Finite Group Databasebeta

← #10705706

Subgroups — #10705706

Order 768.

488 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 87 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#82182C211yesmaximal, characteristic
2C2×C3×#343C211yesmaximal, characteristic, complemented
3#71552C211yesmaximal, characteristic, complemented
4#67496C212yesmaximal, complemented
5#67493C212yesmaximal, complemented
6#11766—31nomaximal, Sylow
7C22×C3×M16—21no—
8C22×C3×C4⋊C4C2211yescharacteristic
9C2×#930C411yescharacteristic
10#5213—21no—
11#5213—21no—
12C4×#217C411yescharacteristic
13#5207—21no—
14#4991C411yescharacteristic
15#4991C2211yescharacteristic
16#4990—21no—
17#4990—21no—
18#4986C411yescharacteristic
19#4986C2211yescharacteristic
20C3×#343—21no—
21C3×#343C2212yes—
22#4528—21no—
23#4528C2212yes—
24#2591—31no—
25C2×#343D311yescharacteristic, complemented
26#1463—31no—
27#1353—32no—
28#1345—32no—
29C23×C3×C4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C22×#67—21no—
33C22×D12C2×C411yescharacteristic
34C2×C3×M16—24no—
35C2×C3×M16—22no—
36C2×C3×M16—41no—
37C2×C3×M16—42no—
38C22×C3×C8—41no—
39C2×C3×C4⋊C4—41no—
40C2×C3×C4⋊C4—41no—
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C4⋊C4C2311yescharacteristic
45C2×#217—21no—
46#930—41no—
47#930—41no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×C4×Dic3—21no—
55C2×#208—41no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95C2×C411yescharacteristic
61C3⋊#95C2×C411yescharacteristic
62#830—22no—
63#830—41no—
64C22×M16—21no—
65C2×#121—31no—
66C22×C4⋊C4C2×D311yescharacteristic
67#407—61no—
68#400—31no—
69#400—31no—
70#394—31no—
71#394—31no—
72#388—61no—
73#388—61no—
74#386—61no—
75#386—61no—
76C4×C22⋊C4—31no—
77#343—21no—
78#343C2×D312yes—
79#324—32no—
80#324—61no—
81C24×C3C22⋊C411yescharacteristic
82C23×D3—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—22no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—81no—
93C22×C3×C4—81no—
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22×C411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C2×#67—42no—
101C2×#67—42no—
102C2×#67—41no—
103C2×#67—41no—
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C2×D12—41no—
108C2×D12—81no—
109C3×M16—24no—
110C3×M16—44no—
111C3×M16—42no—
112C2×C3×C8—44no—
113C2×C3×C8—42no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118#217—42no—
119(C2×D3)⋊C4—42no—
120(C2×D3)⋊C4—42no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—41no—
123(C2×D3)⋊C4—81no—
124(C2×D3)⋊C4—81no—
125C4⋊Dic3—41no—
126C4×Dic3—81no—
127#208—82no—
128C2×C3⋊C8—42no—
129C22×D4—31no—
130C22×D4—61no—
131C23×C4C4×D311yescharacteristic
132C23×C4D1211yescharacteristic
133C23×C4#6711yescharacteristic
134C2×M16—24no—
135C2×M16—22no—
136C2×M16—41no—
137C2×M16—42no—
138C2×M16—121no—
139C22×C8—41no—
140#121—121no—
141#121—121no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4C4×D311yescharacteristic
146C2×C4⋊C4D1211yescharacteristic
147C2×C4⋊C4#6711yescharacteristic
148C2×C4⋊C4C22×D311yescharacteristic
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4—61no—
154C2×C22⋊C4—61no—
155C2×C42—61no—
156C22⋊C8—62no—
157C22⋊C8—121no—
158#95—31no—
159#95—31no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164C23×C3—22no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—41no—
168C23×C3#9911yescharacteristic
169C23×C3#9911yescharacteristic
170C23×C3C2×C22⋊C411yescharacteristic
171C22×D3—41no—
172C22×D3—41no—
173C22×D3—41no—
174C22×D3—41no—
175C22×D3—41no—
176C22×D3—41no—
177C22×D3—41no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
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—21no—
189C2×C3×C4—21no—
190C2×C3×C4—42no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—81no—
196C2×C3×C4—81no—
197C2×C3×C4—81no—
198C2×C3×C4#10412yes—
199C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
200C2×C3×C4#13711yescharacteristic
201#67—82no—
202#67—82no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212D12—82no—
213C3×C8—44no—
214C3⋊C8—82no—
215C24—61no—
216C24(C2×D3)⋊C411yescharacteristic
217C2×D4—121no—
218C2×D4—122no—
219C2×D4—122no—
220C2×D4—121no—
221C2×D4—121no—
222C2×D4—241no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4(C2×D3)⋊C411yescharacteristic
238C22×C4(C2×D3)⋊C411yescharacteristic
239C22×C4(C2×D3)⋊C411yescharacteristic
240C22×C4C2×C4×D311yescharacteristic
241C22×C4C2×D1211yescharacteristic
242C22×C4C2×#6711yescharacteristic
243M16—24no—
244M16—44no—
245M16—42no—
246M16—242no—
247C2×C8—44no—
248C2×C8—42no—
249C2×C8—122no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—121no—
255C22⋊C4—122no—
256C22⋊C4—122no—
257C22⋊C4—122no—
258C22⋊C4—121no—
259C22⋊C4—121no—
260C22⋊C4—241no—
261C22⋊C4—241no—
262C42—241no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—42no—
271C22×C3—42no—
272C22×C3#35512yes—
273C22×C3C2×#9911yescharacteristic
274C22×C3#41711yescharacteristic
275C22×C3C2×#10411yescharacteristic
276C2×D3—81no—
277C2×D3—81no—
278C2×D3—81no—
279C2×D3—81no—
280C2×D3—81no—
281C2×D3—81no—
282C2×D3—81no—
283C3×C4—21no—
284C3×C4—22no—
285C3×C4—21no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—41no—
290Dic3—81no—
291Dic3—81no—
292Dic3—81no—
293C23—21no—
294C23—21no—
295C23—22no—
296C23—41no—
297C23—121no—
298C23—121no—
299C23—121no—
300C23—121no—
301C23—121no—
302C23—121no—
303C23—121no—
304C23#80611yescharacteristic
305C23#80611yescharacteristic
306C23C2×(C2×D3)⋊C411yescharacteristic
307D4—242no—
308D4—242no—
309D4—242no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4#80512yes—
340C2×C4#93211yescharacteristic
341C2×C4#94211yescharacteristic
342C8—44no—
343C8—242no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—22no—
347C2×C3—41no—
348C2×C3#150411yescharacteristic
349C2×C3#212111yescharacteristic
350C2×C3C2×#35511yescharacteristic
351D3—161no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—42no—
360C22—42no—
361C22—241no—
362C22—241no—
363C22—241no—
364C22—241no—
365C22—241no—
366C22—241no—
367C22—241no—
368C22#447712yes—
369C22C2×#80511yescharacteristic
370C22C2×#80611yescharacteristic
371C22#496611yescharacteristic
372C4—21no—
373C4—22no—
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—241no—
380C4—241no—
381C4—241no—
382C3#1176611yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—22no—
386C2—41no—
387C2—481no—
388C2#6939811yescharacteristic
389C2#7009411yescharacteristic
390C2C2×#447711yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC2×C3×#343C2
socleC22×C3C2×#99
prime radicalC2×#930C4
cosocleC2×C3×C4⋊C4C23
hypercenterC2×#343D3
hypocenterC3#11766
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
2. C2×C3×#343C2semidirect
3. #71552C2semidirect
4. #67496C2semidirect
5. #67493C2semidirect
25. C2×#343D3semidirect
382. C3#11766semidirect

Maximal subgroups

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

subgroupindex
#821822
C2×C3×#3432
#715522
#674962
#674932
#117663