Finite Group Databasebeta

← #10691035

Subgroups — #10691035

Order 768.

478 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83212C211yesmaximal, characteristic
2#79049C211yesmaximal, characteristic
3C3×#1548C211yesmaximal, characteristic, complemented
4C2×#4547C211yesmaximal, characteristic, complemented
5#70755C211yesmaximal, characteristic
6#67745C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9167—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#907—21no—
11C4×#235C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5119—21no—
15#5094—21no—
16#5091C412yes—
17#5087—21no—
18#5084C412yes—
19#4966—21no—
20#4964—21no—
21#4924C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24#4839—22no—
25#4837—21no—
26D3×C4⋊C8—21no—
27C3×#333—22no—
28#4547—21no—
29#4547—21no—
30#4459—21no—
31#4456—21no—
32#3079—31no—
33#2906—31no—
34#1548D311yescharacteristic, complemented
35C2×#332—31no—
36#1477—31no—
37#1301—31no—
38#1292—31no—
39C22×C3×Q8—21no—
40C2×#235D411yescharacteristic
41C2×C3×M16—21no—
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×C42C2311yescharacteristic
49C4×#67—21no—
50C4×#67—21no—
51#926D411yescharacteristic
52C2×#208—21no—
53C22×C3⋊C8—21no—
54C22×C3⋊C8—21no—
55#907—42no—
56#907—42no—
57C2×ΓL(1,𝔽25)—22no—
58C2×D3×C8—22no—
59C4×D12C2×C411yescharacteristic
60C4×D12D411yescharacteristic
61#872—21no—
62C42×D3—21no—
63C4×Dic6C2×C411yescharacteristic
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8D412yes—
68C3×C22⋊C8—21no—
69C3×C4×C8—21no—
70Dic3⋊C8—22no—
71Dic3⋊C8—22no—
72#810—22no—
73#809—22no—
74#808—22no—
75#807—22no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)C2×C412yes—
81C4⋊(C3⋊C8)D411yescharacteristic
82C4⋊(C3⋊C8)D411yescharacteristic
83C2×#131—61no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C4×#41—31no—
86#477—32no—
87#476—32no—
88#475—61no—
89#474—61no—
90#431C2×D311yescharacteristic
91#428—61no—
92#424—61no—
93#424—62no—
94#423—31no—
95#423—61no—
96#423C2×D311yescharacteristic
97C2×C4⋊C8—31no—
98C2×C4⋊C8—31no—
99C2×C4⋊C8—61no—
100#417—61no—
101#333—22no—
102#332—61no—
103#332—61no—
104Q8⋊C8—61no—
105D4⋊C8—61no—
106C2×C3×Q8—22no—
107C2×C3×Q8—41no—
108C2×C3×Q8—81no—
109C2×C3×Q8—81no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C2×#67—21no—
115#235—21no—
116#235—21no—
117#235SD1612yes—
118#235Q1612yes—
119C2×D12C22⋊C411yescharacteristic
120C2×C4×D3—21no—
121C2×C4×D3—21no—
122C2×C4×D3—21no—
123C2×Dic6C22⋊C411yescharacteristic
124C3×M16—41no—
125C3×M16—41no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C42C22×C411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136#217—21no—
137(C2×D3)⋊C4—21no—
138(C2×D3)⋊C4—21no—
139C4⋊Dic3C22⋊C411yescharacteristic
140C4⋊Dic3C22⋊C411yescharacteristic
141Dic3⋊C4—21no—
142Dic3⋊C4—21no—
143C4×Dic3—21no—
144C4×Dic3—21no—
145C4×Dic3—21no—
146#208—41no—
147#208—41no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—21no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—81no—
155ΓL(1,𝔽25)—44no—
156D3×C8—44no—
157C2×#41—31no—
158C22×Q8—21no—
159#131—122no—
160#131—122no—
161C2×M16—21no—
162C2×M16—62no—
163C2×M16—61no—
164C22×C8—62no—
165C22×C8—61no—
166C22×C8—61no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8C4×D312yes—
171C4×Q8—31no—
172C4×Q8—31no—
173C4×D4—31no—
174C4×D4—31no—
175C4×D4—61no—
176C4×D4—61no—
177#117—31no—
178#117—61no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C42—61no—
182C2×C42C22×D311yescharacteristic
183#107—62no—
184#106—62no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—32no—
188C4⋊C8—31no—
189C4⋊C8—31no—
190C4⋊C8—62no—
191C4⋊C8—62no—
192C4⋊C8—61no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8#6712yes—
197#103—62no—
198#102—62no—
199C22⋊C8—21no—
200C4×C8—21no—
201C23×C3#12311yescharacteristic
202C22×D3—21no—
203C3×Q8—82no—
204C3×Q8—81no—
205C3×Q8—81no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—82no—
211C2×C3×C4C2×C22⋊C411yescharacteristic
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4C2×SD1611yescharacteristic
218C2×C3×C4C2×Q1611yescharacteristic
219C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
220C2×C3×C4#13711yescharacteristic
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—81no—
237C3⋊C8—42no—
238C3⋊C8—41no—
239C3⋊C8—41no—
240C3⋊C8—81no—
241#41—32no—
242#41—32no—
243#41—61no—
244#41—61no—
245C2×Q8—22no—
246C2×Q8—31no—
247C2×Q8—41no—
248C2×Q8—81no—
249C2×Q8—81no—
250C2×D4—31no—
251C2×D4—61no—
252C22×C4—41no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—61no—
256C22×C4#23511yescharacteristic
257C22×C4D3×D411yescharacteristic
258C22×C4#23711yescharacteristic
259M16—41no—
260M16—41no—
261M16—124no—
262M16—121no—
263M16—121no—
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—62no—
267C2×C8—61no—
268C2×C8—61no—
269C2×C8—124no—
270C2×C8—121no—
271C2×C8—121no—
272C2×C8—121no—
273C2×C8—241no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—31no—
277C4⋊C4—31no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C22⋊C4—61no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C42—61no—
286C42—61no—
287C42—61no—
288C42C2×C4×D311yescharacteristic
289C42D3×D411yescharacteristic
290C42D3×D411yescharacteristic
291C42C2×#6711yescharacteristic
292C22×C3—21no—
293C22×C3—21no—
294C22×C3#38611yescharacteristic
295C22×C3#48311yescharacteristic
296C22×C3#48411yescharacteristic
297C2×D3—22no—
298C2×D3—41no—
299C2×D3—41no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—82no—
304C3×C4C2×#10311yescharacteristic
305C3×C4#41711yescharacteristic
306C3×C4#45011yescharacteristic
307C3×C4#45111yescharacteristic
308C3×C4#47111yescharacteristic
309C3×C4#47211yescharacteristic
310Dic3—22no—
311Dic3—41no—
312Dic3—41no—
313C23—61no—
314C23#88311yescharacteristic
315Q8—32no—
316Q8—61no—
317Q8—82no—
318Q8—81no—
319Q8—81no—
320D4—32no—
321D4—62no—
322D4—62no—
323D4—61no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—61no—
329C2×C4—62no—
330C2×C4—62no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—82no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—241no—
337C2×C4D3×C22⋊C411yescharacteristic
338C2×C4#67⋊C411yescharacteristic
339C2×C4#88411yescharacteristic
340C2×C4D3×SD1611yescharacteristic
341C2×C4#91411yescharacteristic
342C2×C4D3×Q1611yescharacteristic
343C2×C4#91811yescharacteristic
344C2×C4C4×#6711yescharacteristic
345C2×C4#93711yescharacteristic
346C2×C4#93811yescharacteristic
347C8—42no—
348C8—81no—
349C8—121no—
350C8—121no—
351C8—122no—
352C8—241no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3#187211yescharacteristic
356C2×C3#190211yescharacteristic
357C2×C3#190711yescharacteristic
358D3—42no—
359C22—21no—
360C22—21no—
361C22—62no—
362C22—121no—
363C22—121no—
364C22#480511yescharacteristic
365C22#481111yescharacteristic
366C22#495611yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—62no—
371C4—82no—
372C4—121no—
373C4—121no—
374C4#474811yescharacteristic
375C4#474911yescharacteristic
376C4D3×#10311yescharacteristic
377C4#480311yescharacteristic
378C4#516911yescharacteristic
379C4#518611yescharacteristic
380C3#916711yescharacteristic, complemented, Sylow
381C2—21no—
382C2—21no—
383C2—122no—
384C2#7008811yescharacteristic
385C2#7026811yescharacteristic
386C2#7034911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1548D3
hypocenterC3#9167
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. C3×#1548C2semidirect
4. C2×#4547C2semidirect
6. #67745C2semidirect
7. #67711C2semidirect
34. #1548D3semidirect
380. C3#9167semidirect

Maximal subgroups

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

subgroupindex
#832122
#790492
C3×#15482
C2×#45472
#707552
#677452
#677112
#91673