Finite Group Databasebeta

← #11711875

Subgroups — #11711875

Order 768.

336 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 51 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#85343C211yesmaximal, characteristic
2#2988⋊C3C211yesmaximal, characteristic, complemented
3#73072C211yesmaximal, characteristic, complemented
4#73018C211yesmaximal, characteristic, complemented
5C8×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#73009C211yesmaximal, characteristic, complemented
7#226⋊C8C211yesmaximal, characteristic, complemented
8#17927—31nomaximal, Sylow
9C4×#41⋊C3C2211yescharacteristic
10#5427C411yescharacteristic
11#5425C411yescharacteristic
12#5395C411yescharacteristic
13C4×GL(2,𝔽3)C2211yescharacteristic
14#5389C411yescharacteristic
15C4×#226C2211yescharacteristic
16#4839—41nomaximal
17C8×SL(2,𝔽3)C2211yescharacteristic
18C8×SL(2,𝔽3)C2211yescharacteristic
19SL(2,𝔽3)⋊C8C2211yescharacteristic
20SL(2,𝔽3)⋊C8C2211yescharacteristic
21#3102—31no—
22#2993—31no—
23#2988D311yescharacteristic, complemented
24#1587—31no—
25#1586—31no—
26#1583—31no—
27#1582—31no—
28#1580—31no—
29#1579—31no—
30C8×SD16—31no—
31C8×SD16—31no—
32#1570—31no—
33#1489—31no—
34#1483—31no—
35#1453—31no—
36C2×#41⋊C3C2×C411yescharacteristic
37C2×GL(2,𝔽3)C2×C411yescharacteristic
38C2×#226C2×C411yescharacteristic
39C2×ΓL(1,𝔽25)—41no—
40C2×D3×C8—41no—
41#872—41no—
42C4×SL(2,𝔽3)C2×C411yescharacteristic
43C4×SL(2,𝔽3)C2311yescharacteristic
44#859C2×C411yescharacteristic
45#859C2×C411yescharacteristic
46C3×C4⋊C8—41no—
47Dic3⋊C8—41no—
48Dic3⋊C8—41no—
49C4⋊(C3⋊C8)—41no—
50#529—31no—
51C4×#41C2×D311yescharacteristic
52#485—31no—
53#482—31no—
54#479—31no—
55#478—31no—
56#464—31no—
57#463—31no—
58#452—31no—
59#449—31no—
60#446—31no—
61C8×Q8—31no—
62C8×Q8C2×D311yescharacteristic
63C8×Q8C2×D311yescharacteristic
64C4×Q16—31no—
65C4×SD16—31no—
66C4×SD16—31no—
67C4×D8—31no—
68#436—31no—
69#436—61no—
70#435—61no—
71C8×D4—31no—
72C8×D4—31no—
73C8×D4—31no—
74#433—61no—
75#431—31no—
76#431—31no—
77#424—61no—
78#423—31no—
79C2×C4⋊C8—31no—
80C4×M16—31no—
81#334—31no—
82#334—31no—
83#334—31no—
84#334—31no—
85Q8⋊C8—31no—
86Q8⋊C8—31no—
87Q8⋊C8—31no—
88Q8⋊C8—31no—
89D4⋊C8—31no—
90D4⋊C8—31no—
91D4⋊C8—31no—
92D4⋊C8—31no—
93#322—31no—
94#322—31no—
95#322—31no—
96C82—31no—
97C2×C4×D3—41no—
98#41⋊C3—21no—
99C2×SL(2,𝔽3)C22×C411yescharacteristic
100GL(2,𝔽3)—21no—
101#226—21no—
102C2×C3×C8—41no—
103C2×C3×C8—41no—
104C3×C42—41no—
105(C2×D3)⋊C4—41no—
106(C2×D3)⋊C4—41no—
107Dic3⋊C4—41no—
108Dic3⋊C4—41no—
109C4×Dic3—41no—
110C2×C3⋊C8—41no—
111C2×C3⋊C8—41no—
112ΓL(1,𝔽25)—82no—
113D3×C8—82no—
114C2×#41C4×D311yescharacteristic
115C2×Q16—31no—
116C2×SD16—31no—
117C2×SD16—31no—
118C2×D8—31no—
119C2×M16—61no—
120C2×M16—61no—
121C2×M16—61no—
122C22×C8—61no—
123C22×C8—61no—
124#126—61no—
125#125—31no—
126#124—31no—
127#123—61no—
128C22⋊Q8—31no—
129#121—31no—
130C4×Q8—31no—
131C4×Q8C4×D311yescharacteristic
132C4×Q8C22×D311yescharacteristic
133C4×D4—31no—
134C4×D4—31no—
135C4×D4—31no—
136C4×D4—61no—
137#117—31no—
138#117—61no—
139C2×C42—31no—
140#107—31no—
141#107—31no—
142#106—31no—
143#106—31no—
144C4⋊C8—31no—
145C4⋊C8—31no—
146C4⋊C8—31no—
147C4⋊C8—31no—
148C4⋊C8—31no—
149C4⋊C8—31no—
150C4⋊C8—31no—
151C4⋊C8—61no—
152C4⋊C8—61no—
153C4⋊C8—61no—
154C4⋊C8S411yescharacteristic
155#103—31no—
156#103—31no—
157#103—31no—
158#103—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163C22⋊C8—31no—
164C22⋊C8—31no—
165C22⋊C8—31no—
166C22⋊C8—31no—
167C22⋊C8—61no—
168C22⋊C8—61no—
169#97—61no—
170#97—61no—
171C4×C8—31no—
172C4×C8—31no—
173C4×C8—31no—
174C4×C8—31no—
175C4×C8—31no—
176C4×C8—31no—
177C4×C8—61no—
178C22×D3—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C2×Dic3—41no—
185C4×D3—42no—
186C4×D3—42no—
187SL(2,𝔽3)#13111yescharacteristic
188C3×C8—81no—
189C3×C8—81no—
190C3⋊C8—81no—
191C3⋊C8—81no—
192#41—21no—
193#41—61no—
194C2×Q8—31no—
195C2×Q8C2×C4×D311yescharacteristic
196C2×D4—31no—
197C2×D4—31no—
198C2×D4—61no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—61no—
203Q16—61no—
204Q16—61no—
205SD16—62no—
206SD16—61no—
207SD16—61no—
208D8—62no—
209M16—122no—
210M16—241no—
211M16—241no—
212C2×C8—31no—
213C2×C8—31no—
214C2×C8—31no—
215C2×C8—31no—
216C2×C8—31no—
217C2×C8—31no—
218C2×C8—61no—
219C2×C8—61no—
220C2×C8—61no—
221C2×C8—61no—
222C2×C8—241no—
223C2×C8—241no—
224C2×C8C2×S411yescharacteristic
225C2×C8C2×S411yescharacteristic
226C4⋊C4—31no—
227C4⋊C4—31no—
228C4⋊C4—31no—
229C4⋊C4—31no—
230C4⋊C4—61no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C22⋊C4—61no—
235C22⋊C4—61no—
236C22⋊C4—61no—
237C22⋊C4—61no—
238C22⋊C4—61no—
239C22⋊C4—61no—
240C42—31no—
241C42—31no—
242C42—31no—
243C42—61no—
244C42C2×S411yescharacteristic
245C22×C3—41no—
246C2×D3—42no—
247C2×D3—81no—
248C2×D3—81no—
249C3×C4—41no—
250C3×C4—41no—
251C3×C4—81no—
252C3×C4—81no—
253Dic3—42no—
254Dic3—81no—
255Dic3—81no—
256C23—31no—
257C23—61no—
258Q8—31no—
259Q8—61no—
260Q8—61no—
261Q8#90111yescharacteristic
262D4—32no—
263D4—61no—
264D4—61no—
265D4—61no—
266D4—241no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—121no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4C4×S411yescharacteristic
281C2×C4C4×S411yescharacteristic
282C2×C4C22×S411yescharacteristic
283C8—21no—
284C8—21no—
285C8—61no—
286C8—61no—
287C8—61no—
288C8—61no—
289C8—62no—
290C8—61no—
291C8—61no—
292C8—121no—
293C8—121no—
294C8—121no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—41no—
298D3—82no—
299C22—61no—
300C22—61no—
301C22—61no—
302C22—121no—
303C22—121no—
304C22—121no—
305C22C2×C4×S411yescharacteristic
306C4—21no—
307C4—21no—
308C4—31no—
309C4—31no—
310C4—61no—
311C4—61no—
312C4—121no—
313C4—121no—
314C4—121no—
315C4#592411yescharacteristic
316C4#592611yescharacteristic
317C3—41noSylow
318C2—121no—
319C2—241no—
320C2#8536411yescharacteristic
321C2#8543711yescharacteristic
322C2#8545311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
Fitting#2988D3
socleC22C2×C4×S4
prime radical#5427C4
cosocleC4×SL(2,𝔽3)C23
hypercenterC4⋊C8S4
hypocenterSL(2,𝔽3)#131
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. #2988⋊C3C2semidirect
3. #73072C2semidirect
4. #73018C2semidirect
5. C8×GL(2,𝔽3)C2semidirect
6. #73009C2semidirect
7. #226⋊C8C2semidirect
23. #2988D3semidirect

Maximal subgroups

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

subgroupindex
#853432
#2988⋊C32
#730722
#730182
C8×GL(2,𝔽3)2
#730092
#226⋊C82
#179273
#48394