Finite Group Databasebeta

← #10692029

Subgroups — #10692029

Order 768.

480 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83212C211yesmaximal, characteristic
2#79013C211yesmaximal, characteristic
3C3×#1987C211yesmaximal, characteristic, complemented
4#71774C211yesmaximal, characteristic, complemented
5#69494C211yesmaximal, characteristic
6#67705C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9315—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C4×#235C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13C8×#67—22no—
14#5094C412yescomplemented
15#5091—21no—
16#5087C412yescomplemented
17#5084—21no—
18#4966—21no—
19#4964—21no—
20C2×#805—21no—
21#4925C2211yescharacteristic
22#4924C2211yescharacteristic
23#4924C2211yescharacteristic
24#4834—21no—
25#4833—21no—
26D3×C4⋊C8—21no—
27#4830—21no—
28C3×#340—22no—
29#4546—21no—
30#4546—21no—
31#4523—21no—
32#3079—31no—
33#2988—31no—
34#1987D311yescharacteristic, complemented
35#1806—31no—
36#1548—31no—
37#1303—31no—
38#1292—31no—
39C22×C3×Q8—21no—
40C2×#235D411yescharacteristic
41C22×C3×C8D412yes—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8C2×C412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C4×#67—21no—
52C4×#67—21no—
53#926D411yescharacteristic
54C2×#208—21no—
55C2×#208—21no—
56C2×D3×C8—22no—
57C4×D12C2×C411yescharacteristic
58C4×D12D411yescharacteristic
59#872—21no—
60C42×D3—21no—
61C4×Dic6C2×C411yescharacteristic
62C4×Dic6D411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×#95—21no—
66#830—21no—
67(C2×D3)⋊C8—22no—
68Dic3⋊C8—22no—
69C8×Dic3—22no—
70#810—22no—
71#809—22no—
72#808—22no—
73#807—22no—
74#805—42no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)C2×C412yes—
79C4⋊(C3⋊C8)D411yescharacteristic
80C4⋊(C3⋊C8)D411yescharacteristic
81C4×C3⋊C8—21no—
82C2×C4⋊Q8C2×D311yescharacteristic
83C4×#41—31no—
84#477—61no—
85#476—61no—
86#475—32no—
87#474—32no—
88C8×Q8—61no—
89C8×D4—61no—
90C8×D4—62no—
91#431—31no—
92#431—61no—
93#428—61no—
94#423—31no—
95#423—31no—
96C2×C4⋊C8—61no—
97C2×C4⋊C8C2×D311yescharacteristic
98C2×#104—61no—
99#417—61no—
100C4×C4⋊C4C2×D311yescharacteristic
101#340—22no—
102#339—61no—
103#333—61no—
104#333—61no—
105C2×C3×Q8—22no—
106C2×C3×Q8—41no—
107C2×C3×Q8—82no—
108C22×C3×C4—41no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C2×#67—21no—
114#235—21no—
115#235—21no—
116#235SD1612yes—
117#235Q1612yes—
118C2×D12C22⋊C411yescharacteristic
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C2×C4×D3—21no—
122C2×Dic6C22⋊C411yescharacteristic
123C2×C3×C8—22no—
124C2×C3×C8—42no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C42—41no—
132C3×C42—41no—
133C3×C42C22×C411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137#217—21no—
138(C2×D3)⋊C4—21no—
139(C2×D3)⋊C4—21no—
140C4⋊Dic3C22⋊C411yescharacteristic
141C4⋊Dic3C22⋊C411yescharacteristic
142Dic3⋊C4—21no—
143Dic3⋊C4—21no—
144C4×Dic3—21no—
145C4×Dic3—21no—
146C4×Dic3—21no—
147#208—41no—
148#208—41no—
149#208—41no—
150#208—41no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—41no—
155D3×C8—82no—
156C2×#41—31no—
157C22×Q8—21no—
158C2×M16—61no—
159C2×M16—61no—
160C22×C8—62no—
161C22×C8#6712yes—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8C4×D312yes—
166C4×Q8—31no—
167C4×Q8—31no—
168C4×D4—31no—
169C4×D4—31no—
170C4×D4—61no—
171C4×D4—61no—
172#117—31no—
173#117—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C42—21no—
178C2×C42—61no—
179C2×C42C22×D311yescharacteristic
180#107—62no—
181#106—62no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—32no—
185C4⋊C8—31no—
186C4⋊C8—31no—
187C4⋊C8—62no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191#104—122no—
192#103—62no—
193#102—62no—
194C22⋊C8—62no—
195C22⋊C8—61no—
196C4×C8—62no—
197C4×C8—61no—
198#95—21no—
199C23×C3#12111yescharacteristic
200C22×D3—21no—
201C3×Q8—82no—
202C3×Q8—82no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4—81no—
209C2×C3×C4—82no—
210C2×C3×C4—81no—
211C2×C3×C4C2×C22⋊C411yescharacteristic
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12311yescharacteristic
217C2×C3×C4C2×SD1611yescharacteristic
218C2×C3×C4C2×Q1611yescharacteristic
219C2×C3×C4#13511yescharacteristic
220C2×C3×C4#13511yescharacteristic
221#67—22no—
222#67—22no—
223C2×Dic3—21no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226D12—21no—
227D12#10312yes—
228C4×D3—22no—
229C4×D3—22no—
230C4×D3—41no—
231C4×D3—41no—
232C4×D3—81no—
233Dic6—21no—
234Dic6#10312yes—
235C3×C8—42no—
236C3⋊C8—42no—
237C3⋊C8—41no—
238C3⋊C8—41no—
239C3⋊C8—81no—
240#41—32no—
241#41—32no—
242#41—61no—
243#41—61no—
244C2×Q8—22no—
245C2×Q8—31no—
246C2×Q8—41no—
247C2×Q8—82no—
248C2×D4—31no—
249C2×D4—61no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4D3×D411yescharacteristic
256C22×C4#23711yescharacteristic
257C22×C4C2×#6711yescharacteristic
258M16—121no—
259M16—121no—
260M16—121no—
261M16—121no—
262C2×C8—22no—
263C2×C8—42no—
264C2×C8—62no—
265C2×C8—61no—
266C2×C8—61no—
267C2×C8—121no—
268C2×C8—242no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—31no—
273C4⋊C4—31no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C42—41no—
283C42—41no—
284C42—61no—
285C42—61no—
286C42—61no—
287C42C2×C4×D311yescharacteristic
288C42#23511yescharacteristic
289C42D3×D411yescharacteristic
290C42D3×D411yescharacteristic
291C22×C3—21no—
292C22×C3—21no—
293C22×C3#38611yescharacteristic
294C22×C3#46511yescharacteristic
295C22×C3#46711yescharacteristic
296C2×D3—22no—
297C2×D3—41no—
298C2×D3—41no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—82no—
303C3×C4—82no—
304C3×C4C2×#10311yescharacteristic
305C3×C4#41611yescharacteristic
306C3×C4#45511yescharacteristic
307C3×C4#45611yescharacteristic
308C3×C4#46511yescharacteristic
309C3×C4#46711yescharacteristic
310Dic3—22no—
311Dic3—41no—
312Dic3—41no—
313C23—61no—
314C23#93811yescharacteristic
315Q8—32no—
316Q8—61no—
317Q8—82no—
318Q8—82no—
319D4—32no—
320D4—62no—
321D4—62no—
322D4—61no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—61no—
329C2×C4—62no—
330C2×C4—62no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—241no—
339C2×C4D3×C22⋊C411yescharacteristic
340C2×C4#67⋊C411yescharacteristic
341C2×C4#88311yescharacteristic
342C2×C4#88411yescharacteristic
343C2×C4D3×SD1611yescharacteristic
344C2×C4#91611yescharacteristic
345C2×C4D3×Q1611yescharacteristic
346C2×C4#91911yescharacteristic
347C2×C4C4×#6711yescharacteristic
348C2×C4#93711yescharacteristic
349C8—42no—
350C8—122no—
351C8—121no—
352C8—121no—
353C8—241no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3#188211yescharacteristic
357C2×C3#189211yescharacteristic
358C2×C3#190311yescharacteristic
359D3—42no—
360C22—21no—
361C22—21no—
362C22—62no—
363C22—121no—
364C22—121no—
365C22#495611yescharacteristic
366C22#517111yescharacteristic
367C22#518811yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—62no—
372C4—82no—
373C4—82no—
374C4—121no—
375C4—121no—
376C4D3×#10311yescharacteristic
377C4#480411yescharacteristic
378C4#480711yescharacteristic
379C4#481311yescharacteristic
380C4#506111yescharacteristic
381C4#509511yescharacteristic
382C3#931511yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—122no—
386C2#6969311yescharacteristic
387C2#7027011yescharacteristic
388C2#7032911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1987C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1987D3
hypocenterC3#9315
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
3. C3×#1987C2semidirect
4. #71774C2semidirect
6. #67705C2semidirect
7. #67620C2semidirect
14. #5094C4semidirect
16. #5087C4semidirect
34. #1987D3semidirect
382. C3#9315semidirect

Maximal subgroups

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

subgroupindex
#832122
#790132
C3×#19872
#717742
#694942
#677052
#676202
#93153