Finite Group Databasebeta

← #11712886

Subgroups — #11712886

Order 768.

366 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85334C211yesmaximal, characteristic
2#2875⋊C3C211yesmaximal, characteristic
3#73123C211yesmaximal, characteristic
4#73122C211yesmaximal, characteristic
5#73122C211yesmaximal, characteristic
6#73121C211yesmaximal, characteristic
7C4⋊GU(2,𝔽3)C211yesmaximal, characteristic
8#21629—31nomaximal, Sylow
9C2×#131⋊C3C2211yescharacteristic
10C2×#131⋊C3C2211yescharacteristic
11C4×#41⋊C3C2211yescharacteristic
12#5424C2211yescharacteristic
13#5424C2211yescharacteristic
14C2×GU(2,𝔽3)C2211yescharacteristic
15C2×GU(2,𝔽3)C2211yescharacteristic
16#5391C411yescharacteristic, complemented
17#5389C411yescharacteristic, complemented
18#5389C411yescharacteristic, complemented
19#5386C411yescharacteristic, complemented
20#4701—41nomaximal
21#3076—31no—
22#2961—31no—
23#2875D311yescharacteristic
24#1986—31no—
25#1950—31no—
26#1922—31no—
27#1922—31no—
28#1866—31no—
29#1866—31no—
30#1806—31no—
31#1773—31no—
32#1772—31no—
33#1771—31no—
34#1765—31no—
35#1764—31no—
36C2×#41⋊C3C2311yescharacteristic
37C4×Dic6—41no—
38#131⋊C3—21no—
39#131⋊C3Q812yescomplemented
40C4×SL(2,𝔽3)C2×C411yescharacteristic
41C4×SL(2,𝔽3)C2×C411yescharacteristic
42GU(2,𝔽3)—21no—
43GU(2,𝔽3)—21no—
44#859C2×C411yescharacteristic
45#859C2×C411yescharacteristic
46#859C2×C411yescharacteristic
47#859C2×C411yescharacteristic
48C3×#97—41no—
49#815—41no—
50Dic3⋊C8—41no—
51Dic3⋊C8—41no—
52C8×Dic3—41no—
53C4⋊(C3⋊C8)—41no—
54C2×#131C2×D311yescharacteristic
55C2×#131C2×D311yescharacteristic
56#533—31no—
57C4×#41C2×D311yescharacteristic
58#479—31no—
59#479—31no—
60#478—31no—
61#478—31no—
62#477—31no—
63#476—31no—
64#475—31no—
65#474—31no—
66#446—61no—
67#446—61no—
68#446—61no—
69#446—61no—
70C2×#108—31no—
71C2×#108—31no—
72#427—31no—
73#427—31no—
74#424—31no—
75#424—31no—
76#423—31no—
77C2×C4⋊C8—31no—
78C2×#104—31no—
79C2×#104—31no—
80#416—31no—
81#416—31no—
82#405—31no—
83#405—31no—
84#405—31no—
85#405—31no—
86C4×M16—31no—
87C2×#97—31no—
88C2×#97—31no—
89C2×C4×C8—31no—
90C4×C4⋊C4—31no—
91#341—31no—
92#341—31no—
93#339—31no—
94#339—31no—
95#339—31no—
96#339—31no—
97#339—31no—
98#339—31no—
99#336—31no—
100#336—31no—
101C2×Dic6—41no—
102#41⋊C3C2×Q811yescharacteristic
103#41⋊C3#4111yescharacteristic
104C2×SL(2,𝔽3)C22×C411yescharacteristic
105C2×C3×C8—41no—
106C2×C3×C8—41no—
107C3×C42—41no—
108C4⋊Dic3—41no—
109Dic3⋊C4—41no—
110Dic3⋊C4—41no—
111C4×Dic3—41no—
112C4×Dic3—41no—
113C2×C3⋊C8—41no—
114C2×C3⋊C8—41no—
115C2×#41C22×D311yescharacteristic
116#131—21no—
117#131—32no—
118#131—61no—
119#131Dic612yescomplemented
120C2×M16—31no—
121C2×M16—31no—
122C2×M16—31no—
123C2×M16—31no—
124C22×C8—31no—
125C22×C8—31no—
126C22×C8—31no—
127C22×C8—31no—
128C4⋊Q8—31no—
129C4⋊Q8—31no—
130#125—31no—
131#125—31no—
132C22⋊Q8—61no—
133C22⋊Q8—61no—
134C4×Q8—61no—
135C4×Q8—61no—
136C4×Q8C4×D311yescharacteristic
137C4×Q8C4×D311yescharacteristic
138C4×D4—31no—
139C4×D4—31no—
140#117—31no—
141#117—31no—
142#117—31no—
143C2×C4⋊C4—61no—
144C2×C42—31no—
145C2×C42—31no—
146C2×C42—31no—
147#108—62no—
148#108—62no—
149#107—31no—
150#107—31no—
151#107—31no—
152#107—31no—
153#106—31no—
154#106—31no—
155#106—31no—
156#106—31no—
157C4⋊C8—31no—
158C4⋊C8—31no—
159C4⋊C8—31no—
160C4⋊C8—31no—
161C4⋊C8—61no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167#104—61no—
168#104—61no—
169#104—61no—
170#104—61no—
171#103—31no—
172#103—31no—
173#103—31no—
174#103—31no—
175#102—31no—
176#102—31no—
177#102—31no—
178#102—31no—
179#97—31no—
180#97—31no—
181#97—31no—
182#97—62no—
183#97—61no—
184#97—61no—
185#97S411yescharacteristic
186C4×C8—61no—
187C4×C8—62no—
188C4×C8—61no—
189C4×C8—61no—
190#95—61no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198Dic6—82no—
199SL(2,𝔽3)C4×Q811yescharacteristic
200C3×C8—42no—
201C3×C8—81no—
202C3⋊C8—81no—
203C3⋊C8—81no—
204#41—31no—
205#41—31no—
206#41C2×Dic611yescharacteristic
207#41#23511yescharacteristic
208C2×Q8—61no—
209C2×Q8—61no—
210C2×Q8C2×C4×D311yescharacteristic
211C2×D4—31no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—61no—
216C22×C4—61no—
217M16—32no—
218M16—32no—
219M16—61no—
220M16—61no—
221M16—61no—
222M16—61no—
223M16—61no—
224M16—61no—
225C2×C8—31no—
226C2×C8—32no—
227C2×C8—32no—
228C2×C8—31no—
229C2×C8—31no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8—31no—
233C2×C8—61no—
234C2×C8—61no—
235C2×C8—61no—
236C2×C8—61no—
237C2×C8—61no—
238C2×C8—61no—
239C2×C8—61no—
240C2×C8—61no—
241C2×C8C2×S411yescharacteristic
242C2×C8C2×S411yescharacteristic
243C4⋊C4—31no—
244C4⋊C4—31no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—31no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—122no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C42—31no—
260C42—31no—
261C42—31no—
262C42—61no—
263C42—61no—
264C42—61no—
265C42—61no—
266C42—61no—
267C42—61no—
268C42C2×S411yescharacteristic
269C22×C3—41no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—81no—
273C3×C4—81no—
274Dic3—42no—
275Dic3—81no—
276Dic3—81no—
277Dic3—81no—
278C23—31no—
279Q8—31no—
280Q8—241no—
281Q8—241no—
282Q8C4×Dic611yescharacteristic
283D4—31no—
284D4—31no—
285D4—61no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—62no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4C4×S411yescharacteristic
306C2×C4C4×S411yescharacteristic
307C2×C4C22×S411yescharacteristic
308C8—21no—
309C8—32no—
310C8—61no—
311C8—61no—
312C8—61no—
313C8—61no—
314C8—61no—
315C8—121no—
316C8—121no—
317C8A4⋊Q812yes—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C22—31no—
322C22—31no—
323C22—61no—
324C22—61no—
325C22C2×C4×S411yescharacteristic
326C4—21no—
327C4—21no—
328C4—31no—
329C4—31no—
330C4—61no—
331C4—61no—
332C4—122no—
333C4—121no—
334C4—121no—
335C4—121no—
336C4—121no—
337C4—241no—
338C4C2×A4⋊Q811yescharacteristic
339C4#591211yescharacteristic
340C3—41noSylow
341C2—61no—
342C2—61no—
343C2C4×A4⋊Q811yescharacteristic
344C2#8545211yescharacteristic
345C2#8545311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
Fitting#2875D3
socleC22C2×C4×S4
prime radical, cosocleC2×#41⋊C3C23
hypercenter#97S4
hypocenterSL(2,𝔽3)C4×Q8
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ #97

Complements

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

NH ≅ G/Ntype
16. #5391C4semidirect
17. #5389C4semidirect
18. #5389C4semidirect
19. #5386C4semidirect
39. #131⋊C3Q8semidirect
119. #131Dic6semidirect

Maximal subgroups

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

subgroupindex
#853342
#2875⋊C32
#731232
#731222
#731212
C4⋊GU(2,𝔽3)2
#216293
#47014