Finite Group Databasebeta

← #10691847

Subgroups — #10691847

Order 768.

490 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 109 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#83213C211yesmaximal, characteristic
2#78491C211yesmaximal, characteristic
3C3×#1993C211yesmaximal, characteristic, complemented
4#71775C211yesmaximal, characteristic, complemented
5#69545C211yesmaximal, characteristic
6#67660C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9289—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#901—21no—
11#5477C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C3×#382C2211yescharacteristic
14#5227—22no—
15#5110—21no—
16#5085C412yescomplemented
17#5083—22no—
18#5079C412yescomplemented
19#4925C2211yescharacteristic
20#4925C2211yescharacteristic
21#4924C2211yescharacteristic
22#4706—22no—
23#4706—22no—
24C3×#340—21no—
25C3×#340—21no—
26#4547—22no—
27#4451—22no—
28#3206—31no—
29#2932—31no—
30#1993D311yescharacteristic, complemented
31#1549—31no—
32#1517—31no—
33#1286—31no—
34#1275—31no—
35C22×C3×Q8—21no—
36C2×#235—21no—
37C22×C3×C8D411yescharacteristic
38C22×C3×C8D411yescharacteristic
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8C2×C412yes—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#215—24no—
48#930—21no—
49#930D412yes—
50C2×#208—21no—
51#901—42no—
52#901—42no—
53C2×ΓL(1,𝔽25)—22no—
54C2×D3×C8—22no—
55#875—21no—
56#874C2×C411yescharacteristic
57C4×D12—21no—
58C4×D12D412yes—
59#869C2×C411yescharacteristic
60C3×#97—21no—
61C3×#97—21no—
62C3×#95—21no—
63C3×#95—21no—
64#835—24no—
65#830—21no—
66#830—21no—
67(C2×D3)⋊C8—24no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)D412yes—
72C4×C3⋊C8—21no—
73C4×C3⋊C8C2×C412yes—
74C2×#131—61no—
75#532—31no—
76C2×C4⋊Q8C2×D311yescharacteristic
77#492—32no—
78#487—32no—
79#461—62no—
80#448—62no—
81#435—62no—
82#435—62no—
83#431—31no—
84#431—31no—
85#423—31no—
86#408—61no—
87C2×#97C2×D311yescharacteristic
88#382C2×D311yescharacteristic
89#340—21no—
90#340—21no—
91#332—62no—
92#327—62no—
93C2×C3×Q8—22no—
94C2×C3×Q8—41no—
95C2×C3×Q8—82no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4#4111yescharacteristic
101C2×#67—22no—
102#235—41no—
103#235—41no—
104#235—41no—
105#235—41no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12C22⋊C412yes—
109C2×C4×D3—22no—
110C2×Dic6—21no—
111C2×C3×C8—21no—
112C2×C3×C8—21no—
113C2×C3×C8—21no—
114C2×C3×C8—21no—
115C2×C3×C8—21no—
116C2×C3×C8SD1612yescomplemented
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C42C22×C411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126C3×C42#4111yescharacteristic
127#215—84no—
128(C2×D3)⋊C4—22no—
129(C2×D3)⋊C4—22no—
130C4⋊Dic3—21no—
131C4⋊Dic3C22⋊C412yes—
132#208—41no—
133#208—41no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—41no—
137ΓL(1,𝔽25)—44no—
138D3×C8—44no—
139C2×#41—61no—
140C22×Q8—21no—
141C2×SD16—64no—
142#131—122no—
143#131—122no—
144C2×M16—62no—
145C2×M16—61no—
146C22×C8—62no—
147C22×C8#6711yescharacteristic
148C22×C8#6711yescharacteristic
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—31no—
153C4⋊Q8C4×D312yes—
154#127—31no—
155#124—61no—
156#121—32no—
157#121—61no—
158C4×D4—32no—
159C4×D4—61no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C42C22×D311yescharacteristic
164C4⋊C8—32no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168#103—64no—
169C22⋊C8—64no—
170C22⋊C8—61no—
171C22⋊C8—61no—
172#97—21no—
173#97—21no—
174C4×C8—32no—
175C4×C8—61no—
176#95—21no—
177#95—21no—
178C23×C3#12111yescharacteristic
179C22×D3—22no—
180C3×Q8—44no—
181C3×Q8—82no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—42no—
185C2×C3×C4—82no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4—81no—
189C2×C3×C4C2×C22⋊C411yescharacteristic
190C2×C3×C4C4×D411yescharacteristic
191C2×C3×C4C4×D411yescharacteristic
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12311yescharacteristic
195C2×C3×C4C2×SD1611yescharacteristic
196C2×C3×C4#13511yescharacteristic
197C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
198C2×C3×C4#13711yescharacteristic
199#67—42no—
200#67—42no—
201C2×Dic3—22no—
202D12—42no—
203D12—41no—
204D12—41no—
205D12—81no—
206C4×D3—42no—
207C4×D3—42no—
208Dic6—41no—
209Dic6—41no—
210C3×C8—22no—
211C3×C8—41no—
212C3⋊C8—42no—
213C3⋊C8—42no—
214C3⋊C8—81no—
215#41—121no—
216#41—121no—
217#41—121no—
218#41—121no—
219C2×Q8—22no—
220C2×Q8—41no—
221C2×Q8—61no—
222C2×Q8—82no—
223C2×D4—32no—
224C2×D4—62no—
225C2×D4—61no—
226C2×D4—61no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—62no—
230C22×C4D3×D411yescharacteristic
231C22×C4#23711yescharacteristic
232C22×C4C2×#6711yescharacteristic
233SD16—244no—
234M16—124no—
235M16—121no—
236M16—121no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—62no—
243C2×C8—62no—
244C2×C8—124no—
245C2×C8—121no—
246C2×C8#21512yescomplemented
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—32no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—61no—
255C22⋊C4—62no—
256C22⋊C4—62no—
257C42C2×C4×D311yescharacteristic
258C42#23511yescharacteristic
259C42D3×D411yescharacteristic
260C42D3×D411yescharacteristic
261C22×C3—21no—
262C22×C3—21no—
263C22×C3#38611yescharacteristic
264C22×C3#46111yescharacteristic
265C22×C3#46311yescharacteristic
266C2×D3—42no—
267C2×D3—42no—
268C2×D3—42no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—82no—
274C3×C4—161no—
275C3×C4C4×SD1611yescharacteristic
276C3×C4#44011yescharacteristic
277C3×C4#45011yescharacteristic
278C3×C4#45211yescharacteristic
279Dic3—42no—
280C23—62no—
281C23#93811yescharacteristic
282Q8—44no—
283Q8—82no—
284Q8—121no—
285Q8—121no—
286D4—122no—
287D4—122no—
288D4—122no—
289D4—121no—
290D4—121no—
291D4—241no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—62no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—122no—
301C2×C4—122no—
302C2×C4D3×C22⋊C411yescharacteristic
303C2×C4#67⋊C411yescharacteristic
304C2×C4#88311yescharacteristic
305C2×C4#88411yescharacteristic
306C2×C4C4×#6711yescharacteristic
307C2×C4#93711yescharacteristic
308C2×C4C2×#21511yescharacteristic
309C2×C4#94211yescharacteristic
310C2×C4#94911yescharacteristic
311C2×C4#95011yescharacteristic
312C8—22no—
313C8—41no—
314C8—122no—
315C8—122no—
316C8—241no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3#187711yescharacteristic
320C2×C3#190211yescharacteristic
321C2×C3#191011yescharacteristic
322D3—82no—
323C22—21no—
324C22—21no—
325C22—122no—
326C22—122no—
327C22—122no—
328C22#495611yescharacteristic
329C22#505711yescharacteristic
330C22#508311yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—82no—
336C4—122no—
337C4—161no—
338C4C4×#21511yescharacteristic
339C4#502611yescharacteristic
340C4#504611yescharacteristic
341C4#504711yescharacteristic
342C3#928911yescharacteristic, complemented, Sylow
343C2—21no—
344C2—21no—
345C2—242no—
346C2#6968711yescharacteristic
347C2#7033011yescharacteristic
348C2#7034911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1993C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1993D3
hypocenterC3#9289
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
3. C3×#1993C2semidirect
4. #71775C2semidirect
6. #67660C2semidirect
7. #67631C2semidirect
16. #5085C4semidirect
18. #5079C4semidirect
30. #1993D3semidirect
116. C2×C3×C8SD16semidirect
246. C2×C8#215semidirect
342. C3#9289semidirect

Maximal subgroups

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

subgroupindex
#832132
#784912
C3×#19932
#717752
#695452
#676602
#676312
#92893