Finite Group Databasebeta

← #10707621

Subgroups — #10707621

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
2C3×#1610C211yesmaximal, characteristic
3#71877C211yesmaximal, characteristic
4#71874C211yesmaximal, characteristic
5#67650C211yesmaximal, characteristic, complemented
6#67458C211yesmaximal, characteristic, complemented
7#67452C211yesmaximal, characteristic, complemented
8#12157—31nomaximal, Sylow
9C3×#537C2211yescharacteristic
10C2×#889—21no—
11C3×#480—21no—
12C3×#398C2211yescharacteristic
13C2×#837—21no—
14#5232—21no—
15#5033—21no—
16#4982—21no—
17#4982—21no—
18#4981C2211yescharacteristic
19#4977C2211yescharacteristic
20#4660—21no—
21#4658—21no—
22#4657—21no—
23C3×#327—21no—
24C3×#324C2211yescharacteristic
25#4537—21no—
26#4530C2211yescharacteristic
27#4528C2211yescharacteristic
28#2822—31no—
29#2040—31no—
30#1610—31no—
31#1610D311yescharacteristic
32#1346—31no—
33#1346—31no—
34#1324—31no—
35C2×C3×#41—21no—
36C3×C22⋊Q8—41no—
37C3×#121—21no—
38C3×#121—21no—
39C3×#121—41no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3D411yescharacteristic
45C2×C4⋊Dic3D411yescharacteristic
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×C4×Dic3D411yescharacteristic
52C2×#208—21no—
53#889—42no—
54#889—42no—
55C3×#107—41no—
56C3×#102—41no—
57C3×C22⋊C8—21no—
58C3×#95—21no—
59C3×#95D411yescharacteristic
60#837—42no—
61#835—41no—
62#832—41no—
63#832—41no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95D411yescharacteristic
71#830—21no—
72#807—41no—
73#537C2×D311yescharacteristic
74C2×#125—61no—
75#480—21no—
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#343—31no—
88#327—21no—
89#327—61no—
90#324—31no—
91#324C2×D311yescharacteristic
92C3×#41—41no—
93C3×#41—41no—
94C3×#41—41no—
95C3×#41—41no—
96C2×C3×Q8—21no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4—21no—
100C2×C3×D4—41no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—41no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C2×C3×C8—41no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—41no—
120C4⋊Dic3—42no—
121C4⋊Dic3—41no—
122C4⋊Dic3—41no—
123Dic3⋊C4—42no—
124Dic3⋊C4—42no—
125Dic3⋊C4—42no—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128Dic3⋊C4—81no—
129C4×Dic3—21no—
130C4×Dic3D812yes—
131#208—41no—
132#208—41no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8—41no—
135C2×#41—21no—
136C2×M16—61no—
137#125—122no—
138#125—122no—
139C22⋊Q8—41no—
140#121—21no—
141#121—21no—
142#121—41no—
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⋊C4#6711yescharacteristic
153C2×C4⋊C4C22×D311yescharacteristic
154C2×C42—31no—
155#107—41no—
156#107—121no—
157#104—122no—
158#103—121no—
159#102—41no—
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#95#6711yescharacteristic
173C23×C3—41no—
174C23×C3#12011yescharacteristic
175C3×Q8—41no—
176C3×Q8—41no—
177C3×D4—41no—
178C3×D4—41no—
179C3×D4—81no—
180C3×D4—81no—
181C3×D4—81no—
182C3×D4—81no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—81no—
197C2×C3×C4—81no—
198C2×C3×C4C2×D811yescharacteristic
199C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
200C2×Dic3—22no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—42no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C3×C8—81no—
212C3⋊C8—81no—
213C3⋊C8—81no—
214#41—41no—
215#41—41no—
216#41—41no—
217#41—41no—
218C2×Q8—21no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4D3×D411yescharacteristic
231C22×C4D3×D411yescharacteristic
232C22×C4C2×#6711yescharacteristic
233M16—121no—
234M16—121no—
235C2×C8—41no—
236C2×C8—121no—
237C2×C8—121no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
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—41no—
254C22⋊C4—41no—
255C42—32no—
256C42—61no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—81no—
260C22×C3—81no—
261C22×C3—81no—
262C22×C3#44711yescharacteristic
263C22×C3#45411yescharacteristic
264C22×C3B(4,𝔽2)11yescharacteristic
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4—81no—
270C3×C4—81no—
271C3×C4—81no—
272Dic3—42no—
273Dic3—81no—
274Dic3—81no—
275Dic3—81no—
276C23—41no—
277C23#93711yescharacteristic
278Q8—41no—
279Q8—41no—
280D4—41no—
281D4—41no—
282D4—81no—
283D4—81no—
284D4—81no—
285D4—81no—
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—41no—
299C2×C4—62no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—122no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4D3×D811yescharacteristic
313C2×C4#91111yescharacteristic
314C8—81no—
315C8—241no—
316C8—241no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—161no—
320C2×C3#159811yescharacteristic
321C2×C3#220111yescharacteristic
322C2×C3#220311yescharacteristic
323C22—21no—
324C22—21no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22#503211yescharacteristic
329C22#515611yescharacteristic
330C22#520111yescharacteristic
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—122no—
339C4—241no—
340C4—241no—
341C4—241no—
342C3#1215711yescharacteristic, complemented, Sylow
343C2—21no—
344C2—21no—
345C2—161no—
346C2#7097111yescharacteristic
347C2#7099311yescharacteristic
348C2#7100311yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1610C2
socleC22×C3B(4,𝔽2)
prime radicalC3×#537C22
hypercenter#1610D3
hypocenterC3#12157
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1610

Complements

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

NH ≅ G/Ntype
1. #77927C2semidirect
5. #67650C2semidirect
6. #67458C2semidirect
7. #67452C2semidirect
342. C3#12157semidirect

Maximal subgroups

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

subgroupindex
#779272
C3×#16102
#718772
#718742
#676502
#674582
#674522
#121573