Finite Group Databasebeta

← #10707897

Subgroups — #10707897

Order 768.

380 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 55 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#78323C211yesmaximal, characteristic, complemented
2C3×#1616C211yesmaximal, characteristic
3#71881C211yesmaximal, characteristic
4#71874C211yesmaximal, characteristic
5#67652C211yesmaximal, characteristic, complemented
6#67458C211yesmaximal, characteristic, complemented
7#67455C211yesmaximal, characteristic, complemented
8#12201—31nomaximal, Sylow
9C3×#537C2211yescharacteristic
10C2×#951—21no—
11C3×#481—21no—
12C3×#400C2211yescharacteristic
13C2×#837—21no—
14#5033—21no—
15#4986—21no—
16#4981C2211yescharacteristic
17#4973C2211yescharacteristic
18C3⋊#401—21no—
19#4661—21no—
20#4660—21no—
21#4656—21no—
22#4652—21no—
23C3×#327—21no—
24C3×#324C2211yescharacteristic
25#4537—21no—
26#4530C2211yescharacteristic
27#4528C2211yescharacteristic
28#2833—31no—
29#2007—31no—
30#1616D311yescharacteristic
31#1610—31no—
32#1353—31no—
33#1346—31no—
34#1319—31no—
35C2×C3×#41—21no—
36C22×Dic6D411yescharacteristic
37C3×C22⋊Q8—41no—
38C3×#121—21no—
39C3×#121—21no—
40C3×#121—41no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
43#951—42no—
44#951—42no—
45C2×#216—41no—
46C2×#214—41no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3D411yescharacteristic
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3D411yescharacteristic
53C2×#208—21no—
54C3×#106—41no—
55C3×#102—41no—
56C3×C22⋊C8—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95D411yescharacteristic
60#837—42no—
61#832—41no—
62C3⋊#95—21no—
63C3⋊#95—21no—
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—
73C2×#137—61no—
74#537C2×D311yescharacteristic
75#481—21no—
76#480—61no—
77C2×#104—61no—
78#401—61no—
79#400—61no—
80#400—61no—
81#400C2×D311yescharacteristic
82#398—31no—
83#398—61no—
84#395—61no—
85#391—31no—
86#387—61no—
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×Dic6—21no—
112C2×Dic6—21no—
113C2×Dic6—41no—
114C2×Dic6—41no—
115C2×C3×C8—41no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—41no—
122#216—44no—
123#214—44no—
124C4⋊Dic3—41no—
125C4⋊Dic3—81no—
126C4⋊Dic3—81no—
127Dic3⋊C4—41no—
128Dic3⋊C4—41no—
129Dic3⋊C4—81no—
130C4×Dic3—41no—
131#208—22no—
132#208—41no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8—41no—
135C2×#41—21no—
136C22×Q8—31no—
137#137—122no—
138#137—122no—
139C2×Q16—121no—
140C2×SD16—121no—
141C2×M16—61no—
142C22⋊Q8—41no—
143#121—21no—
144#121—21no—
145#121—41no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4#6711yescharacteristic
152C2×C4⋊C4C22×D311yescharacteristic
153C2×C42—31no—
154#107—121no—
155#106—41no—
156#104—122no—
157#102—41no—
158#102—121no—
159C22⋊C8—21no—
160C22⋊C8—61no—
161#95—21no—
162#95—21no—
163#95—31no—
164#95—61no—
165#95—61no—
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—42no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—81no—
196C2×C3×C4—81no—
197C2×C3×C4#13511yescharacteristic
198C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211Dic6—42no—
212Dic6—81no—
213C3×C8—81no—
214C3⋊C8—42no—
215C3⋊C8—81no—
216#41—41no—
217#41—41no—
218#41—41no—
219#41—41no—
220C2×Q8—21no—
221C2×Q8—61no—
222C2×Q8—61no—
223C2×Q8—121no—
224C2×Q8—121no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—41no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4D3×D411yescharacteristic
237C22×C4D3×D411yescharacteristic
238C22×C4C2×#6711yescharacteristic
239Q16—124no—
240SD16—124no—
241M16—62no—
242M16—121no—
243C2×C8—41no—
244C2×C8—121no—
245C2×C8—121no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4—241no—
254C4⋊C4—241no—
255C4⋊C4—241no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C42—121no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—81no—
262C22×C3—81no—
263C22×C3—81no—
264C22×C3#44911yescharacteristic
265C22×C3#45311yescharacteristic
266C22×C3B(4,𝔽2)11yescharacteristic
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—81no—
272C3×C4—81no—
273C3×C4—81no—
274Dic3—81no—
275Dic3—81no—
276Dic3—81no—
277Dic3—81no—
278C23—41no—
279C23#93711yescharacteristic
280Q8—41no—
281Q8—41no—
282Q8—122no—
283Q8—241no—
284D4—41no—
285D4—41no—
286D4—81no—
287D4—81no—
288D4—81no—
289D4—81no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4#91411yescharacteristic
317C2×C4#91611yescharacteristic
318C8—81no—
319C8—122no—
320C8—241no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—161no—
324C2×C3#160011yescharacteristic
325C2×C3#220111yescharacteristic
326C2×C3#220311yescharacteristic
327C22—21no—
328C22—21no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22#503211yescharacteristic
333C22#517211yescharacteristic
334C22#519811yescharacteristic
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—241no—
343C4—241no—
344C4—241no—
345C4—241no—
346C3#1220111yescharacteristic, complemented, Sylow
347C2—21no—
348C2—21no—
349C2—161no—
350C2#7097711yescharacteristic
351C2#7098911yescharacteristic
352C2#7100311yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1616C2
socleC22×C3B(4,𝔽2)
prime radicalC3×#537C22
hypercenter#1616D3
hypocenterC3#12201
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1616

Complements

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

NH ≅ G/Ntype
1. #78323C2semidirect
5. #67652C2semidirect
6. #67458C2semidirect
7. #67455C2semidirect
346. C3#12201semidirect

Maximal subgroups

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

subgroupindex
#783232
C3×#16162
#718812
#718742
#676522
#674582
#674552
#122013