Finite Group Databasebeta

← #11713032

Subgroups — #11713032

Order 768.

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

normal subgroups only

This group has 441 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#85452C212yesmaximal
2C2×#5404C211yesmaximal, characteristic
3#2916⋊C3C211yesmaximal, characteristic, complemented
4#73129C212yesmaximal, complemented
5#73120C211yesmaximal, characteristic, complemented
6#22694—31nomaximal, Sylow
7C2×#985C2211yescharacteristic
8#568⋊C3C2212yes—
9C2×#131⋊C3C2211yescharacteristic
10#5423C2212yes—
11C2×GU(2,𝔽3)C2211yescharacteristic
12#5405—22no—
13#5404—22no—
14#5404C411yescharacteristic, complemented
15#5404C411yescharacteristic, complemented
16#5404C411yescharacteristic, complemented
17#5404C411yescharacteristic, complemented
18D3×#108—41nomaximal
19#2957—32no—
20C2×#443—31no—
21#2916—31no—
22#2916D311yescharacteristic, complemented
23#1953—32no—
24#1861—32no—
25#1861—32no—
26#1848—31no—
27#1814—31no—
28#1814—31no—
29#1772—31no—
30C2×#41⋊C3C2311yescharacteristic
31#985C2×C411yescharacteristic
32#985C2×C411yescharacteristic
33#985D411yescharacteristic
34#985D411yescharacteristic
35C2×GL(2,𝔽3)Q811yescharacteristic
36C2×#226Q811yescharacteristic
37D3×M16—42no—
38C2×D3×C8—41no—
39#131⋊C3—22no—
40#131⋊C3C2×C411yescharacteristic
41#131⋊C3C2×C411yescharacteristic
42GU(2,𝔽3)—22no—
43GU(2,𝔽3)C2×C411yescharacteristic
44GU(2,𝔽3)C2×C411yescharacteristic
45C3×#108—41no—
46#822—42no—
47#819—41no—
48C2×#135—31no—
49#568—32no—
50#568C2×D312yes—
51C2×#131—31no—
52C2×#131C2×D311yescharacteristic
53#444—62no—
54#444—62no—
55#443—31no—
56#443—31no—
57#443—31no—
58#443—31no—
59#443—31no—
60#443—31no—
61#443—31no—
62#443—31no—
63#443—62no—
64#443—62no—
65#430—32no—
66#430—32no—
67#430—62no—
68C2×#108—31no—
69C2×#108—31no—
70C2×#108—31no—
71C2×#108—31no—
72C2×#108—32no—
73C2×#108—61no—
74C2×#108—61no—
75#427—31no—
76#423—32no—
77#421—32no—
78#421—32no—
79C2×#104—31no—
80C2×#104—31no—
81#405—32no—
82C2×C4×C8—31no—
83#344—32no—
84#344—32no—
85#339—32no—
86C2×C4×D3—41no—
87#41⋊C3C22×C411yescharacteristic
88#41⋊C3C2×D411yescharacteristic
89C2×SL(2,𝔽3)C2×Q811yescharacteristic
90GL(2,𝔽3)C4⋊C411yescharacteristic
91GL(2,𝔽3)C4⋊C411yescharacteristic
92#226C4⋊C411yescharacteristic
93#226C4⋊C411yescharacteristic
94C3×M16—42no—
95C2×C3×C8—41no—
96#208—42no—
97C2×C3⋊C8—41no—
98ΓL(1,𝔽25)—82no—
99D3×C8—41no—
100D3×C8—41no—
101D3×C8—41no—
102D3×C8—41no—
103D3×C8—82no—
104C2×#41—31no—
105C2×#41C22×D311yescharacteristic
106#135—31no—
107#135—31no—
108#135—31no—
109#135—31no—
110#135—31no—
111#135—31no—
112#135—31no—
113#135—31no—
114C2×Q16—31no—
115C2×SD16—31no—
116C2×SD16—31no—
117C2×D8—31no—
118#131—22no—
119#131—31no—
120#131—31no—
121#131—31no—
122#131—31no—
123#131—31no—
124#131—31no—
125#131—61no—
126#131—61no—
127#131—62no—
128#131—62no—
129#131—62no—
130#131—122no—
131#131C4×D311yescharacteristic
132#131C4×D311yescharacteristic
133C2×M16—31no—
134C2×M16—31no—
135C2×M16—32no—
136C2×M16—32no—
137C2×M16—32no—
138C2×M16—61no—
139C2×M16—62no—
140C2×M16—62no—
141C22×C8—31no—
142C22×C8—31no—
143C22×C8—61no—
144#117—32no—
145C2×C42—31no—
146#108—31no—
147#108—31no—
148#108—31no—
149#108—31no—
150#108—32no—
151#108—31no—
152#108—61no—
153#108—61no—
154#108—62no—
155#108—62no—
156#108—62no—
157#108—62no—
158#108—61no—
159#108S411yescharacteristic
160#107—61no—
161#106—61no—
162C4⋊C8—122no—
163#104—31no—
164#104—31no—
165#104—31no—
166#104—31no—
167#104—31no—
168#104—31no—
169#104—31no—
170#104—31no—
171#104—62no—
172#104—62no—
173#104—62no—
174#104—62no—
175#97—62no—
176C4×C8—31no—
177C4×C8—31no—
178C4×C8—31no—
179C4×C8—31no—
180C4×C8—62no—
181C22×D3—41no—
182C2×C3×C4—41no—
183C2×Dic3—41no—
184C4×D3—41no—
185C4×D3—41no—
186C4×D3—41no—
187C4×D3—41no—
188SL(2,𝔽3)C2×C4⋊C411yescharacteristic
189C3×C8—41no—
190C3×C8—41no—
191C3×C8—82no—
192C3⋊C8—41no—
193C3⋊C8—41no—
194C3⋊C8—82no—
195#41—31no—
196#41—31no—
197#41—31no—
198#41—31no—
199#41—31no—
200#41—31no—
201#41—61no—
202#41—61no—
203#41C2×C4×D311yescharacteristic
204#41D3×D411yescharacteristic
205C2×Q8—31no—
206C2×Q8D3×Q811yescharacteristic
207C2×D4—31no—
208C2×D4—31no—
209C2×D4—61no—
210C22×C4—31no—
211C22×C4—61no—
212C22×C4—61no—
213Q16—31no—
214Q16—31no—
215Q16—31no—
216Q16—31no—
217SD16—31no—
218SD16—31no—
219SD16—31no—
220SD16—31no—
221SD16—31no—
222SD16—31no—
223SD16—31no—
224SD16—31no—
225D8—31no—
226D8—31no—
227D8—31no—
228D8—31no—
229M16—31no—
230M16—31no—
231M16—31no—
232M16—31no—
233M16—32no—
234M16—62no—
235M16—61no—
236M16—61no—
237M16—61no—
238M16—61no—
239M16—61no—
240M16—61no—
241M16—62no—
242M16—62no—
243M16—62no—
244M16—122no—
245M16—122no—
246M16C2×S412yes—
247C2×C8—31no—
248C2×C8—31no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8—31no—
252C2×C8—31no—
253C2×C8—31no—
254C2×C8—31no—
255C2×C8—31no—
256C2×C8—31no—
257C2×C8—31no—
258C2×C8—62no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—62no—
265C2×C8—62no—
266C2×C8—122no—
267C2×C8—122no—
268C2×C8C2×S411yescharacteristic
269C4⋊C4—62no—
270C22⋊C4—62no—
271C42—31no—
272C42—31no—
273C42—61no—
274C42—62no—
275C22×C3—41no—
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—82no—
279C3×C4—41no—
280C3×C4—41no—
281Dic3—41no—
282Dic3—41no—
283C23—31no—
284C23—61no—
285Q8—31no—
286Q8—31no—
287Q8—31no—
288Q8—61no—
289Q8D3×C4⋊C411yescharacteristic
290D4—31no—
291D4—31no—
292D4—31no—
293D4—31no—
294D4—61no—
295D4—61no—
296D4—61no—
297D4—61no—
298D4—61no—
299D4—61no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—122no—
313C2×C4—122no—
314C2×C4C22×S411yescharacteristic
315C8—22no—
316C8—31no—
317C8—31no—
318C8—31no—
319C8—31no—
320C8—31no—
321C8—31no—
322C8—61no—
323C8—61no—
324C8—62no—
325C8—122no—
326C8C4×S411yescharacteristic
327C8C4×S411yescharacteristic
328C2×C3—41no—
329C2×C3—81no—
330D3—42no—
331D3—81no—
332C22—31no—
333C22—31no—
334C22—61no—
335C22—61no—
336C22—121no—
337C22—241no—
338C22Q8×S411yescharacteristic
339C4—31no—
340C4—31no—
341C4—62no—
342C4—61no—
343C4—61no—
344C4—121no—
345C4—242no—
346C4C2×C4×S411yescharacteristic
347C4D4×S411yescharacteristic
348C3—41noSylow
349C2—21no—
350C2—61no—
351C2—61no—
352C2—121no—
353C2—121no—
354C2C4⋊C4×S411yescharacteristic
subgroupG/H
centerC4D4×S4
commutator#41⋊C3C22×C4
FrattiniC2×C4C22×S4
Fitting#2916D3
socleC2C4⋊C4×S4
prime radicalC2×#985C22
cosocleC2×#41⋊C3C23
hypercenter#108S4
hypocenterSL(2,𝔽3)C2×C4⋊C4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ #108

Complements

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

NH ≅ G/Ntype
3. #2916⋊C3C2semidirect
4. #73129C2semidirect
5. #73120C2semidirect
14. #5404C4semidirect
15. #5404C4semidirect
16. #5404C4semidirect
17. #5404C4semidirect
22. #2916D3semidirect

Maximal subgroups

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

subgroupindex
#854522
C2×#54042
#2916⋊C32
#731292
#731202
#226943
D3×#1084