Finite Group Databasebeta

← #10765327

Subgroups — #10765327

Order 768.

638 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79316C211yesmaximal, characteristic, complemented
2#79298C211yesmaximal, characteristic
3#79286C211yesmaximal, characteristic
4C3×#1572C211yesmaximal, characteristic, complemented
5#70799C211yesmaximal, characteristic
6#70769C211yesmaximal, characteristic, complemented
7#70743C211yesmaximal, characteristic, complemented
8#68561C212yesmaximal, complemented
9#68558C212yesmaximal, complemented
10#68519C212yesmaximal, complemented
11#68516C212yesmaximal, complemented
12#17924—31nomaximal, Sylow
13C2×C4×D12C2211yescharacteristic
14#5470C2211yescharacteristic
15C3×#432C2211yescharacteristic
16C3×#431C2211yescharacteristic
17C3×#431C2211yescharacteristic
18#4925C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4842C412yes—
21#4841C412yes—
22#4834C2212yes—
23#4834C2212yes—
24#4831C2212yescomplemented
25#4830C2212yescomplemented
26#4731C412yes—
27D3×C22⋊C8—22no—
28#4721C412yes—
29#4720—22no—
30#4701—22no—
31C4×#204C2212yes—
32C4×#204C2212yes—
33C8×D12—22no—
34C3×#335C2212yescomplemented
35C3×#334C2212yescomplemented
36#4484C2212yescomplemented
37#4484C2212yescomplemented
38#4483C2212yes—
39#4483C2212yes—
40#4455C2211yescharacteristic, complemented
41#4454C2211yescharacteristic, complemented
42#4454C2211yescharacteristic, complemented
43C8×C3⋊C8C2211yescharacteristic, complemented
44#3101—31no—
45#2985—31no—
46#2972—31no—
47#1585—32no—
48#1582—32no—
49SD16⋊C8—32no—
50C8×SD16—32no—
51#1572D311yescharacteristic, complemented
52#1490—31no—
53#1485—31no—
54#1453—31no—
55C22×D12C2×C411yescharacteristic
56C22×C4×D3—21no—
57C2×C3×C42C2311yescharacteristic
58C2×(C2×D3)⋊C4—21no—
59#926—21no—
60C2×C4⋊Dic3C2×C411yescharacteristic
61#924—21no—
62#924C2×C412yes—
63C2×#208—21no—
64C2×#204C2×C412yes—
65C2×#204C2×C412yes—
66C2×D3×C8—22no—
67C2×D3×C8—22no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12C2×C412yes—
71C4×D12C2312yes—
72#870—21no—
73#869C2×C411yescharacteristic
74#869C2×C411yescharacteristic
75C4×Dic6—21no—
76C4×Dic6C2312yes—
77C3×C4⋊C8C2×C412yes—
78C3×C4⋊C8C2312yes—
79C3×C4⋊C8C2312yes—
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8C2×C412yes—
83C3×#97—21no—
84C3×C4×C8—21no—
85C3×C4×C8C2311yescharacteristic
86C3×C4×C8C2311yescharacteristic
87#830D412yes—
88#821C2×C412yes—
89#821C2×C412yes—
90(C2×D3)⋊C8—22no—
91(C2×D3)⋊C8—22no—
92#817C2×C412yes—
93#817C2×C412yes—
94#816C2×C412yes—
95#816C2×C412yes—
96#815—22no—
97Dic3⋊C8—22no—
98Dic3⋊C8—22no—
99C8×Dic3—22no—
100C4⋊(C3⋊C8)D412yes—
101#803—21no—
102C4×C3⋊C8C2311yescharacteristic
103C4×C3⋊C8C2311yescharacteristic
104C4×C3⋊C8C2311yescharacteristic
105C4×C3⋊C8C2311yescharacteristic
106#533—31no—
107C2×C4×D4—31no—
108#483—32no—
109#476—32no—
110#461—32no—
111#450—32no—
112#446—32no—
113#446—62no—
114C8×Q8—32no—
115C4×SD16—32no—
116C4×SD16—32no—
117C8×D4—32no—
118C8×D4—32no—
119C8×D4—62no—
120#433—62no—
121#432C2×D311yescharacteristic
122#431—31no—
123#431C2×D311yescharacteristic
124#431C2×D311yescharacteristic
125C2×C22⋊C8—62no—
126C4×M16—31no—
127#335C2×D312yescomplemented
128#334C2×D312yescomplemented
129Q8⋊C8—32no—
130Q8⋊C8—32no—
131D4⋊C8—32no—
132D4⋊C8—32no—
133#322—31no—
134#322—31no—
135#322—31no—
136C82—31no—
137C23×D3—21no—
138C22×C3×C4C22×C411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×Dic3—21no—
142C2×D12—21no—
143C2×D12—24no—
144C2×D12C22×C412yes—
145C2×C4×D3—22no—
146C2×C4×D3—22no—
147C2×C4×D3—82no—
148C2×Dic6—21no—
149C2×Dic6C22×C412yes—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8C22×C412yes—
153C2×C3×C8C22×C412yes—
154C3×C42C22×C411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4111yescharacteristic
157C3×C42C2411yescharacteristic
158#217—22no—
159(C2×D3)⋊C4—22no—
160(C2×D3)⋊C4—22no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3C22×C412yes—
166C4⋊Dic3C22×C412yes—
167Dic3⋊C4—22no—
168Dic3⋊C4—22no—
169C4×Dic3—22no—
170#208—42no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8C2×D412yes—
174C2×C3⋊C8C2×D412yes—
175#204—24no—
176#204—24no—
177D3×C8—82no—
178D3×C8—82no—
179C22×D4—31no—
180C23×C4—61no—
181C2×SD16—32no—
182C2×SD16—32no—
183C2×M16—61no—
184C22×C8—62no—
185C22×C8—62no—
186C4⋊Q8—31no—
187C4⋊Q8—31no—
188#125—61no—
189C22⋊Q8—32no—
190C22⋊Q8—61no—
191C4×Q8—32no—
192C4×Q8—61no—
193C4×D4—32no—
194C4×D4—32no—
195C4×D4—61no—
196C4×D4—61no—
197#117—61no—
198C2×C4⋊C4—31no—
199C2×C22⋊C4—61no—
200C2×C42C22×D311yescharacteristic
201#106—32no—
202#106—32no—
203C4⋊C8—32no—
204C4⋊C8—62no—
205C4⋊C8—62no—
206C4⋊C8C4×D312yes—
207C4⋊C8C22×D312yes—
208C4⋊C8C22×D312yes—
209#103—32no—
210#103—32no—
211#102—32no—
212#102—32no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—32no—
216C22⋊C8—62no—
217C22⋊C8—62no—
218C22⋊C8C4×D312yes—
219#97—21no—
220#97—62no—
221#97—61no—
222C4×C8—21no—
223C4×C8—31no—
224C4×C8—31no—
225C4×C8—31no—
226C4×C8—31no—
227C4×C8—62no—
228C4×C8C22×D311yescharacteristic
229C4×C8C22×D311yescharacteristic
230C23×C3C4×D411yescharacteristic
231C22×D3—22no—
232C22×D3—24no—
233C22×D3—81no—
234C22×D3—81no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—41no—
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4C4×D411yescharacteristic
241C2×C3×C4C23×C411yescharacteristic
242C2×C3×C4C22×D411yescharacteristic
243C2×C3×C4C2×#4111yescharacteristic
244C2×Dic3—22no—
245C2×Dic3—22no—
246C2×Dic3—22no—
247C2×Dic3—81no—
248D12—24no—
249D12—22no—
250D12#13114yes—
251C4×D3—42no—
252C4×D3—42no—
253C4×D3—82no—
254Dic6—22no—
255Dic6—22no—
256Dic6—81no—
257C3×C8—22no—
258C3×C8—22no—
259C3×C8—42no—
260C3×C8—42no—
261C3⋊C8—22no—
262C3⋊C8—22no—
263C3⋊C8—22no—
264C3⋊C8—41no—
265C24—61no—
266C2×Q8—32no—
267C2×Q8—61no—
268C2×D4—32no—
269C2×D4—64no—
270C2×D4—61no—
271C22×C4—61no—
272C22×C4—62no—
273C22×C4—62no—
274C22×C4—242no—
275C22×C4C2×C4×D311yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4#23911yescharacteristic
278SD16—64no—
279SD16—64no—
280M16—122no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—32no—
284C2×C8—32no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—242no—
288C2×C8—242no—
289C2×C8C2×C4×D312yes—
290C2×C8C2×C4×D312yes—
291C4⋊C4—32no—
292C4⋊C4—32no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—62no—
297C4⋊C4—61no—
298C4⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C42—62no—
303C42C2×C4×D311yescharacteristic
304C42D3×D411yescharacteristic
305C42#23911yescharacteristic
306C42C23×D311yescharacteristic
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3C2×C4×D411yescharacteristic
311C2×D3—24no—
312C2×D3—42no—
313C2×D3—42no—
314C2×D3—84no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4C2×#13112yes—
320C3×C4#57712yes—
321Dic3—42no—
322Dic3—42no—
323Dic3—42no—
324C23—64no—
325C23—62no—
326C23—241no—
327C23—241no—
328C23#89311yescharacteristic
329Q8—62no—
330Q8—62no—
331Q8—241no—
332D4—34no—
333D4—64no—
334D4—62no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—242no—
344C2×C4—241no—
345C2×C4#89311yescharacteristic
346C2×C4#89311yescharacteristic
347C2×C4#89311yescharacteristic
348C2×C4C22×C4×D311yescharacteristic
349C2×C4C2×D3×D411yescharacteristic
350C2×C4C2×#23911yescharacteristic
351C8—22no—
352C8—22no—
353C8—42no—
354C8—42no—
355C8—62no—
356C8—62no—
357C8—62no—
358C8—121no—
359C2×C3—41no—
360C2×C3#292911yescharacteristic
361C2×C3#294911yescharacteristic
362C2×C3#295611yescharacteristic
363D3—44no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—64no—
368C22—122no—
369C22—122no—
370C22—244no—
371C22C2×#89311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—122no—
377C4—122no—
378C4—122no—
379C4D3×#13112yes—
380C4#577012yes—
381C3#1792411yescharacteristic, complemented, Sylow
382C2—41no—
383C2—124no—
384C2#7907311yescharacteristic
385C2#7916811yescharacteristic
386C2#7925811yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1572C2
socleC22×C3C2×C4×D4
prime radicalC22×D12C2×C4
cosocleC3×C42C24
hypercenter#1572D3
hypocenterC3#17924
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1572

Complements

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

NH ≅ G/Ntype
1. #79316C2semidirect
4. C3×#1572C2semidirect
6. #70769C2semidirect
7. #70743C2semidirect
8. #68561C2semidirect
9. #68558C2semidirect
10. #68519C2semidirect
11. #68516C2semidirect
24. #4831C22semidirect
25. #4830C22semidirect
34. C3×#335C22semidirect
35. C3×#334C22semidirect
36. #4484C22semidirect
37. #4484C22semidirect
40. #4455C22semidirect
41. #4454C22semidirect
42. #4454C22semidirect
43. C8×C3⋊C8C22semidirect
51. #1572D3semidirect
127. #335C2×D3semidirect
128. #334C2×D3semidirect
381. C3#17924semidirect

Maximal subgroups

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

subgroupindex
#793162
#792982
#792862
C3×#15722
#707992
#707692
#707432
#685612
#685582
#685192
#685162
#179243