Finite Group Databasebeta

← #10692039

Subgroups — #10692039

Order 768.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79060C211yesmaximal, characteristic
2#79013C211yesmaximal, characteristic
3#70529C211yesmaximal, characteristic
4#70508C211yesmaximal, characteristic, complemented
5#70456C211yesmaximal, characteristic, complemented
6C3×#1292C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#9317—31nomaximal, Sylow
9C4×#235C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×#825—21no—
15#5133—21no—
16#5121—21no—
17C8×#67—22no—
18C2×#816—22no—
19C4×C4⋊Dic3C2211yescharacteristic
20#4925C2211yescharacteristic
21#4851—21no—
22#4850C412yescomplemented
23#4846—21no—
24#4842C412yescomplemented
25#4834—21no—
26#4833—21no—
27D3×C4⋊C8—21no—
28#4830—21no—
29#4555—21no—
30#4486—21no—
31#4486—21no—
32#3079—31no—
33#2988—31no—
34#1981—31no—
35#1806—31no—
36#1549—31no—
37#1303—31no—
38#1292D311yescharacteristic, complemented
39C2×#235D411yescharacteristic
40C22×Dic6—21no—
41C2×C3×M16—21no—
42C2×C3×M16—21no—
43C22×C3×C8D412yes—
44C2×C3×C42C2311yescharacteristic
45C4×#67—21no—
46C4×#67—21no—
47#926D411yescharacteristic
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4×Dic3—21no—
52C2×D3×C8—22no—
53C4×D12C2×C411yescharacteristic
54C4×D12D411yescharacteristic
55#872—21no—
56C42×D3—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869C2×C412yes—
61C4×Dic6C2×C411yescharacteristic
62C4×Dic6D411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8C2×C412yes—
69C3⋊#95—21no—
70#830—21no—
71#825—42no—
72#821—22no—
73(C2×D3)⋊C8—22no—
74#818—22no—
75#817—22no—
76#816—22no—
77#816—44no—
78Dic3⋊C8—22no—
79C8×Dic3—22no—
80C4⋊(C3⋊C8)D411yescharacteristic
81C4⋊(C3⋊C8)D411yescharacteristic
82C4×C3⋊C8—21no—
83C2×C4⋊Q8—31no—
84C4×#41—31no—
85#477—32no—
86#476—32no—
87#475—61no—
88#474—61no—
89C8×Q8—61no—
90C8×D4—61no—
91C8×D4—62no—
92#431—31no—
93#431—61no—
94#428—61no—
95#423C2×D311yescharacteristic
96#423C2×D311yescharacteristic
97C2×C4⋊C8—61no—
98C2×C4⋊C8C2×D311yescharacteristic
99C2×#104—61no—
100#417—61no—
101C2×#103—62no—
102C4×C4⋊C4—31no—
103#339—61no—
104#332—61no—
105#332—61no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×#67—21no—
110C22×Dic3—41no—
111C22×Dic3—41no—
112#235—21no—
113#235—21no—
114#235SD1612yes—
115#235Q1612yes—
116C2×D12C22⋊C411yescharacteristic
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×Dic6—22no—
121C2×Dic6—42no—
122C2×Dic6—42no—
123C2×Dic6—41no—
124C2×Dic6C22⋊C411yescharacteristic
125C3×M16—41no—
126C3×M16—41no—
127C3×M16—41no—
128C3×M16—41no—
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C2×C3×C8—42no—
132C2×C3×C8—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⋊Dic3—22no—
141C4⋊Dic3—22no—
142C4⋊Dic3—22no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4⋊Dic3C22⋊C411yescharacteristic
147C4⋊Dic3C22⋊C411yescharacteristic
148Dic3⋊C4—21no—
149Dic3⋊C4—21no—
150C4×Dic3—21no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—21no—
157D3×C8—82no—
158C2×#41—31no—
159C22×Q8—61no—
160C2×M16—21no—
161C2×M16—21no—
162C22×C8—62no—
163C22×C8D1212yes—
164C4⋊Q8—32no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4×Q8—31no—
169C4×Q8—31no—
170C4×D4—31no—
171C4×D4—31no—
172C4×D4—61no—
173C4×D4—61no—
174#117—31no—
175#117—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C42—61no—
180C2×C42—61no—
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#106—62no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—31no—
190C4⋊C8—31no—
191C4⋊C8—62no—
192C4⋊C8C4×D312yes—
193#104—122no—
194#103—62no—
195#103—124no—
196#102—62no—
197C22⋊C8—62no—
198C22⋊C8—61no—
199C4×C8—62no—
200C4×C8—61no—
201#95—61no—
202C23×C3#12111yescharacteristic
203C22×D3—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—41no—
207C2×C3×C4C2×C22⋊C411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12311yescharacteristic
213C2×C3×C4C2×SD1611yescharacteristic
214C2×C3×C4C2×Q1611yescharacteristic
215C2×C3×C4#13511yescharacteristic
216C2×C3×C4#13511yescharacteristic
217#67—22no—
218#67—22no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—82no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227D12—21no—
228D12#10312yescomplemented
229C4×D3—22no—
230C4×D3—22no—
231C4×D3—41no—
232C4×D3—41no—
233C4×D3—81no—
234Dic6—21no—
235Dic6—44no—
236Dic6—81no—
237Dic6—81no—
238Dic6#10312yescomplemented
239C3×C8—42no—
240C3×C8—42no—
241C3×C8—81no—
242C3⋊C8—41no—
243C3⋊C8—41no—
244#41—32no—
245#41—32no—
246#41—61no—
247#41—61no—
248C2×Q8—31no—
249C2×Q8—62no—
250C2×Q8—122no—
251C2×Q8—122no—
252C2×Q8—121no—
253C2×D4—31no—
254C2×D4—61no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—121no—
259C22×C4—121no—
260C22×C4C2×D1211yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4#23911yescharacteristic
263M16—41no—
264M16—41no—
265M16—41no—
266M16—41no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—61no—
272C2×C8—61no—
273C2×C8—242no—
274C4⋊C4—31no—
275C4⋊C4—31no—
276C4⋊C4—62no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—62no—
280C4⋊C4—62no—
281C4⋊C4—121no—
282C4⋊C4—121no—
283C4⋊C4—121no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C42—61no—
288C42—61no—
289C42—61no—
290C42—121no—
291C42—121no—
292C42C2×C4×D311yescharacteristic
293C42#23511yescharacteristic
294C42D3×D411yescharacteristic
295C42D3×D411yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3#38611yescharacteristic
299C22×C3#46511yescharacteristic
300C22×C3#46711yescharacteristic
301C2×D3—22no—
302C2×D3—41no—
303C2×D3—41no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4C2×#10311yescharacteristic
308C3×C4#41611yescharacteristic
309C3×C4#45311yescharacteristic
310C3×C4#45411yescharacteristic
311C3×C4#46511yescharacteristic
312C3×C4#46711yescharacteristic
313Dic3—22no—
314Dic3—41no—
315Dic3—41no—
316Dic3—82no—
317Dic3—82no—
318C23—61no—
319C23#89511yescharacteristic
320Q8—32no—
321Q8—61no—
322Q8—124no—
323Q8—241no—
324Q8—241no—
325D4—32no—
326D4—62no—
327D4—62no—
328D4—61no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—61no—
333C2×C4—62no—
334C2×C4—62no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—241no—
342C2×C4—242no—
343C2×C4—241no—
344C2×C4—241no—
345C2×C4C4×D1211yescharacteristic
346C2×C4D3×C22⋊C411yescharacteristic
347C2×C4#88211yescharacteristic
348C2×C4#88411yescharacteristic
349C2×C4#89311yescharacteristic
350C2×C4#89411yescharacteristic
351C2×C4#91211yescharacteristic
352C2×C4D3×SD1611yescharacteristic
353C2×C4#91611yescharacteristic
354C2×C4D3×Q1611yescharacteristic
355C8—42no—
356C8—42no—
357C8—81no—
358C8—121no—
359C8—121no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3#188311yescharacteristic
363C2×C3#189211yescharacteristic
364C2×C3#190211yescharacteristic
365D3—42no—
366C22—21no—
367C22—21no—
368C22—62no—
369C22—121no—
370C22—121no—
371C22#466511yescharacteristic
372C22#486411yescharacteristic
373C22#488711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—62no—
378C4—121no—
379C4—121no—
380C4—242no—
381C4—242no—
382C4#474711yescharacteristic
383C4#475311yescharacteristic
384C4#477211yescharacteristic
385C4#477811yescharacteristic
386C4D3×#10311yescharacteristic
387C4#481311yescharacteristic
388C3#931711yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—122no—
392C2#6872511yescharacteristic
393C2#6888211yescharacteristic
394C2#6900511yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1292C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1292D3
hypocenterC3#9317
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
4. #70508C2semidirect
5. #70456C2semidirect
6. C3×#1292C2semidirect
7. #67715C2semidirect
22. #4850C4semidirect
24. #4842C4semidirect
38. #1292D3semidirect
228. D12#103semidirect
238. Dic6#103semidirect
388. C3#9317semidirect

Maximal subgroups

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

subgroupindex
#790602
#790132
#705292
#705082
#704562
C3×#12922
#677152
#93173