Finite Group Databasebeta

← #10691069

Subgroups — #10691069

Order 768.

510 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83209C211yesmaximal, characteristic
2#79012C211yesmaximal, characteristic
3C2×C3×#332C211yesmaximal, characteristic, complemented
4#71773C211yesmaximal, characteristic, complemented
5#70425C211yesmaximal, characteristic
6#67707C211yesmaximal, characteristic, complemented
7#67705C211yesmaximal, characteristic, complemented
8#9171—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#942—21no—
11C2×#901—21no—
12C4×#235C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15#5225—21no—
16#5102—21no—
17#5090—21no—
18#5090C412yes—
19#5083—21no—
20#5083C412yes—
21#4925C2211yescharacteristic
22#4924C2211yescharacteristic
23#4924C2211yescharacteristic
24#4839—22no—
25#4837—21no—
26D3×C4⋊C8—21no—
27C3×#332—21no—
28C3×#332—21no—
29#4547—22no—
30#4484—21no—
31#4483—21no—
32#3039—31no—
33#2906—31no—
34#1546—31no—
35C2×#332D311yescharacteristic, complemented
36#1477—31no—
37#1301—31no—
38#1292—31no—
39C22×C3×Q8—21no—
40C2×#235D411yescharacteristic
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8C2×C412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#216—22no—
51#942—42no—
52#942—42no—
53C2×#215—22no—
54C4×#67—21no—
55C4×#67—21no—
56#926D411yescharacteristic
57C2×#208—21no—
58C2×#208—21no—
59#901—42no—
60#901—42no—
61C2×ΓL(1,𝔽25)—22no—
62C2×D3×C8—22no—
63C4×D12C2×C411yescharacteristic
64C4×D12D411yescharacteristic
65#872—21no—
66C42×D3—21no—
67C4×Dic6C2×C411yescharacteristic
68C4×Dic6D411yescharacteristic
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D411yescharacteristic
73C3×C4⋊C8D411yescharacteristic
74#835—22no—
75#835—22no—
76#830—21no—
77Dic3⋊C8—22no—
78Dic3⋊C8—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)C2×C412yes—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85C2×#137—61no—
86C2×#131—61no—
87C2×C4⋊Q8C2×D311yescharacteristic
88C4×#41—31no—
89#462—32no—
90#462—61no—
91#461—32no—
92#461—61no—
93#431—31no—
94#424—62no—
95#424—61no—
96#423—31no—
97#423—31no—
98#423—61no—
99C2×C4⋊C8—61no—
100C2×C4⋊C8C2×D311yescharacteristic
101C2×C4⋊C8C2×D311yescharacteristic
102#419—61no—
103#332—21no—
104#332—21no—
105#332—62no—
106Q8⋊C8—61no—
107D4⋊C8—61no—
108C2×C3×Q8—22no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C2×#67—21no—
119#235—21no—
120#235—21no—
121#235D812yes—
122#235SD1612yes—
123C2×D12C22⋊C411yescharacteristic
124C2×C4×D3—21no—
125C2×C4×D3—21no—
126C2×C4×D3—21no—
127C2×Dic6C22⋊C411yescharacteristic
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—81no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C42C22×C411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142#217—21no—
143#216—44no—
144#215—44no—
145(C2×D3)⋊C4—21no—
146(C2×D3)⋊C4—21no—
147C4⋊Dic3C22⋊C411yescharacteristic
148C4⋊Dic3C22⋊C411yescharacteristic
149Dic3⋊C4—21no—
150Dic3⋊C4—21no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153C4×Dic3—21no—
154#208—41no—
155#208—41no—
156#208—41no—
157#208—41no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161ΓL(1,𝔽25)—44no—
162D3×C8—44no—
163C2×#41—31no—
164C22×Q8—21no—
165#137—122no—
166#137—122no—
167C2×Q16—62no—
168C2×SD16—62no—
169#131—122no—
170#131—122no—
171C2×M16—62no—
172C2×M16—61no—
173C2×M16—61no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—62no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8C4×D312yes—
181C4×Q8—31no—
182C4×Q8—31no—
183C4×D4—31no—
184C4×D4—31no—
185C4×D4—61no—
186C4×D4—61no—
187#117—31no—
188#117—61no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C42—61no—
192C2×C42C22×D311yescharacteristic
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—32no—
197C4⋊C8—32no—
198C4⋊C8—62no—
199C4⋊C8—62no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203C4⋊C8#6711yescharacteristic
204C4⋊C8#6711yescharacteristic
205#103—62no—
206#103—62no—
207C22⋊C8—61no—
208C4×C8—61no—
209C23×C3#12311yescharacteristic
210C22×D3—21no—
211C3×Q8—42no—
212C3×Q8—42no—
213C3×Q8—81no—
214C3×Q8—81no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4C2×C22⋊C411yescharacteristic
222C2×C3×C4C4×D411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4C2×D811yescharacteristic
228C2×C3×C4C2×SD1611yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×C3×C4#13711yescharacteristic
231#67—22no—
232#67—22no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236D12—21no—
237D12#10212yes—
238C4×D3—22no—
239C4×D3—22no—
240C4×D3—41no—
241C4×D3—41no—
242C4×D3—81no—
243Dic6—21no—
244Dic6#10212yes—
245C3×C8—41no—
246C3×C8—41no—
247C3×C8—81no—
248C3⋊C8—42no—
249C3⋊C8—42no—
250C3⋊C8—81no—
251#41—32no—
252#41—32no—
253#41—61no—
254#41—61no—
255C2×Q8—22no—
256C2×Q8—31no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×D4—31no—
263C2×D4—61no—
264C22×C4—41no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4#23511yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4#23711yescharacteristic
271Q16—124no—
272SD16—124no—
273M16—124no—
274M16—121no—
275M16—121no—
276M16—121no—
277M16—121no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—62no—
284C2×C8—62no—
285C2×C8—81no—
286C2×C8—124no—
287C2×C8—121no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—31no—
291C4⋊C4—31no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C42—61no—
300C42—61no—
301C42—61no—
302C42C2×C4×D311yescharacteristic
303C42D3×D411yescharacteristic
304C42D3×D411yescharacteristic
305C42C2×#6711yescharacteristic
306C22×C3—21no—
307C22×C3—21no—
308C22×C3#38611yescharacteristic
309C22×C3#48011yescharacteristic
310C22×C3#48111yescharacteristic
311C2×D3—22no—
312C2×D3—41no—
313C2×D3—41no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—82no—
318C3×C4C2×#10211yescharacteristic
319C3×C4#41711yescharacteristic
320C3×C4#44711yescharacteristic
321C3×C4#44811yescharacteristic
322C3×C4#47111yescharacteristic
323C3×C4#47311yescharacteristic
324Dic3—22no—
325Dic3—41no—
326Dic3—41no—
327C23—61no—
328C23#88311yescharacteristic
329Q8—32no—
330Q8—42no—
331Q8—42no—
332Q8—61no—
333Q8—81no—
334Q8—81no—
335D4—32no—
336D4—62no—
337D4—62no—
338D4—61no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—61no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—241no—
353C2×C4D3×C22⋊C411yescharacteristic
354C2×C4#67⋊C411yescharacteristic
355C2×C4#88411yescharacteristic
356C2×C4D3×D811yescharacteristic
357C2×C4#91111yescharacteristic
358C2×C4D3×SD1611yescharacteristic
359C2×C4#91511yescharacteristic
360C2×C4C4×#6711yescharacteristic
361C2×C4#93711yescharacteristic
362C2×C4#93811yescharacteristic
363C8—41no—
364C8—41no—
365C8—81no—
366C8—122no—
367C8—122no—
368C8—241no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3#187411yescharacteristic
372C2×C3#190211yescharacteristic
373C2×C3#190811yescharacteristic
374D3—42no—
375C22—21no—
376C22—21no—
377C22—62no—
378C22—121no—
379C22—121no—
380C22#477411yescharacteristic
381C22#477711yescharacteristic
382C22#495611yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—62no—
387C4—82no—
388C4—121no—
389C4—121no—
390C4D3×#10211yescharacteristic
391C4#477111yescharacteristic
392C4#515611yescharacteristic
393C4#517011yescharacteristic
394C4#517711yescharacteristic
395C4#519211yescharacteristic
396C3#917111yescharacteristic, complemented, Sylow
397C2—21no—
398C2—21no—
399C2—122no—
400C2#7012211yescharacteristic
401C2#7020011yescharacteristic
402C2#7028611yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenterC2×#332D3
hypocenterC3#9171
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. C2×C3×#332C2semidirect
4. #71773C2semidirect
6. #67707C2semidirect
7. #67705C2semidirect
35. C2×#332D3semidirect
396. C3#9171semidirect

Maximal subgroups

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

subgroupindex
#832092
#790122
C2×C3×#3322
#717732
#704252
#677072
#677052
#91713