Finite Group Databasebeta

← #10707624

Subgroups — #10707624

Order 768.

372 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 57 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#77927C211yesmaximal, characteristic, complemented
2#69712C211yesmaximal, characteristic
3#68977C211yesmaximal, characteristic
4#68135C211yesmaximal, characteristic
5C3×#1346C211yesmaximal, characteristic, complemented
6#67495C211yesmaximal, characteristic, complemented
7#67452C211yesmaximal, characteristic, complemented
8#12157—31nomaximal, Sylow
9#5510C2211yescharacteristic
10C2×#889—21no—
11C3×#398C2211yescharacteristic
12C2×#825—21no—
13#5133—21no—
14#4982—21no—
15#4982—21no—
16#4981C2211yescharacteristic
17#4977C2211yescharacteristic
18#4967—21no—
19#4736—21no—
20#4660—21no—
21#4658—21no—
22#4657—21no—
23C3×#343C2211yescharacteristic
24C3×#324C2211yescharacteristic
25#4528C2211yescharacteristic
26#4470—21no—
27#4451—21no—
28#2822—31no—
29#2040—31no—
30#1610—31no—
31#1610—31no—
32#1346—31no—
33#1346D311yescharacteristic, complemented
34#1324—31no—
35C2×#235—21no—
36C2×C3×M16—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C4⋊C4C2311yescharacteristic
40#930—21no—
41#930—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3D411yescharacteristic
44C2×C4⋊Dic3D411yescharacteristic
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×C4×Dic3D411yescharacteristic
51#897—41no—
52#895—41no—
53#889—42no—
54#889—42no—
55C3×C22⋊C8—21no—
56C3×#95—21no—
57C3×#95D411yescharacteristic
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95D411yescharacteristic
65#830—21no—
66#825—42no—
67#821—41no—
68#821—41no—
69#818—41no—
70#816—41no—
71#809—41no—
72#807—41no—
73#537—31no—
74C2×#125—61no—
75#480—61no—
76#480—61no—
77C2×#104—61no—
78#417—61no—
79#400—61no—
80#400—61no—
81#398—31no—
82#398—61no—
83#398—61no—
84#398—61no—
85#398C2×D311yescharacteristic
86#382—31no—
87#343C2×D311yescharacteristic
88#327—61no—
89#327—61no—
90#324—31no—
91#324C2×D311yescharacteristic
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C2×#67—41no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102#235—41no—
103#235—41no—
104#235—41no—
105#235—41no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12—21no—
109C2×C4×D3—41no—
110C2×Dic6—21no—
111C3×M16—41no—
112C3×M16—41no—
113C2×C3×C8—41no—
114C2×C3×C8—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119(C2×D3)⋊C4—41no—
120(C2×D3)⋊C4—41no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—42no—
124C4⋊Dic3—41no—
125C4⋊Dic3—41no—
126Dic3⋊C4—42no—
127Dic3⋊C4—42no—
128Dic3⋊C4—42no—
129Dic3⋊C4—41no—
130Dic3⋊C4—41no—
131Dic3⋊C4—81no—
132C4×Dic3—21no—
133C4×Dic3D812yes—
134C2×C3⋊C8—41no—
135C2×#41—61no—
136C2×M16—21no—
137#125—122no—
138#125—122no—
139C22⋊Q8—121no—
140#121—61no—
141#121—61no—
142#121—121no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4—61no—
152C2×C4⋊C4D1211yescharacteristic
153C2×C4⋊C4C22×D311yescharacteristic
154C2×C42—31no—
155#107—121no—
156#107—121no—
157#104—122no—
158#103—121no—
159#102—121no—
160#102—121no—
161#102—121no—
162C22⋊C8—21no—
163C22⋊C8—61no—
164#95—21no—
165#95—31no—
166#95—61no—
167#95—61no—
168#95—61no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95D1211yescharacteristic
173C23×C3#12011yescharacteristic
174C22×D3—41no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4C2×D811yescharacteristic
188C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
189#67—81no—
190#67—81no—
191C2×Dic3—22no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—42no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203D12—41no—
204D12—41no—
205D12—81no—
206D12—81no—
207C4×D3—81no—
208C4×D3—81no—
209Dic6—41no—
210Dic6—41no—
211C3×C8—81no—
212C3×C8—81no—
213C3⋊C8—81no—
214#41—121no—
215#41—121no—
216#41—121no—
217#41—121no—
218C2×Q8—61no—
219C2×D4—61no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4—121no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—121no—
230C22×C4C2×D1211yescharacteristic
231C22×C4D3×D411yescharacteristic
232C22×C4D3×D411yescharacteristic
233M16—41no—
234M16—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—121no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—121no—
245C4⋊C4—121no—
246C4⋊C4—122no—
247C4⋊C4—122no—
248C4⋊C4—122no—
249C4⋊C4—122no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—241no—
253C22⋊C4—121no—
254C22⋊C4—121no—
255C42—32no—
256C42—61no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3#44711yescharacteristic
260C22×C3#45411yescharacteristic
261C22×C3B(4,𝔽2)11yescharacteristic
262C2×D3—81no—
263C2×D3—81no—
264C2×D3—81no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4—81no—
270C3×C4—81no—
271Dic3—42no—
272Dic3—81no—
273Dic3—81no—
274Dic3—81no—
275Dic3—81no—
276C23—121no—
277C23#88211yescharacteristic
278Q8—121no—
279Q8—121no—
280D4—121no—
281D4—121no—
282D4—241no—
283D4—241no—
284D4—241no—
285D4—241no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—62no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—122no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—241no—
311C2×C4—241no—
312C2×C4D3×D811yescharacteristic
313C2×C4#91411yescharacteristic
314C8—81no—
315C8—81no—
316C8—241no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3#159811yescharacteristic
320C2×C3#220111yescharacteristic
321C2×C3#220311yescharacteristic
322D3—161no—
323C22—21no—
324C22—21no—
325C22—241no—
326C22—241no—
327C22—241no—
328C22#474111yescharacteristic
329C22#477311yescharacteristic
330C22#482711yescharacteristic
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—81no—
337C4—122no—
338C4—241no—
339C4—241no—
340C4—241no—
341C4—241no—
342C3#1215711yescharacteristic, complemented, Sylow
343C2—21no—
344C2—21no—
345C2—481no—
346C2#6829311yescharacteristic
347C2#6914111yescharacteristic
348C2#6916711yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1346C2
socleC22×C3B(4,𝔽2)
prime radical#5510C22
hypercenter#1346D3
hypocenterC3#12157
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1346

Complements

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

NH ≅ G/Ntype
1. #77927C2semidirect
5. C3×#1346C2semidirect
6. #67495C2semidirect
7. #67452C2semidirect
33. #1346D3semidirect
342. C3#12157semidirect

Maximal subgroups

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

subgroupindex
#779272
#697122
#689772
#681352
C3×#13462
#674952
#674522
#121573