Finite Group Databasebeta

← #10692841

Subgroups — #10692841

Order 768.

516 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#2921C211yesmaximal, characteristic, complemented
2#82257C211yesmaximal, characteristic
3#71594C211yesmaximal, characteristic, complemented
4#69978C212yesmaximal
5#67682C212yesmaximal, complemented
6#9446—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C3×#433—24no—
10C3×#423—21no—
11C3×#407C2212yes—
12C3×C4×M16—21no—
13#5211C2212yes—
14C4×#217C2211yescharacteristic
15#4988C411yescharacteristic, complemented
16#4988C411yescharacteristic, complemented
17#4977C411yescharacteristic, complemented
18#4975C411yescharacteristic, complemented
19#4959—21no—
20#4959—21no—
21#4940—21no—
22#4940—21no—
23#4526—21no—
24#4526—21no—
25#4470—24no—
26#2921D311yescharacteristic, complemented
27#2500—31no—
28#1820—31no—
29#1510—32no—
30#1284—32no—
31C23×C3×C4C2311yescharacteristic
32C22×#67—21no—
33C22×D12C2×C411yescharacteristic
34C2×C3×M16—22no—
35C3×#117—22no—
36C3×#117—22no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C22⋊C4D411yescharacteristic
40C2×C3×C22⋊C4D411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×#217—21no—
44C2×#217—22no—
45#930—22no—
46#930—22no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×C4⋊Dic3C2×C411yescharacteristic
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×C4×Dic3C2×C411yescharacteristic
56C2×C4×Dic3C2×C411yescharacteristic
57C3×C4⋊C8—24no—
58C3×C22⋊C8—22no—
59C3×C22⋊C8D414yes—
60C3×#97—22no—
61C3×C4×C8—22no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—22no—
66C3⋊#95—22no—
67C2×#121—31no—
68C2×#117C2×D311yescharacteristic
69#433—24no—
70#423—21no—
71#407C2×D312yes—
72C4×M16—21no—
73#394—61no—
74#394—61no—
75#390—31no—
76#390—31no—
77#388—61no—
78#388—61no—
79#384—31no—
80#382—31no—
81#381—32no—
82C4×C22⋊C4—31no—
83#342—61no—
84#342—61no—
85#327—64no—
86C24×C3C2×D411yescharacteristic
87C23×D3—21no—
88C22×C3×C4—41no—
89C22×C3×C4—41no—
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D412yes—
97C22×C3×C4#4112yes—
98C2×#67—42no—
99C2×#67—81no—
100C2×#67—81no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C22×Dic3—42no—
105C2×D12—22no—
106C2×D12—41no—
107C2×D12—41no—
108C3×M16—42no—
109C3×M16—42no—
110C2×C3×C8—24no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—22no—
113C3×C22⋊C4—22no—
114C3×C22⋊C4—22no—
115C3×C42—21no—
116C3×C42—21no—
117C3×C42—21no—
118C3×C42—21no—
119#217—42no—
120#217—42no—
121#217—42no—
122(C2×D3)⋊C4—42no—
123(C2×D3)⋊C4—42no—
124(C2×D3)⋊C4—41no—
125(C2×D3)⋊C4—41no—
126(C2×D3)⋊C4—81no—
127(C2×D3)⋊C4—81no—
128C4⋊Dic3—22no—
129Dic3⋊C4—81no—
130Dic3⋊C4—81no—
131C4×Dic3—41no—
132C4×Dic3—41no—
133C22×D4—31no—
134C22×D4—61no—
135C23×C4C22×D311yescharacteristic
136C2×M16—22no—
137#121—62no—
138#121—62no—
139#117—22no—
140#117—22no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4C4×D311yescharacteristic
145C2×C4⋊C4C4×D311yescharacteristic
146C2×C22⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—62no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4D1211yescharacteristic
154C2×C22⋊C4D1211yescharacteristic
155C2×C42—31no—
156C2×C42—31no—
157C2×C42#6711yescharacteristic
158C2×C42#6711yescharacteristic
159C4⋊C8—24no—
160C22⋊C8—22no—
161C22⋊C8#6714yes—
162#97—22no—
163C4×C8—22no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95—62no—
168#95—62no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3C2×C22⋊C411yescharacteristic
174C23×C3#12112yes—
175C22×D3—41no—
176C22×D3—41no—
177C22×D3—41no—
178C22×D3—41no—
179C22×D3—41no—
180C22×D3—41no—
181C22×D3—41no—
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—42no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4C4×D412yes—
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12311yescharacteristic
199#67—84no—
200C2×Dic3—22no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—82no—
210C2×Dic3—82no—
211C2×Dic3—82no—
212D12—82no—
213C3×C8—44no—
214C24—61no—
215C24C2×D1211yescharacteristic
216C2×D4—62no—
217C2×D4—121no—
218C2×D4—121no—
219C2×D4—122no—
220C2×D4—241no—
221C2×D4—241no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—122no—
228C22×C4(C2×D3)⋊C411yescharacteristic
229C22×C4(C2×D3)⋊C411yescharacteristic
230C22×C4(C2×D3)⋊C411yescharacteristic
231C22×C4(C2×D3)⋊C411yescharacteristic
232C22×C4C2×C4×D311yescharacteristic
233C22×C4#23512yes—
234C22×C4C2×#6711yescharacteristic
235C22×C4C2×#6712yes—
236M16—42no—
237M16—42no—
238C2×C8—24no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—62no—
242C4⋊C4—241no—
243C4⋊C4—241no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—122no—
247C22⋊C4—122no—
248C22⋊C4—122no—
249C22⋊C4—121no—
250C22⋊C4—121no—
251C22⋊C4—122no—
252C22⋊C4—122no—
253C22⋊C4—241no—
254C22⋊C4—241no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—121no—
260C42—121no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—81no—
267C22×C3—81no—
268C22×C3#38611yescharacteristic
269C22×C3#41011yescharacteristic
270C22×C3#41011yescharacteristic
271C2×D3—81no—
272C2×D3—81no—
273C2×D3—81no—
274C2×D3—81no—
275C2×D3—81no—
276C2×D3—81no—
277C2×D3—81no—
278C3×C4—21no—
279C3×C4—22no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4—41no—
285Dic3—42no—
286Dic3—81no—
287Dic3—81no—
288Dic3—162no—
289C23—21no—
290C23—21no—
291C23—41no—
292C23—41no—
293C23—121no—
294C23—121no—
295C23—121no—
296C23—121no—
297C23—121no—
298C23—121no—
299C23—121no—
300C23C2×(C2×D3)⋊C411yescharacteristic
301C23#93012yes—
302D4—242no—
303D4—244no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—62no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—242no—
328C2×C4—242no—
329C2×C4—242no—
330C2×C4C4×#6712yes—
331C2×C4#92911yescharacteristic
332C2×C4#95311yescharacteristic
333C8—44no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—42no—
337C2×C3#188411yescharacteristic
338C2×C3#189911yescharacteristic
339C2×C3#191011yescharacteristic
340D3—161no—
341C22—21no—
342C22—21no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—81no—
347C22—81no—
348C22—241no—
349C22—241no—
350C22—241no—
351C22—241no—
352C22—241no—
353C22—241no—
354C22—241no—
355C22#500611yescharacteristic
356C22#500611yescharacteristic
357C22#521311yescharacteristic
358C4—21no—
359C4—22no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—122no—
366C4—241no—
367C4—241no—
368C4—482no—
369C3#944611yescharacteristic, complemented, Sylow
370C2—21no—
371C2—21no—
372C2—42no—
373C2—481no—
374C2#7159511yescharacteristic
375C2#7202111yescharacteristic
376C2#7202511yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2921C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#2921D3
hypocenterC3#9446
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2921

Complements

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

NH ≅ G/Ntype
1. C3×#2921C2semidirect
3. #71594C2semidirect
5. #67682C2semidirect
15. #4988C4semidirect
16. #4988C4semidirect
17. #4977C4semidirect
18. #4975C4semidirect
26. #2921D3semidirect
369. C3#9446semidirect

Maximal subgroups

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

subgroupindex
C3×#29212
#822572
#715942
#699782
#676822
#94463