Finite Group Databasebeta

← #10708346

Subgroups — #10708346

Order 768.

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

normal subgroups only

This group has 372 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#78198C211yesmaximal, characteristic
2C3×#1617C211yesmaximal, characteristic, complemented
3#71903C211yesmaximal, characteristic
4#71898C211yesmaximal, characteristic, complemented
5#67652C211yesmaximal, characteristic, complemented
6#67458C211yesmaximal, characteristic, complemented
7#67454C211yesmaximal, characteristic
8#12272—31nomaximal, Sylow
9C3×#544C2211yescharacteristic
10C3×#484—21no—
11C3×#400C2211yescharacteristic
12#5043—21no—
13#5001—21no—
14#4990—21no—
15#4988C2211yescharacteristic
16#4981C2211yescharacteristic
17#4672—21no—
18#4670—21no—
19#4669—21no—
20#4669—21no—
21#4649—21no—
22C3×#328—21no—
23C3×#324C2211yescharacteristic
24#4538—21no—
25#4532—21no—
26#4530C2211yescharacteristic
27#4528C2211yescharacteristic
28#2593—31no—
29#2027—31no—
30#1617D311yescharacteristic, complemented
31#1611—31no—
32#1353—31no—
33#1346—31no—
34#1318—31no—
35C22×D12D411yescharacteristic
36C3×C4⋊Q8—41no—
37C3×#125—41no—
38C3×C22⋊Q8—21no—
39C3×C22⋊Q8—21no—
40C3×#117—21no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×C4⋊Dic3D411yescharacteristic
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3—21no—
54C2×C4×Dic3D411yescharacteristic
55C2×#208—21no—
56C3×#107—41no—
57C3×#103—41no—
58C3×C22⋊C8—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95D411yescharacteristic
62#835—41no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95D411yescharacteristic
66#830—21no—
67#809—41no—
68#809—41no—
69#808—41no—
70#544C2×D311yescharacteristic
71#484—21no—
72#483—61no—
73#418—61no—
74#400C2×D311yescharacteristic
75#398—31no—
76#394—61no—
77#390—31no—
78#388—61no—
79#388—61no—
80#388—61no—
81#388—61no—
82C4×C4⋊C4—61no—
83#343—31no—
84#339—61no—
85#328—21no—
86#328—61no—
87#324—31no—
88#324C2×D311yescharacteristic
89C23×D3—21no—
90C2×C3×Q8—21no—
91C2×C3×Q8—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C2×D12—21no—
101C2×D12—21no—
102C2×D12—41no—
103C2×D12—41no—
104C2×C3×C8—41no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C22⋊C4—41no—
114C3×C42—41no—
115(C2×D3)⋊C4—42no—
116(C2×D3)⋊C4—42no—
117(C2×D3)⋊C4—42no—
118(C2×D3)⋊C4—42no—
119(C2×D3)⋊C4—81no—
120(C2×D3)⋊C4—81no—
121C4⋊Dic3—41no—
122Dic3⋊C4—42no—
123Dic3⋊C4—41no—
124Dic3⋊C4—41no—
125C4×Dic3—42no—
126C4×Dic3—41no—
127C4×Dic3—81no—
128#208—41no—
129#208—41no—
130C2×C3⋊C8—41no—
131C2×C3⋊C8—41no—
132C22×D4—31no—
133C2×M16—61no—
134C4⋊Q8—41no—
135#125—41no—
136C22⋊Q8—21no—
137C22⋊Q8—21no—
138#117—21no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4#6711yescharacteristic
143C2×C4⋊C4C22×D311yescharacteristic
144C2×C22⋊C4—61no—
145C2×C22⋊C4—61no—
146C2×C22⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C42—31no—
151C2×C42—61no—
152C2×C42—61no—
153#107—41no—
154#106—121no—
155#103—41no—
156#103—121no—
157#102—121no—
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#6711yescharacteristic
167C23×C3#12011yescharacteristic
168C22×D3—41no—
169C22×D3—41no—
170C22×D3—41no—
171C22×D3—41no—
172C22×D3—41no—
173C22×D3—41no—
174C22×D3—41no—
175C3×Q8—81no—
176C3×Q8—81no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
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×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4#13511yescharacteristic
192C2×C3×C4#13711yescharacteristic
193C2×Dic3—22no—
194C2×Dic3—22no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202D12—42no—
203D12—81no—
204C3×C8—81no—
205C3⋊C8—81no—
206C3⋊C8—81no—
207C24—61no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×D4—61no—
211C2×D4—61no—
212C2×D4—121no—
213C2×D4—121no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4—61no—
219C22×C4D3×D411yescharacteristic
220C22×C4D3×D411yescharacteristic
221C22×C4C2×#6711yescharacteristic
222M16—121no—
223M16—121no—
224C2×C8—41no—
225C2×C8—121no—
226C2×C8—121no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—121no—
236C4⋊C4—122no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C22⋊C4—41no—
240C22⋊C4—122no—
241C22⋊C4—122no—
242C22⋊C4—122no—
243C22⋊C4—122no—
244C22⋊C4—241no—
245C22⋊C4—241no—
246C42—41no—
247C42—121no—
248C42—122no—
249C42—241no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3#45211yescharacteristic
253C22×C3#45511yescharacteristic
254C22×C3B(4,𝔽2)11yescharacteristic
255C2×D3—81no—
256C2×D3—81no—
257C2×D3—81no—
258C2×D3—81no—
259C2×D3—81no—
260C2×D3—81no—
261C2×D3—81no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—81no—
267C3×C4—81no—
268C3×C4—81no—
269C3×C4—81no—
270Dic3—42no—
271Dic3—42no—
272Dic3—81no—
273C23—121no—
274C23—121no—
275C23—121no—
276C23—121no—
277C23—121no—
278C23—121no—
279C23—121no—
280C23#93711yescharacteristic
281Q8—81no—
282Q8—81no—
283D4—122no—
284D4—241no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
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—62no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4#91811yescharacteristic
309C2×C4#91911yescharacteristic
310C8—81no—
311C8—241no—
312C8—241no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3#160611yescharacteristic
316C2×C3#219811yescharacteristic
317C2×C3#220411yescharacteristic
318D3—161no—
319C22—21no—
320C22—21no—
321C22—241no—
322C22—241no—
323C22—241no—
324C22—241no—
325C22—241no—
326C22—241no—
327C22—241no—
328C22#503211yescharacteristic
329C22#518711yescharacteristic
330C22#520211yescharacteristic
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—122no—
340C4—122no—
341C4—241no—
342C3#1227211yescharacteristic, complemented, Sylow
343C2—21no—
344C2—21no—
345C2—481no—
346C2#7098111yescharacteristic
347C2#7098811yescharacteristic
348C2#7100211yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1617C2
socleC22×C3B(4,𝔽2)
prime radicalC22×D12D4
hypercenter#1617D3
hypocenterC3#12272
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1617

Complements

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

NH ≅ G/Ntype
2. C3×#1617C2semidirect
4. #71898C2semidirect
5. #67652C2semidirect
6. #67458C2semidirect
30. #1617D3semidirect
342. C3#12272semidirect

Maximal subgroups

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

subgroupindex
#781982
C3×#16172
#719032
#718982
#676522
#674582
#674542
#122723