Finite Group Databasebeta

← #10795149

Subgroups — #10795149

Order 768.

1262 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×Q8C211yesmaximal, characteristic
2#78517C212yesmaximal
3C3×C4×#101C211yesmaximal, characteristic, complemented
4C4×#836C211yesmaximal, characteristic, complemented
5#70113C212yesmaximal
6#70097C212yesmaximal, complemented
7C4×#824C212yesmaximal, complemented
8#68997C212yesmaximal, complemented
9#68990C212yesmaximal
10#20727—31nomaximal, Sylow
11C22×D3×Q8C2211yescharacteristic
12C2×C3×C4×Q8C2211yescharacteristic
13C2×Q8×Dic3C2211yescharacteristic
14C4×D3×Q8—22no—
15C4×D3×Q8—22no—
16C4×D3×Q8—22no—
17C4×D3×Q8—22no—
18C2×D3×C4⋊C4C412yes—
19C2×D3×C4⋊C4C2211yescharacteristic
20C2×#887C412yes—
21C2×#887C2211yescharacteristic
22C2×C42×D3C412yes—
23C2×C42×D3C2211yescharacteristic
24C2×C4×Dic6C412yes—
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×#101C2212yes—
27C3×C4×M16C2212yes—
28C2×#836C2212yes—
29C2×#824C2214yes—
30#5126C2214yes—
31#5118C2214yes—
32C4×#208C2212yes—
33#4705—22no—
34#4705—22no—
35#4685—22no—
36#4685—22no—
37C3×#343C2212yes—
38#4554C2214yes—
39#4530C2212yes—
40C22×C4×Q8—31no—
41#2920—32no—
42#1821—32no—
43#1801—32no—
44#1801—32no—
45#1784—32no—
46C4×#101—32no—
47C4×#101—31no—
48C4×#101D311yescharacteristic, complemented
49C22×C3×Q8C2311yescharacteristic
50C2×D3×Q8—24no—
51C2×D3×Q8—22no—
52C2×D3×Q8—22no—
53C2×D3×Q8—24no—
54C22×C4×D3C2×C412yes—
55C22×C4×D3C2×C411yescharacteristic
56C22×C4×D3C2×C412yes—
57C22×C4×D3C2×C411yescharacteristic
58C22×C4×D3C2311yescharacteristic
59C22×Dic6C2×C412yes—
60C22×Dic6C2311yescharacteristic
61C2×C3×M16C2314yes—
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C2×C3×C4⋊C4C2×C412yes—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C42C2×C412yes—
67C2×C3×C42C2311yescharacteristic
68Q8×Dic3—22no—
69Q8×Dic3—22no—
70C2×C4⋊Dic3C2×C412yes—
71C2×C4⋊Dic3C2311yescharacteristic
72C2×Dic3⋊C4C2×C412yes—
73C2×Dic3⋊C4C2×C412yes—
74C2×Dic3⋊C4C2×C412yes—
75C2×C4×Dic3C2×C412yes—
76C2×C4×Dic3C2×C412yes—
77C2×C4×Dic3C2×C412yes—
78C2×C4×Dic3C2×C411yescharacteristic
79C2×C4×Dic3C2×C411yescharacteristic
80C2×C4×Dic3C2311yescharacteristic
81C2×#208C2314yes—
82D3×C4⋊C4—24no—
83D3×C4⋊C4—24no—
84D3×C4⋊C4—22no—
85D3×C4⋊C4—22no—
86#887—24no—
87#887—24no—
88#887—22no—
89#887—22no—
90C42×D3—22no—
91C42×D3—22no—
92C42×D3—22no—
93C42×D3—22no—
94C42×D3D412yes—
95C42×D3D412yes—
96C42×D3D412yes—
97C42×D3D412yes—
98C4×Dic6—22no—
99C4×Dic6—22no—
100C4×Dic6—24no—
101C4×Dic6—24no—
102C3×#101—24no—
103C3×#97—22no—
104C3×C4×C8—22no—
105#836—24no—
106#824—28no—
107(C2×D3)⋊C8—24no—
108(C2×D3)⋊C8—24no—
109Dic3⋊C8—24no—
110Dic3⋊C8—24no—
111#803—22no—
112C4×C3⋊C8—22no—
113C23×Q8—31no—
114C2×C4×Q8—31no—
115C2×C4×Q8—32no—
116C2×C4×Q8—31no—
117C2×C4×Q8—32no—
118C2×C4×Q8—31no—
119C2×C4×Q8—62no—
120C2×C4×Q8—62no—
121C2×C4×Q8—62no—
122C2×C4×Q8—62no—
123C2×C4×Q8C2×D311yescharacteristic
124C22×C4⋊C4—32no—
125C22×C4⋊C4—31no—
126C22×C42—31no—
127C22×C42—32no—
128#432—62no—
129#432—62no—
130#431—62no—
131#431—62no—
132#423—34no—
133C2×#101—32no—
134C2×#101—34no—
135C2×#101C2×D312yes—
136#407—34no—
137C4×M16—32no—
138C4×M16C2×D312yes—
139#343—34no—
140#343—32no—
141#343C2×D312yes—
142C23×D3C22×C411yescharacteristic
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8#4114yes—
146C22×C3×C4C22×C411yescharacteristic
147C22×C3×C4C22×C412yes—
148C22×C3×C4C22×C412yes—
149C22×C3×C4C22×C411yescharacteristic
150C22×C3×C4C2411yescharacteristic
151C22×Dic3C22×C411yescharacteristic
152C22×Dic3C22×C411yescharacteristic
153C22×Dic3C22×C412yes—
154C22×Dic3C22×C412yes—
155C22×Dic3C22×C411yescharacteristic
156D3×Q8—216no—
157D3×Q8—216no—
158C2×C4×D3—22no—
159C2×C4×D3—22no—
160C2×C4×D3—24no—
161C2×C4×D3—22no—
162C2×C4×D3—22no—
163C2×C4×D3—24no—
164C2×C4×D3—24no—
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—42no—
168C2×C4×D3—44no—
169C2×C4×D3C22⋊C411yescharacteristic
170C2×C4×D3C22⋊C411yescharacteristic
171C2×C4×D3C22⋊C411yescharacteristic
172C2×C4×D3C22⋊C412yes—
173C2×C4×D3C22⋊C412yes—
174C2×C4×D3C22⋊C411yescharacteristic
175C2×C4×D3C2×D412yes—
176C2×C4×D3C2×D412yes—
177C2×Dic6—24no—
178C2×Dic6—22no—
179C2×Dic6—22no—
180C2×Dic6—24no—
181C2×Dic6—24no—
182C2×Dic6#4114yes—
183C3×M16—24no—
184C3×M16—24no—
185C2×C3×C8—24no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—22no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42C2×D412yes—
191C3×C42C2×D412yes—
192C4⋊Dic3—22no—
193C4⋊Dic3—24no—
194Dic3⋊C4—24no—
195Dic3⋊C4—24no—
196Dic3⋊C4—24no—
197C4×Dic3—22no—
198C4×Dic3—22no—
199C4×Dic3—22no—
200C4×Dic3—22no—
201C4×Dic3—22no—
202C4×Dic3—22no—
203C4×Dic3C22⋊C412yes—
204C4×Dic3C22⋊C412yes—
205C4×Dic3C22⋊C412yes—
206C4×Dic3C22⋊C412yes—
207C4×Dic3C2×D412yes—
208C4×Dic3C2×D412yes—
209#208—24no—
210#208—24no—
211C2×C3⋊C8—24no—
212C22×Q8—32no—
213C22×Q8—31no—
214C22×Q8—62no—
215C22×Q8—62no—
216C22×Q8—64no—
217C22×Q8—64no—
218C22×Q8C22×D311yescharacteristic
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—31no—
222C23×C4—32no—
223C23×C4—32no—
224C2×M16—34no—
225C2×M16C22×D314yes—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—64no—
229C4×Q8—64no—
230C4×Q8—64no—
231C4×Q8—64no—
232C4×Q8—62no—
233C4×Q8—62no—
234C4×Q8—62no—
235C4×Q8—62no—
236C4×Q8—62no—
237C4×Q8—62no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—32no—
241C2×C4⋊C4—32no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—64no—
244C2×C4⋊C4—64no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4C4×D312yes—
248C2×C4⋊C4C22×D311yescharacteristic
249C2×C42—31no—
250C2×C42—31no—
251C2×C42—31no—
252C2×C42—32no—
253C2×C42—32no—
254C2×C42—32no—
255C2×C42—32no—
256C2×C42—32no—
257C2×C42—32no—
258C2×C42—32no—
259C2×C42—62no—
260C2×C42—62no—
261C2×C42—62no—
262C2×C42—62no—
263C2×C42C4×D312yes—
264C2×C42C22×D311yescharacteristic
265C4⋊C8—64no—
266C4⋊C8—64no—
267#101—24no—
268#101—68no—
269#101—64no—
270C22⋊C8—64no—
271C22⋊C8—64no—
272#97—22no—
273#97—62no—
274C4×C8—22no—
275C4×C8—62no—
276C23×C3C23×C411yescharacteristic
277C22×D3—21no—
278C22×D3—21no—
279C22×D3—22no—
280C22×D3—22no—
281C22×D3C2×C22⋊C411yescharacteristic
282C22×D3C2×C22⋊C411yescharacteristic
283C3×Q8—28no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—41no—
289C2×C3×C4—42no—
290C2×C3×C4C2×C22⋊C411yescharacteristic
291C2×C3×C4C2×C22⋊C412yes—
292C2×C3×C4C2×C22⋊C411yescharacteristic
293C2×C3×C4#11714yes—
294C2×C3×C4C22×D412yes—
295C2×C3×C4C2×#4112yes—
296C2×C3×C4C2×#4112yes—
297C2×Dic3—22no—
298C2×Dic3—21no—
299C2×Dic3—22no—
300C2×Dic3—21no—
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—41no—
304C2×Dic3—42no—
305C2×Dic3C2×C22⋊C411yescharacteristic
306C2×Dic3C2×C22⋊C412yes—
307C2×Dic3C2×C22⋊C412yes—
308C2×Dic3C2×C22⋊C411yescharacteristic
309C2×Dic3#11714yes—
310C4×D3—28no—
311C4×D3—28no—
312C4×D3—22no—
313C4×D3—24no—
314C4×D3—24no—
315C4×D3—24no—
316C4×D3—22no—
317C4×D3—22no—
318C4×D3—42no—
319C4×D3—44no—
320C4×D3—44no—
321C4×D3#10114yescomplemented
322Dic6—216no—
323Dic6—28no—
324C3×C8—44no—
325C3⋊C8—44no—
326C24—31no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—34no—
330C2×Q8—64no—
331C2×Q8—64no—
332C2×Q8—616no—
333C2×Q8—616no—
334C2×Q8—64no—
335C2×Q8—62no—
336C2×Q8—62no—
337C2×Q8#23514yes—
338C22×C4—31no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—32no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—32no—
346C22×C4—31no—
347C22×C4—32no—
348C22×C4—32no—
349C22×C4—32no—
350C22×C4—32no—
351C22×C4—64no—
352C22×C4—64no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—62no—
356C22×C4—62no—
357C22×C4—64no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—122no—
361C22×C4—124no—
362C22×C4C2×C4×D312yes—
363C22×C4C2×C4×D311yescharacteristic
364C22×C4C2×C4×D312yes—
365C22×C4C2×C4×D311yescharacteristic
366C22×C4C23×D311yescharacteristic
367M16—24no—
368M16—24no—
369M16—64no—
370M16—64no—
371C2×C8—24no—
372C2×C8—64no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—64no—
376C4⋊C4—64no—
377C4⋊C4—64no—
378C4⋊C4—64no—
379C4⋊C4—62no—
380C42—22no—
381C42—22no—
382C42—32no—
383C42—32no—
384C42—32no—
385C42—32no—
386C42—32no—
387C42—32no—
388C42—62no—
389C42—62no—
390C42—62no—
391C42—62no—
392C42—62no—
393C42—62no—
394C42D3×D412yes—
395C42D3×D412yes—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3C22×C22⋊C411yescharacteristic
399C22×C3C2×#11711yescharacteristic
400C22×C3C2×#11711yescharacteristic
401C2×D3—24no—
402C2×D3—24no—
403C2×D3—21no—
404C2×D3—21no—
405C2×D3—21no—
406C2×D3—21no—
407C2×D3C2×#10112yes—
408C2×D3#41312yes—
409C3×C4—21no—
410C3×C4—22no—
411C3×C4—24no—
412C3×C4—42no—
413C3×C4C2×#10112yes—
414C3×C4#52514yes—
415Dic3—21no—
416Dic3—22no—
417Dic3—22no—
418Dic3—24no—
419Dic3—42no—
420Dic3C2×#10112yes—
421C23—31no—
422C23—31no—
423C23—62no—
424C23—62no—
425C23—61no—
426C23—61no—
427C23C22×C4×D311yescharacteristic
428Q8—28no—
429Q8—616no—
430Q8—68no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—31no—
436C2×C4—34no—
437C2×C4—31no—
438C2×C4—32no—
439C2×C4—32no—
440C2×C4—34no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—64no—
446C2×C4—68no—
447C2×C4—68no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—61no—
451C2×C4—62no—
452C2×C4—61no—
453C2×C4—62no—
454C2×C4—64no—
455C2×C4—64no—
456C2×C4—62no—
457C2×C4—122no—
458C2×C4—124no—
459C2×C4—124no—
460C2×C4—121no—
461C2×C4—122no—
462C2×C4#87214yes—
463C2×C4D3×C22⋊C411yescharacteristic
464C2×C4D3×C22⋊C412yes—
465C2×C4D3×C22⋊C411yescharacteristic
466C2×C4C2×#23512yes—
467C2×C4C2×D3×D412yes—
468C2×C4D3×#4112yes—
469C8—44no—
470C8—124no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3#228011yescharacteristic
474C2×C3C22×#10111yescharacteristic
475C2×C3C2×#41311yescharacteristic
476D3—22no—
477D3—22no—
478C22—21no—
479C22—21no—
480C22—32no—
481C22—32no—
482C22—64no—
483C22—64no—
484C22—61no—
485C22—61no—
486C22—61no—
487C22—61no—
488C22C2×#87211yescharacteristic
489C22C2×D3×C22⋊C411yescharacteristic
490C22D3×#11711yescharacteristic
491C4—24no—
492C4—21no—
493C4—22no—
494C4—32no—
495C4—42no—
496C4—62no—
497C4—64no—
498C4—61no—
499C4—62no—
500C4—122no—
501C4D3×#10112yes—
502C4#554714yes—
503C3#2072711yescharacteristic, complemented, Sylow
504C2—21no—
505C2—21no—
506C2—62no—
507C2—62no—
508C2#7356111yescharacteristic
509C2C2×D3×#10111yescharacteristic
510C2D3×#41311yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#101C2
socleC22×C3C22×C22⋊C4
prime radicalC23×D3C22×C4
cosocleC22×C3×C4C24
hypercenterC4×#101D3
hypocenterC3#20727
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
3. C3×C4×#101C2semidirect
4. C4×#836C2semidirect
6. #70097C2semidirect
7. C4×#824C2semidirect
8. #68997C2semidirect
48. C4×#101D3semidirect
321. C4×D3#101semidirect
503. C3#20727semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×Q82
#785172
C3×C4×#1012
C4×#8362
#701132
#700972
C4×#8242
#689972
#689902
#207273