Finite Group Databasebeta

← #10690041

Subgroups — #10690041

Order 768.

462 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3104C211yesmaximal, characteristic
2Q8×#208C211yesmaximal, characteristic
3#71841C211yesmaximal, characteristic
4#71826C211yesmaximal, characteristic
5C4×#835C211yesmaximal, characteristic
6#67746C211yesmaximal, characteristic
7#67739C211yesmaximal, characteristic
8#8990—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C3×#484—21no—
12C3×#477C411yescharacteristic, complemented
13C3×#477C411yescharacteristic, complemented
14C3×#462C411yescharacteristic, complemented
15C3×#462C411yescharacteristic, complemented
16C3×#451—21no—
17C3×C4×Q16—21no—
18C3×C4×Q16—21no—
19C3×#431C2211yescharacteristic
20C2×#835—21no—
21#5024—21no—
22#5024—22no—
23Q8×C3⋊C8—21no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4924—21no—
26#4924C2211yescharacteristic
27C4×#208—21no—
28C4×#208C2211yescharacteristic
29C2×C4×C3⋊C8C2211yescharacteristic
30#4535—22no—
31#4532—21no—
32#4532—21no—
33#4529—21no—
34#3104D311yescharacteristic
35Q8×M16—31no—
36#1986—31no—
37C4×#103—31no—
38#1493—31no—
39#1303—31no—
40#1278—31no—
41C22×C3×Q8D411yescharacteristic
42C2×C3×Q16—21no—
43C2×C3×Q16—21no—
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8C2×C411yescharacteristic
54C3×C4×Q8C2×C411yescharacteristic
55C3×C4×Q8D411yescharacteristic
56C3×C4×Q8D411yescharacteristic
57C3×#117—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×C4⋊Dic3—21no—
63C2×C4×Dic3—21no—
64C2×#208—21no—
65C2×#208D411yescharacteristic
66C2×#208D411yescharacteristic
67C22×C3⋊C8—21no—
68C3×#107—21no—
69C3×#107—21no—
70C3×C4⋊C8C2×C411yescharacteristic
71C3×C4⋊C8C2×C411yescharacteristic
72C3×#103—21no—
73C3×#103—21no—
74C3×#103—21no—
75C3×#103—21no—
76C3×C22⋊C8—21no—
77C3×C4×C8—21no—
78#835—24no—
79C3⋊#95—21no—
80C4⋊(C3⋊C8)—22no—
81C4⋊(C3⋊C8)—22no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84#803—22no—
85#803—21no—
86C4×C3⋊C8—22no—
87C4×C3⋊C8—21no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8—21no—
90#533C2×D311yescharacteristic
91C2×C4×Q8C2×D311yescharacteristic
92#484—21no—
93#477Dic311yescharacteristic, complemented
94#477Dic311yescharacteristic, complemented
95#462Dic311yescharacteristic, complemented
96#462Dic311yescharacteristic, complemented
97#451—21no—
98#446—62no—
99#446—61no—
100C8×Q8—61no—
101C4×Q16—21no—
102C4×Q16—21no—
103#431C2×D311yescharacteristic
104#423—31no—
105#423—61no—
106C2×#103—61no—
107C4×M16—31no—
108C4×M16—61no—
109C2×C4×C8—31no—
110C4×C4⋊C4—31no—
111#340—61no—
112#339—61no—
113#339—61no—
114Q8⋊C8—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—22no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—41no—
120C2×C3×Q8C22⋊C411yescharacteristic
121C2×C3×Q8C22⋊C411yescharacteristic
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C22×Dic3—41no—
128C3×Q16—42no—
129C3×Q16—42no—
130C2×C3×C8—21no—
131C2×C3×C8—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4C22⋊C411yescharacteristic
144C3×C4⋊C4C22⋊C411yescharacteristic
145C3×C22⋊C4—41no—
146C3×C42—21no—
147C3×C42—21no—
148C3×C42—21no—
149C3×C42—21no—
150C3×C42—41no—
151C3×C42C22×C411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42#4111yescharacteristic
155C4⋊Dic3—22no—
156C4⋊Dic3—41no—
157C4×Dic3—41no—
158C4×Dic3—41no—
159#208—21no—
160#208—21no—
161#208—21no—
162#208—41no—
163#208—41no—
164#208Q1612yescomplemented
165C2×C3⋊C8—21no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—41no—
170C2×C3⋊C8—41no—
171C22×Q8#6711yescharacteristic
172C2×Q16—21no—
173C2×Q16—21no—
174C2×M16—31no—
175C2×M16—31no—
176C2×M16—61no—
177C22×C8—61no—
178C4⋊Q8C2×Dic311yescharacteristic
179C4⋊Q8C2×Dic311yescharacteristic
180#125—21no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8C2×Dic311yescharacteristic
188C4×Q8C2×Dic311yescharacteristic
189C4×Q8#6711yescharacteristic
190C4×Q8#6711yescharacteristic
191#117—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4#6711yescharacteristic
195C2×C42—21no—
196C2×C42—61no—
197C2×C42C22×D311yescharacteristic
198#107—21no—
199#107—21no—
200C4⋊C8—32no—
201C4⋊C8—62no—
202C4⋊C8—62no—
203C4⋊C8—61no—
204C4⋊C8C2×Dic311yescharacteristic
205C4⋊C8C2×Dic311yescharacteristic
206#103—21no—
207#103—21no—
208#103—21no—
209#103—21no—
210#103—64no—
211C22⋊C8—21no—
212#97—62no—
213#97—61no—
214C4×C8—21no—
215C4×C8—62no—
216C4×C8—61no—
217C4×C8—61no—
218C4×C8—61no—
219#95—61no—
220C23×C3#12111yescharacteristic
221C3×Q8—21no—
222C3×Q8—21no—
223C3×Q8—21no—
224C3×Q8—42no—
225C3×Q8—42no—
226C3×Q8—41no—
227C3×Q8—41no—
228C3×Q8#10412yes—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—81no—
240C2×C3×C4C2×C22⋊C411yescharacteristic
241C2×C3×C4C4×D411yescharacteristic
242C2×C3×C4C4×D411yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12311yescharacteristic
246C2×C3×C4C2×Q1611yescharacteristic
247C2×C3×C4#13511yescharacteristic
248C2×C3×C4#13711yescharacteristic
249C2×C3×C4#13711yescharacteristic
250C2×Dic3—42no—
251C2×Dic3—81no—
252C2×Dic3—81no—
253C3×C8—41no—
254C3×C8—41no—
255C3⋊C8—22no—
256C3⋊C8—42no—
257C3⋊C8—41no—
258C3⋊C8—42no—
259C2×Q8—21no—
260C2×Q8—22no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—41no—
264C2×Q8#21711yescharacteristic
265C2×Q8#21711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—121no—
269C22×C4D3×D411yescharacteristic
270C22×C4#23711yescharacteristic
271C22×C4C2×#6711yescharacteristic
272Q16—42no—
273Q16—42no—
274M16—32no—
275M16—61no—
276M16—61no—
277M16—61no—
278M16—121no—
279M16—121no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—61no—
283C2×C8—62no—
284C2×C8—62no—
285C2×C8—61no—
286C2×C8—121no—
287C2×C8—121no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—62no—
300C4⋊C4—121no—
301C4⋊C4#21711yescharacteristic
302C4⋊C4#21711yescharacteristic
303C22⋊C4—41no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—41no—
309C42—121no—
310C42—121no—
311C42D3×D411yescharacteristic
312C42#23711yescharacteristic
313C42C22×Dic311yescharacteristic
314C42C2×#6711yescharacteristic
315C22×C3—21no—
316C22×C3—21no—
317C22×C3#38611yescharacteristic
318C22×C3#46211yescharacteristic
319C22×C3#46411yescharacteristic
320C3×C4—21no—
321C3×C4—22no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—41no—
325C3×C4—41no—
326C3×C4—41no—
327C3×C4—81no—
328C3×C4—81no—
329C3×C4C2×#10411yescharacteristic
330C3×C4#42111yescharacteristic
331C3×C4C4×Q1611yescharacteristic
332C3×C4#44111yescharacteristic
333C3×C4#45111yescharacteristic
334C3×C4#45211yescharacteristic
335Dic3—82no—
336C23#93911yescharacteristic
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—41no—
343Q8—41no—
344Q8#83712yes—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—122no—
357C2×C4—241no—
358C2×C4—241no—
359C2×C4D3×Q1611yescharacteristic
360C2×C4#91811yescharacteristic
361C2×C4#91811yescharacteristic
362C2×C4#91911yescharacteristic
363C2×C4D4×Dic311yescharacteristic
364C2×C4D4×Dic311yescharacteristic
365C2×C4#93511yescharacteristic
366C2×C4#93711yescharacteristic
367C2×C4#93911yescharacteristic
368C2×C4C2×#21711yescharacteristic
369C8—41no—
370C8—41no—
371C8—62no—
372C8—122no—
373C8—121no—
374C8—122no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3#187111yescharacteristic
378C2×C3#187811yescharacteristic
379C2×C3#188111yescharacteristic
380C22—21no—
381C22—21no—
382C22#518211yescharacteristic
383C22#518511yescharacteristic
384C22#522011yescharacteristic
385C4—21no—
386C4—22no—
387C4—21no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—81no—
393C4—81no—
394C4—242no—
395C4Dic3×Q1611yescharacteristic
396C4#518411yescharacteristic
397C4#518611yescharacteristic
398C4#518711yescharacteristic
399C4C2×#83711yescharacteristic
400C4#523611yescharacteristic
401C3#899011yescharacteristic, complemented, Sylow
402C2—21no—
403C2—21no—
404C2#7171311yescharacteristic
405C2#7206511yescharacteristic
406C2#7211611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3104C2
socleC22×C3#386
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#3104D3
hypocenterC3#8990
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
12. C3×#477C4semidirect
13. C3×#477C4semidirect
14. C3×#462C4semidirect
15. C3×#462C4semidirect
93. #477Dic3semidirect
94. #477Dic3semidirect
95. #462Dic3semidirect
96. #462Dic3semidirect
164. #208Q16semidirect
401. C3#8990semidirect

Maximal subgroups

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

subgroupindex
C3×#31042
Q8×#2082
#718412
#718262
C4×#8352
#677462
#677392
#89903