Finite Group Databasebeta

← #10690063

Subgroups — #10690063

Order 768.

476 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83349C211yesmaximal, characteristic
2#79012C211yesmaximal, characteristic
3C3×#1481C211yesmaximal, characteristic, complemented
4#71826C211yesmaximal, characteristic, complemented
5#70754C211yesmaximal, characteristic
6#67745C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#8993—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#901—21no—
11C4×#235C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5102—21no—
15#5094C411yescharacteristic
16#5091C411yescharacteristic
17#5087C411yescharacteristic
18#5084C411yescharacteristic
19#5066—21no—
20#5063—21no—
21#5002—21no—
22#4998—21no—
23#4924C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26#4839—22no—
27#4837—21no—
28D3×C4⋊C8—21no—
29C3×Q8⋊C8—21no—
30C3×Q8⋊C8—21no—
31#4535—22no—
32#4484—21no—
33#4483—21no—
34#3076—31no—
35#2906—31no—
36#1493—31no—
37#1481D311yescharacteristic, complemented
38#1476—31no—
39#1301—31no—
40#1277—31no—
41C2×#235D411yescharacteristic
42C22×C3×C8—21no—
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C42C2311yescharacteristic
52C4×#67—21no—
53C4×#67—21no—
54#926D411yescharacteristic
55C2×#208—21no—
56C22×C3⋊C8—21no—
57C22×C3⋊C8—21no—
58#901—42no—
59#901—42no—
60C2×ΓL(1,𝔽25)—22no—
61C2×D3×C8—22no—
62C4×D12C2×C411yescharacteristic
63C4×D12D411yescharacteristic
64#872—21no—
65C42×D3—21no—
66C4×Dic6C2×C411yescharacteristic
67C4×Dic6D411yescharacteristic
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8D411yescharacteristic
71C3×C4⋊C8D411yescharacteristic
72C3×C22⋊C8—21no—
73C3×C4×C8—21no—
74Dic3⋊C8—22no—
75Dic3⋊C8—22no—
76#810—21no—
77#810—21no—
78#809—21no—
79#809—21no—
80#808—21no—
81#808—21no—
82#807—21no—
83#807—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)C2×C411yescharacteristic
87C4⋊(C3⋊C8)C2×C411yescharacteristic
88C4⋊(C3⋊C8)D412yes—
89C4×C3⋊C8—21no—
90C4×C3⋊C8—21no—
91C2×#131—61no—
92#533C2×D311yescharacteristic
93C4×#41—31no—
94#479—61no—
95#478—61no—
96#477—31no—
97#476—31no—
98#475—31no—
99#474—31no—
100#431C2×D311yescharacteristic
101#427—61no—
102#424—62no—
103#424—61no—
104#423—31no—
105#423—61no—
106C2×C4⋊C8—31no—
107C2×C4⋊C8—61no—
108C2×C4⋊C8C2×D311yescharacteristic
109#416—61no—
110C2×C4×C8—31no—
111Q8⋊C8—21no—
112Q8⋊C8—21no—
113Q8⋊C8—61no—
114Q8⋊C8—62no—
115D4⋊C8—61no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×#67—21no—
122#235—21no—
123#235—21no—
124#235SD1612yes—
125#235Q1612yes—
126C2×D12C22⋊C411yescharacteristic
127C2×C4×D3—21no—
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×Dic6C22⋊C411yescharacteristic
131C2×C3×C8—21no—
132C2×C3×C8—21no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42—41no—
144C3×C42C22×C411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148#217—21no—
149(C2×D3)⋊C4—21no—
150(C2×D3)⋊C4—21no—
151C4⋊Dic3C22⋊C411yescharacteristic
152C4⋊Dic3C22⋊C411yescharacteristic
153Dic3⋊C4—21no—
154Dic3⋊C4—21no—
155C4×Dic3—21no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158#208—41no—
159#208—41no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—41no—
168ΓL(1,𝔽25)—44no—
169D3×C8—44no—
170C2×#41—31no—
171#131—122no—
172#131—122no—
173C2×M16—62no—
174C2×M16—61no—
175C22×C8—21no—
176C22×C8—62no—
177C22×C8—61no—
178C22×C8—61no—
179C4⋊Q8C4×D311yescharacteristic
180C4⋊Q8C4×D311yescharacteristic
181#125—21no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—31no—
187C4×Q8—31no—
188C4×D4—31no—
189C4×D4—31no—
190C4×D4—61no—
191C4×D4—61no—
192#117—21no—
193#117—31no—
194#117—61no—
195C2×C42—61no—
196C2×C42C22×D311yescharacteristic
197#107—61no—
198#107—61no—
199#106—61no—
200#106—61no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—31no—
204C4⋊C8—31no—
205C4⋊C8—32no—
206C4⋊C8—62no—
207C4⋊C8—62no—
208C4⋊C8—61no—
209C4⋊C8—61no—
210C4⋊C8#6711yescharacteristic
211C4⋊C8#6711yescharacteristic
212#103—61no—
213#103—61no—
214#102—61no—
215#102—61no—
216C22⋊C8—21no—
217C4×C8—21no—
218C4×C8—61no—
219C4×C8—61no—
220C23×C3#12311yescharacteristic
221C22×D3—21no—
222C3×Q8—42no—
223C3×Q8—41no—
224C3×Q8—41no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4C2×C22⋊C411yescharacteristic
231C2×C3×C4C4×D411yescharacteristic
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4C2×SD1611yescharacteristic
237C2×C3×C4C2×Q1611yescharacteristic
238C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
239C2×C3×C4#13711yescharacteristic
240#67—22no—
241#67—22no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245D12—21no—
246D12#10312yes—
247C4×D3—22no—
248C4×D3—22no—
249C4×D3—41no—
250C4×D3—41no—
251C4×D3—81no—
252Dic6—21no—
253Dic6#10312yes—
254C3×C8—41no—
255C3×C8—41no—
256C3×C8—81no—
257C3⋊C8—42no—
258C3⋊C8—41no—
259C3⋊C8—41no—
260C3⋊C8—42no—
261#41—32no—
262#41—32no—
263#41—61no—
264#41—61no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—31no—
268C2×D4—31no—
269C2×D4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4#23511yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4#23711yescharacteristic
276M16—124no—
277M16—121no—
278M16—121no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—62no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—62no—
288C2×C8—124no—
289C2×C8—121no—
290C2×C8—121no—
291C2×C8—121no—
292C2×C8—121no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—31no—
298C4⋊C4—31no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C22⋊C4—41no—
304C22⋊C4—61no—
305C22⋊C4—61no—
306C22⋊C4—61no—
307C42—41no—
308C42—61no—
309C42—61no—
310C42—61no—
311C42C2×C4×D311yescharacteristic
312C42D3×D411yescharacteristic
313C42D3×D411yescharacteristic
314C42C2×#6711yescharacteristic
315C22×C3—21no—
316C22×C3—21no—
317C22×C3#38611yescharacteristic
318C22×C3#48311yescharacteristic
319C22×C3#48411yescharacteristic
320C2×D3—22no—
321C2×D3—41no—
322C2×D3—41no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4C2×#10311yescharacteristic
329C3×C4#41711yescharacteristic
330C3×C4#44311yescharacteristic
331C3×C4#44411yescharacteristic
332C3×C4#45011yescharacteristic
333C3×C4#45111yescharacteristic
334Dic3—22no—
335Dic3—41no—
336Dic3—41no—
337C23—61no—
338C23#88311yescharacteristic
339Q8—32no—
340Q8—42no—
341Q8—41no—
342Q8—41no—
343Q8—61no—
344D4—32no—
345D4—62no—
346D4—62no—
347D4—61no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—61no—
354C2×C4—62no—
355C2×C4—62no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—241no—
361C2×C4D3×C22⋊C411yescharacteristic
362C2×C4#67⋊C411yescharacteristic
363C2×C4#88411yescharacteristic
364C2×C4D3×SD1611yescharacteristic
365C2×C4#91411yescharacteristic
366C2×C4D3×Q1611yescharacteristic
367C2×C4#91811yescharacteristic
368C2×C4C4×#6711yescharacteristic
369C2×C4#93711yescharacteristic
370C2×C4#93811yescharacteristic
371C8—41no—
372C8—41no—
373C8—81no—
374C8—122no—
375C8—122no—
376C8—121no—
377C8—121no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3#187211yescharacteristic
381C2×C3#187611yescharacteristic
382C2×C3#188111yescharacteristic
383D3—42no—
384C22—21no—
385C22—21no—
386C22—62no—
387C22—121no—
388C22—121no—
389C22#480511yescharacteristic
390C22#481111yescharacteristic
391C22#495611yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—62no—
396C4—81no—
397C4—81no—
398C4—121no—
399C4—121no—
400C4D3×#10311yescharacteristic
401C4#480311yescharacteristic
402C4#514411yescharacteristic
403C4#514511yescharacteristic
404C4#516911yescharacteristic
405C4#518611yescharacteristic
406C3#899311yescharacteristic, complemented, Sylow
407C2—21no—
408C2—21no—
409C2—122no—
410C2#7001111yescharacteristic
411C2#7026811yescharacteristic
412C2#7041611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1481C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1481D3
hypocenterC3#8993
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
3. C3×#1481C2semidirect
4. #71826C2semidirect
6. #67745C2semidirect
7. #67703C2semidirect
37. #1481D3semidirect
406. C3#8993semidirect

Maximal subgroups

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

subgroupindex
#833492
#790122
C3×#14812
#718262
#707542
#677452
#677032
#89933