Finite Group Databasebeta

← #10706111

Subgroups — #10706111

Order 768.

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

normal subgroups only

This group has 438 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#82182C211yesmaximal, characteristic
2C2×C3×#324C211yesmaximal, characteristic, complemented
3#71562C211yesmaximal, characteristic, complemented
4#67459C212yesmaximal, complemented
5#67456C212yesmaximal, complemented
6#11852—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#930C411yescharacteristic
9C2×C3×C22⋊C8—21no—
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×#324—21no—
21C3×#324C2212yes—
22#4530—21no—
23#4530C2212yes—
24#2591—31no—
25#1821—31no—
26C2×#324D311yescharacteristic, complemented
27#1353—32no—
28#1345—32no—
29C23×C3×C4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C22×#67—21no—
33C22×D12C2×C411yescharacteristic
34C22×C3×C8—41no—
35C2×C3×C4⋊C4—41no—
36C2×C3×C4⋊C4—41no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C4⋊C4C2311yescharacteristic
41C2×#217—21no—
42#930—41no—
43#930—41no—
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⋊Dic3C2×C411yescharacteristic
50C2×C4×Dic3—21no—
51C2×#208—22no—
52C3×C22⋊C8—22no—
53C3×C22⋊C8—41no—
54C3⋊#95—21no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95C2×C411yescharacteristic
59C3⋊#95C2×C411yescharacteristic
60#830—41no—
61#830—41no—
62C2×#121—31no—
63C22×C4⋊C4C2×D311yescharacteristic
64#407—61no—
65C2×C22⋊C8—21no—
66#400—31no—
67#400—31no—
68#394—31no—
69#394—31no—
70#388—61no—
71#388—61no—
72#386—61no—
73#386—61no—
74C4×C22⋊C4—31no—
75#343—32no—
76#343—61no—
77#324—21no—
78#324C2×D312yes—
79C24×C3C22⋊C411yescharacteristic
80C23×D3—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—22no—
88C22×C3×C4—41no—
89C22×C3×C4—81no—
90C22×C3×C4—81no—
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C2×#67—42no—
98C2×#67—42no—
99C2×#67—41no—
100C2×#67—41no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C2×D12—41no—
105C2×D12—81no—
106C2×C3×C8—42no—
107C2×C3×C8—82no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112#217—42no—
113(C2×D3)⋊C4—42no—
114(C2×D3)⋊C4—42no—
115(C2×D3)⋊C4—42no—
116(C2×D3)⋊C4—81no—
117(C2×D3)⋊C4—81no—
118C4⋊Dic3—41no—
119C4×Dic3—81no—
120#208—42no—
121#208—42no—
122C2×C3⋊C8—42no—
123C22×D4—31no—
124C22×D4—61no—
125C23×C4C4×D311yescharacteristic
126C23×C4D1211yescharacteristic
127C23×C4#6711yescharacteristic
128C2×M16—62no—
129C22×C8—41no—
130#121—121no—
131#121—121no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4C4×D311yescharacteristic
136C2×C4⋊C4D1211yescharacteristic
137C2×C4⋊C4#6711yescharacteristic
138C2×C4⋊C4C22×D311yescharacteristic
139C2×C22⋊C4—61no—
140C2×C22⋊C4—61no—
141C2×C22⋊C4—61no—
142C2×C22⋊C4—61no—
143C2×C22⋊C4—61no—
144C2×C22⋊C4—61no—
145C2×C42—61no—
146C22⋊C8—22no—
147C22⋊C8—41no—
148C22⋊C8—121no—
149C22⋊C8—121no—
150#95—31no—
151#95—31no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156C23×C3—22no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—41no—
160C23×C3#9911yescharacteristic
161C23×C3#9911yescharacteristic
162C23×C3C2×C22⋊C411yescharacteristic
163C22×D3—41no—
164C22×D3—41no—
165C22×D3—41no—
166C22×D3—41no—
167C22×D3—41no—
168C22×D3—41no—
169C22×D3—41no—
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—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—81no—
187C2×C3×C4—81no—
188C2×C3×C4—81no—
189C2×C3×C4#13512yes—
190#67—82no—
191#67—82no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201D12—82no—
202C3×C8—82no—
203C3⋊C8—82no—
204C24—61no—
205C24(C2×D3)⋊C411yescharacteristic
206C2×D4—121no—
207C2×D4—122no—
208C2×D4—122no—
209C2×D4—121no—
210C2×D4—121no—
211C2×D4—241no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—41no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4(C2×D3)⋊C411yescharacteristic
226C22×C4(C2×D3)⋊C411yescharacteristic
227C22×C4(C2×D3)⋊C411yescharacteristic
228C22×C4C2×C4×D311yescharacteristic
229C22×C4C2×D1211yescharacteristic
230C22×C4C2×#6711yescharacteristic
231M16—122no—
232M16—122no—
233C2×C8—42no—
234C2×C8—82no—
235C2×C8—122no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—121no—
241C22⋊C4—122no—
242C22⋊C4—122no—
243C22⋊C4—122no—
244C22⋊C4—122no—
245C22⋊C4—241no—
246C22⋊C4—241no—
247C42—241no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3#35512yes—
258C22×C3C2×#9911yescharacteristic
259C22×C3#41611yescharacteristic
260C22×C3#42111yescharacteristic
261C2×D3—81no—
262C2×D3—81no—
263C2×D3—81no—
264C2×D3—81no—
265C2×D3—81no—
266C2×D3—81no—
267C2×D3—81no—
268C3×C4—22no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274Dic3—81no—
275Dic3—81no—
276Dic3—81no—
277C23—21no—
278C23—22no—
279C23—21no—
280C23—41no—
281C23—121no—
282C23—121no—
283C23—121no—
284C23—121no—
285C23—121no—
286C23—121no—
287C23—121no—
288C23#80611yescharacteristic
289C23#80611yescharacteristic
290C23C2×(C2×D3)⋊C411yescharacteristic
291D4—242no—
292D4—242no—
293D4—242no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4#90412yes—
323C8—82no—
324C8—242no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—22no—
328C2×C3—41no—
329C2×C3#151611yescharacteristic
330C2×C3#212111yescharacteristic
331C2×C3C2×#35511yescharacteristic
332D3—161no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—42no—
341C22—42no—
342C22—241no—
343C22—241no—
344C22—241no—
345C22—241no—
346C22—241no—
347C22—241no—
348C22—241no—
349C22#447712yes—
350C22C2×#80611yescharacteristic
351C22#511311yescharacteristic
352C22#513911yescharacteristic
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—241no—
360C4—241no—
361C4—241no—
362C3#1185211yescharacteristic, complemented, Sylow
363C2—21no—
364C2—21no—
365C2—22no—
366C2—41no—
367C2—481no—
368C2#6994311yescharacteristic
369C2#7001511yescharacteristic
370C2C2×#447711yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC2×C3×#324C2
socleC22×C3C2×#99
prime radicalC2×#930C4
cosocleC2×C3×C4⋊C4C23
hypercenterC2×#324D3
hypocenterC3#11852
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. C2×C3×#324C2semidirect
3. #71562C2semidirect
4. #67459C2semidirect
5. #67456C2semidirect
26. C2×#324D3semidirect
362. C3#11852semidirect

Maximal subgroups

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

subgroupindex
#821822
C2×C3×#3242
#715622
#674592
#674562
#118523