacteristic
400C2×C4#91212yes—
401C2×C4#91512yes—
402C2×C4#91612yes—
403C2×C4#91812yes—
404C2×C4C22×C4×D311yescharacteristic
405C2×C4C2×D3×D411yescharacteristic
406C2×C4C2×D3×D411yescharacteristic
407C8—42no—
408C8—42no—
409C8—122no—
410C8—122no—
411C2×C3#155414yescomplemented
412C2×C3C2×#41311yescharacteristic
413C2×C3C2×#41611yescharacteristic
414C2×C3C2×#41911yescharacteristic
415D3—42no—
416C22—62no—
417C22—62no—
418C22—62no—
419C22—124no—
420C22#475112yes—
421C22#477212yes—
422C22#480212yes—
423C22C2×D3×C22⋊C411yescharacteristic
424C4—21no—
425C4—21no—
426C4—44no—
427C4—62no—
428C4—62no—
429C4—62no—
430C4—124no—
431C4#477212yes—
432C4#480212yes—
433C4C2×#91211yescharacteristic
434C4C2×#91511yescharacteristic
435C4C2×#91611yescharacteristic
436C4C2×#91811yescharacteristic
437C3C2×#155411yescharacteristic, complemented, Sylow
438C2—122no—
439C2#6842714yesdirect product factor
440C2C2×#475111yescharacteristic
441C2C2×#477211yescharacteristic
442C2C2×#480211yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC2×C3×#332C2
socleC23×C3C2×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC3×C42C24
hypercenterC2×#332D3
hypocenterC3C2×#1554
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
4. C2×C3×#332C2semidirect
5. C2×#4547C2semidirect
6. C2×#4481C2semidirect
7. C2×#4481C2semidirect
8. #68427C2direct (N × H)
35. C3×#332C22semidirect
36. #4547C22semidirect
37. #4481C22semidirect
38. #4481C22semidirect
46. C2×#332D3semidirect
132. #332C2×D3semidirect
411. C2×C3#1554semidirect
437. C3C2×#1554semidirect
439. C2#68427direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57282
C2×#48352
C2×#48332
C2×C3×#3322
C2×#45472
C2×#44812
#684272
C2×#15543