Finite Group Databasebeta

← #10705704

Subgroups — #10705704

Order 768.

470 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 83 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×#1464C211yesmaximal, characteristic, complemented
3#71587C211yesmaximal, characteristic, complemented
4#67457C212yesmaximal, complemented
5#67456C212yesmaximal, complemented
6#11765—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C22×#208—21no—
9C2×#875C411yescharacteristic
10C3×#432—21no—
11#5213—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#1742—31no—
26#1464D311yescharacteristic, 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—24no—
50C2×#208—22no—
51C2×#208—41no—
52C2×#208—42no—
53C22×C3⋊C8—41no—
54#875—41no—
55#875—41no—
56C3×C4⋊C8—41no—
57C3×C22⋊C8—22no—
58C3×#97—41no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95C2×C411yescharacteristic
63C3⋊#95C2×C411yescharacteristic
64C22×M16—61no—
65C2×#124—31no—
66C2×#117C2×D311yescharacteristic
67#432—21no—
68#395—31no—
69#394—31no—
70#393—31no—
71#392—31no—
72#390—61no—
73#388—61no—
74#388—61no—
75#386—61no—
76#376—31no—
77#343—32no—
78#343—61no—
79#324C2×D312yes—
80#323—21no—
81C24×C3C22⋊C411yescharacteristic
82C23×D3—21no—
83C22×C3×C4—22no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C2×#67—42no—
95C2×#67—42no—
96C2×#67—81no—
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C2×D12—41no—
101C2×D12—81no—
102C2×Dic6—41no—
103C2×Dic6—81no—
104C2×C3×C8—42no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—41no—
107C3×C22⋊C4—41no—
108C3×C22⋊C4—41no—
109C3×C42—21no—
110C3×C42—21no—
111C3×C42—21no—
112C3×C42—21no—
113#217—81no—
114(C2×D3)⋊C4—42no—
115(C2×D3)⋊C4—42no—
116(C2×D3)⋊C4—41no—
117(C2×D3)⋊C4—41no—
118(C2×D3)⋊C4—41no—
119(C2×D3)⋊C4—41no—
120(C2×D3)⋊C4—81no—
121Dic3⋊C4—81no—
122C4×Dic3—42no—
123#208—24no—
124#208—44no—
125#208—42no—
126C2×C3⋊C8—44no—
127C2×C3⋊C8—42no—
128C22×Q8—31no—
129C22×D4—31no—
130C22×D4—61no—
131C23×C4D1211yescharacteristic
132C2×M16—64no—
133C2×M16—62no—
134C2×M16—121no—
135C2×M16—122no—
136C22×C8—121no—
137#124—121no—
138#124—121no—
139#117—41no—
140#117—41no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4D1211yescharacteristic
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×C22⋊C4C4×D311yescharacteristic
151C2×C22⋊C4#6711yescharacteristic
152C2×C42—61no—
153C2×C42C4×D311yescharacteristic
154C2×C42#6711yescharacteristic
155C4⋊C8—41no—
156C22⋊C8—22no—
157#97—41no—
158#95—31no—
159#95—31no—
160#95—61no—
161#95—61no—
162#95—61no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—41no—
166C23×C3—41no—
167C23×C3#10412yes—
168C23×C3C2×C22⋊C411yescharacteristic
169C22×D3—41no—
170C22×D3—41no—
171C22×D3—41no—
172C22×D3—41no—
173C22×D3—41no—
174C22×D3—41no—
175C22×D3—41no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4#9911yescharacteristic
193C2×C3×C4#9911yescharacteristic
194C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
195C2×C3×C4#13711yescharacteristic
196#67—84no—
197C2×Dic3—22no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206D12—82no—
207Dic6—82no—
208C3×C8—82no—
209C3⋊C8—44no—
210C24—61no—
211C24(C2×D3)⋊C411yescharacteristic
212C2×Q8—121no—
213C2×Q8—241no—
214C2×D4—121no—
215C2×D4—122no—
216C2×D4—122no—
217C2×D4—241no—
218C2×D4—241no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4(C2×D3)⋊C411yescharacteristic
228C22×C4(C2×D3)⋊C411yescharacteristic
229C22×C4(C2×D3)⋊C411yescharacteristic
230C22×C4C2×C4×D311yescharacteristic
231C22×C4C2×D1211yescharacteristic
232C22×C4C2×#6711yescharacteristic
233M16—64no—
234M16—124no—
235M16—122no—
236C2×C8—42no—
237C2×C8—124no—
238C2×C8—122no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—241no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—122no—
245C22⋊C4—122no—
246C22⋊C4—121no—
247C22⋊C4—121no—
248C22⋊C4—121no—
249C22⋊C4—121no—
250C22⋊C4—241no—
251C22⋊C4—241no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—122no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—22no—
260C22×C3—42no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—81no—
264C22×C3C2×#9911yescharacteristic
265C22×C3#41711yescharacteristic
266C22×C3C2×#10411yescharacteristic
267C2×D3—81no—
268C2×D3—81no—
269C2×D3—81no—
270C2×D3—81no—
271C2×D3—81no—
272C2×D3—81no—
273C2×D3—81no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—41no—
282Dic3—42no—
283Dic3—81no—
284Dic3—81no—
285C23—21no—
286C23—21no—
287C23—41no—
288C23—41no—
289C23—121no—
290C23—121no—
291C23—121no—
292C23—121no—
293C23—121no—
294C23—121no—
295C23—121no—
296C23#82512yes—
297C23C2×(C2×D3)⋊C411yescharacteristic
298Q8—242no—
299D4—242no—
300D4—244no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—62no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4#80611yescharacteristic
327C2×C4#80611yescharacteristic
328C2×C4#90811yescharacteristic
329C2×C4#90911yescharacteristic
330C8—82no—
331C8—124no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—42no—
335C2×C3#150411yescharacteristic
336C2×C3#212211yescharacteristic
337C2×C3#213211yescharacteristic
338D3—161no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—241no—
347C22—241no—
348C22—241no—
349C22—241no—
350C22—241no—
351C22—241no—
352C22—241no—
353C22C2×#80611yescharacteristic
354C22#513311yescharacteristic
355C22C2×#82511yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—122no—
365C4—241no—
366C4—241no—
367C3#1176511yescharacteristic, complemented, Sylow
368C2—21no—
369C2—21no—
370C2—42no—
371C2—481no—
372C2#6997611yescharacteristic
373C2#7005711yescharacteristic
374C2#7014811yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1464C2
socleC22×C3C2×#99
prime radical#4960C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1464D3
hypocenterC3#11765
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. C3×#1464C2semidirect
3. #71587C2semidirect
4. #67457C2semidirect
5. #67456C2semidirect
26. #1464D3semidirect
367. C3#11765semidirect

Maximal subgroups

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

subgroupindex
#822602
C3×#14642
#715872
#674572
#674562
#117653