Finite Group Databasebeta

← #10706106

Subgroups — #10706106

Order 768.

418 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 79 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#82260C211yesmaximal, characteristic
2C3×#1458C211yesmaximal, characteristic, complemented
3#71589C211yesmaximal, characteristic, complemented
4#67457C212yesmaximal, complemented
5#67456C212yesmaximal, complemented
6#11851—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#875C411yescharacteristic
9C3×#431—21no—
10#5213—21no—
11#5207—21no—
12#4991C2211yescharacteristic
13#4990—21no—
14#4979C2211yescharacteristic
15#4960C411yescharacteristic
16#4957—21no—
17#4947C411yescharacteristic
18#4940—21no—
19#4932C411yescharacteristic
20C3×#324C2212yes—
21C3×#323—21no—
22#4530—21no—
23#4530C2212yes—
24#2610—31no—
25#1822—31no—
26#1458D311yescharacteristic, complemented
27#1354—32no—
28#1345—32no—
29C23×C3×C4D411yescharacteristic
30C22×#67—21no—
31C22×D12C2×C411yescharacteristic
32C22×Dic6C2×C411yescharacteristic
33C3×#117—41no—
34C3×#117—41no—
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C4⋊C4C2311yescharacteristic
37C2×C3×C22⋊C4C2×C411yescharacteristic
38C2×C3×C22⋊C4D411yescharacteristic
39C2×C3×C42C2×C411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×#217—21no—
42C2×(C2×D3)⋊C4—21no—
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—21no—
49C2×#208—22no—
50#875—41no—
51#875—41no—
52C3×C4⋊C8—41no—
53C3×C22⋊C8—22no—
54C3×C4×C8—41no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95C2×C411yescharacteristic
59C3⋊#95C2×C411yescharacteristic
60#830—41no—
61#830—41no—
62C2×#124—31no—
63C2×#117C2×D311yescharacteristic
64#431—21no—
65#407—61no—
66#395—31no—
67#394—31no—
68#393—31no—
69#392—31no—
70#390—61no—
71#388—61no—
72#388—61no—
73#386—61no—
74#376—31no—
75#343—32no—
76#343—61no—
77#324C2×D312yes—
78#323—21no—
79C24×C3C22⋊C411yescharacteristic
80C23×D3—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—22no—
84C22×C3×C4—41no—
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22×C411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C2×#67—42no—
92C2×#67—42no—
93C2×#67—81no—
94C22×Dic3—21no—
95C22×Dic3—21no—
96C22×Dic3—21no—
97C2×D12—41no—
98C2×D12—81no—
99C2×Dic6—41no—
100C2×Dic6—81no—
101C2×C3×C8—42no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—41no—
106C3×C42—21no—
107C3×C42—21no—
108C3×C42—21no—
109C3×C42—21no—
110#217—81no—
111(C2×D3)⋊C4—42no—
112(C2×D3)⋊C4—42no—
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4—41no—
115(C2×D3)⋊C4—42no—
116(C2×D3)⋊C4—81no—
117Dic3⋊C4—81no—
118C4×Dic3—42no—
119#208—42no—
120#208—42no—
121C2×C3⋊C8—42no—
122C22×Q8—31no—
123C22×D4—31no—
124C22×D4—61no—
125C23×C4D1211yescharacteristic
126C2×M16—62no—
127#124—121no—
128#124—121no—
129#117—41no—
130#117—41no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4D1211yescharacteristic
133C2×C4⋊C4C22×D311yescharacteristic
134C2×C22⋊C4—61no—
135C2×C22⋊C4—61no—
136C2×C22⋊C4—61no—
137C2×C22⋊C4—61no—
138C2×C22⋊C4—61no—
139C2×C22⋊C4—61no—
140C2×C22⋊C4C4×D311yescharacteristic
141C2×C22⋊C4#6711yescharacteristic
142C2×C42—61no—
143C2×C42C4×D311yescharacteristic
144C2×C42#6711yescharacteristic
145C4⋊C8—41no—
146C22⋊C8—22no—
147C22⋊C8—121no—
148C22⋊C8—121no—
149C4×C8—41no—
150#95—31no—
151#95—31no—
152#95—61no—
153#95—61no—
154#95—61no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—21no—
158C23×C3—41no—
159C23×C3—41no—
160C23×C3C2×C22⋊C411yescharacteristic
161C22×D3—41no—
162C22×D3—41no—
163C22×D3—41no—
164C22×D3—41no—
165C22×D3—41no—
166C22×D3—41no—
167C22×D3—41no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4#9911yescharacteristic
183C2×C3×C4#9911yescharacteristic
184C2×C3×C4#13512yes—
185#67—84no—
186C2×Dic3—22no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195D12—82no—
196Dic6—82no—
197C3×C8—82no—
198C3⋊C8—82no—
199C24—61no—
200C24(C2×D3)⋊C411yescharacteristic
201C2×Q8—121no—
202C2×Q8—241no—
203C2×D4—121no—
204C2×D4—122no—
205C2×D4—122no—
206C2×D4—241no—
207C2×D4—241no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—41no—
212C22×C4—61no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4(C2×D3)⋊C411yescharacteristic
216C22×C4(C2×D3)⋊C411yescharacteristic
217C22×C4(C2×D3)⋊C411yescharacteristic
218C22×C4C2×C4×D311yescharacteristic
219C22×C4C2×D1211yescharacteristic
220C22×C4C2×#6711yescharacteristic
221M16—122no—
222M16—122no—
223C2×C8—42no—
224C2×C8—122no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—241no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—122no—
231C22⋊C4—122no—
232C22⋊C4—121no—
233C22⋊C4—121no—
234C22⋊C4—122no—
235C22⋊C4—241no—
236C22⋊C4—241no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—122no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—42no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—81no—
249C22×C3C2×#9911yescharacteristic
250C22×C3#41611yescharacteristic
251C22×C3#42111yescharacteristic
252C2×D3—81no—
253C2×D3—81no—
254C2×D3—81no—
255C2×D3—81no—
256C2×D3—81no—
257C2×D3—81no—
258C2×D3—81no—
259C3×C4—22no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—41no—
266Dic3—42no—
267Dic3—81no—
268Dic3—81no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—41no—
273C23—41no—
274C23—121no—
275C23—121no—
276C23—121no—
277C23—121no—
278C23—121no—
279C23—121no—
280C23—121no—
281C23C2×(C2×D3)⋊C411yescharacteristic
282Q8—242no—
283D4—242no—
284D4—244no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—62no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4#80611yescharacteristic
309C2×C4#80611yescharacteristic
310C2×C4#90412yes—
311C8—82no—
312C8—242no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—81no—
316C2×C3#151611yescharacteristic
317C2×C3#212211yescharacteristic
318C2×C3#213211yescharacteristic
319D3—161no—
320C22—21no—
321C22—21no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—241no—
328C22—241no—
329C22—241no—
330C22—241no—
331C22—241no—
332C22—241no—
333C22—241no—
334C22C2×#80611yescharacteristic
335C22#511311yescharacteristic
336C22#513911yescharacteristic
337C4—22no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—122no—
345C4—241no—
346C4—241no—
347C3#1185111yescharacteristic, complemented, Sylow
348C2—21no—
349C2—21no—
350C2—81no—
351C2—481no—
352C2#6994311yescharacteristic
353C2#7005711yescharacteristic
354C2#7014811yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1458C2
socleC22×C3C2×#99
prime radical#4960C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1458D3
hypocenterC3#11851
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. C3×#1458C2semidirect
3. #71589C2semidirect
4. #67457C2semidirect
5. #67456C2semidirect
26. #1458D3semidirect
347. C3#11851semidirect

Maximal subgroups

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

subgroupindex
#822602
C3×#14582
#715892
#674572
#674562
#118513