Finite Group Databasebeta

← #10684359

Subgroups — #10684359

Order 768.

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

normal subgroups only

This group has 404 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
1C3×#2727C211yesmaximal, characteristic
2#69640C211yesmaximal, characteristic
3#69611C211yesmaximal, characteristic, complemented
4#69608C211yesmaximal, characteristic, complemented
5#67639C211yesmaximal, characteristic
6#67638C211yesmaximal, characteristic, complemented
7#67638C211yesmaximal, characteristic, complemented
8#7808—31nomaximal, Sylow
9C2×C3×#126—21no—
10C3×#399—21no—
11C3×#396—21no—
12C3×#394—21no—
13C3×#390C2211yescharacteristic
14C3×#389—21no—
15C3×#389C2211yescharacteristic
16C3×#389C2211yescharacteristic
17C3×#388—21no—
18C2×#832—21no—
19C2×#808—21no—
20#4935C2211yescharacteristic
21#4529—21no—
22#4529—21no—
23#4529—21no—
24#4529—21no—
25#4524C2211yescharacteristic
26#4524C2211yescharacteristic
27#4524C2211yescharacteristic
28#2727D311yescharacteristic
29#2099—31no—
30#2084—31no—
31#2071—31no—
32#1271—31no—
33#1271—31no—
34#1270—31no—
35C22×C3×D4D411yescharacteristic
36C3×#126—42no—
37C3×#126—42no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C42—21no—
54C2×C3×C42C2311yescharacteristic
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C22×C3⋊C8—21no—
59C22×C3⋊C8—21no—
60C22×C3⋊C8—21no—
61C3×#95—21no—
62C3×#95—21no—
63C3×#95—21no—
64#832—42no—
65#808—42no—
66C2×#126—21no—
67C2×#106—61no—
68C2×#102—61no—
69#399—21no—
70#396—21no—
71#394—21no—
72#390C2×D311yescharacteristic
73#389—21no—
74#389C2×D311yescharacteristic
75#389C2×D311yescharacteristic
76#388—21no—
77#384—31no—
78#340—61no—
79#340—61no—
80#340—61no—
81#340—61no—
82#336—31no—
83#336—31no—
84#336—31no—
85C24×C3—21no—
86C2×C3×D4—21no—
87C2×C3×D4—21no—
88C2×C3×D4—41no—
89C2×C3×D4—41no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—41no—
99C3×C4⋊C4—21no—
100C3×C4⋊C4—21no—
101C3×C4⋊C4—21no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—42no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—81no—
111C3×C4⋊C4SD1612yes—
112C3×C22⋊C4—42no—
113C3×C22⋊C4—42no—
114C3×C22⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—41no—
118C3×C22⋊C4—81no—
119C3×C22⋊C4—81no—
120C3×C42—41no—
121C3×C42—42no—
122C4⋊Dic3—42no—
123C4⋊Dic3—41no—
124C4⋊Dic3—41no—
125C4⋊Dic3—41no—
126C4⋊Dic3—41no—
127C2×C3⋊C8—22no—
128C2×C3⋊C8—41no—
129C2×C3⋊C8—41no—
130C2×C3⋊C8—41no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134C2×C3⋊C8—41no—
135C2×C3⋊C8—41no—
136C22×D4#6711yescharacteristic
137C22×C8—61no—
138C22×C8—61no—
139C22×C8—61no—
140#126—42no—
141#126—42no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4#6711yescharacteristic
150C2×C4⋊C4#6711yescharacteristic
151C2×C4⋊C4#6711yescharacteristic
152C2×C4⋊C4#6711yescharacteristic
153C2×C4⋊C4#6711yescharacteristic
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C42—21no—
161C2×C42C22×D311yescharacteristic
162#106—122no—
163#102—122no—
164#95—21no—
165#95—21no—
166#95—21no—
167C23×C3—41no—
168C23×C3—41no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3#12011yescharacteristic
175C3×D4—42no—
176C3×D4—81no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—22no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4C2×SD1611yescharacteristic
199C2×C3×C4#13511yescharacteristic
200C2×C3×C4#13511yescharacteristic
201C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
202C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
203C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
204C2×Dic3—81no—
205C2×Dic3—81no—
206C2×Dic3—81no—
207C3⋊C8—42no—
208C3⋊C8—81no—
209C3⋊C8—81no—
210C24—21no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—41no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—121no—
221C22×C4C2×#6711yescharacteristic
222C22×C4C2×#6711yescharacteristic
223C22×C4C2×#6711yescharacteristic
224C2×C8—62no—
225C2×C8—121no—
226C2×C8—121no—
227C2×C8—121no—
228C2×C8—121no—
229C2×C8—121no—
230C2×C8—121no—
231C2×C8—121no—
232C2×C8—121no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—81no—
245C4⋊C4—122no—
246C4⋊C4—121no—
247C4⋊C4—121no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C4⋊C4#21412yes—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—81no—
258C22⋊C4—81no—
259C42—41no—
260C42—42no—
261C22×C3—81no—
262C22×C3—81no—
263C22×C3—81no—
264C22×C3—81no—
265C22×C3—81no—
266C22×C3—81no—
267C22×C3—81no—
268C22×C3#44911yescharacteristic
269C22×C3#44911yescharacteristic
270C22×C3#45011yescharacteristic
271C22×C3#45411yescharacteristic
272C22×C3#45511yescharacteristic
273C22×C3#45611yescharacteristic
274C22×C3#45811yescharacteristic
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—42no—
282C3×C4—81no—
283C3×C4—81no—
284C3×C4—81no—
285C3×C4—81no—
286Dic3—161no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#95311yescharacteristic
295D4—42no—
296D4—81no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—241no—
319C2×C4—241no—
320C2×C4—241no—
321C2×C4#93211yescharacteristic
322C2×C4C2×#21411yescharacteristic
323C2×C4#94911yescharacteristic
324C2×C4#94911yescharacteristic
325C2×C4#95011yescharacteristic
326C2×C4#95011yescharacteristic
327C8—122no—
328C8—241no—
329C8—241no—
330C2×C3—161no—
331C2×C3#160911yescharacteristic
332C2×C3#161211yescharacteristic
333C2×C3#161611yescharacteristic
334C2×C3#162811yescharacteristic
335C2×C3#164211yescharacteristic
336C2×C3#165411yescharacteristic
337C2×C3#166411yescharacteristic
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22#521811yescharacteristic
346C22#523811yescharacteristic
347C22#523811yescharacteristic
348C22#524211yescharacteristic
349C22#524311yescharacteristic
350C22#524511yescharacteristic
351C22#524611yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—481no—
364C3#780811yescharacteristic, complemented, Sylow
365C2—161no—
366C2#7187511yescharacteristic
367C2#7188211yescharacteristic
368C2#7188611yescharacteristic
369C2#7191311yescharacteristic
370C2#7192111yescharacteristic
371C2#7193111yescharacteristic
372C2#7195711yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2727C2
socleC23×C3#120
prime radicalC22×C3×D4D4
hypercenter#2727D3
hypocenterC3#7808
LCSG ▷ C2×C3×C42 ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2727

Complements

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

NH ≅ G/Ntype
3. #69611C2semidirect
4. #69608C2semidirect
6. #67638C2semidirect
7. #67638C2semidirect
364. C3#7808semidirect

Maximal subgroups

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

subgroupindex
C3×#27272
#696402
#696112
#696082
#676392
#676382
#78083