Finite Group Databasebeta

← #10698684

Subgroups — #10698684

Order 768.

402 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 89 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#83055C211yesmaximal, characteristic
2C3×#1992C211yesmaximal, characteristic
3C2×#4543C211yesmaximal, characteristic
4C2×#4465C211yesmaximal, characteristic
5#69583C211yesmaximal, characteristic
6#69578C211yesmaximal, characteristic
7#67659C211yesmaximal, characteristic
8#10429—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×C3×#97C2211yescharacteristic
12C3×#384C2211yescharacteristic
13#4986—21no—
14#4985—21no—
15#4983—21no—
16#4982—21no—
17#4982—21no—
18#4981—21no—
19C2×#808—21no—
20C2×#807—21no—
21#4936C2211yescharacteristic
22C2×#803C2211yescharacteristic
23C2×#803C2211yescharacteristic
24C3×#340—21no—
25C3×#340—21no—
26#4543—22no—
27#4529—21no—
28#4529—21no—
29#4465—22no—
30#2822—31no—
31#1992—31no—
32#1992D311yescharacteristic
33#1847—31no—
34C2×#330—31no—
35C2×#330—31no—
36#1295—31no—
37C22×C3×C8—21no—
38C3×#125—21no—
39C3×#125—21no—
40C3×#125—21no—
41C3×#125—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C4⋊C4D411yescharacteristic
49C2×C3×C42C2311yescharacteristic
50C2×C4⋊Dic3D411yescharacteristic
51C2×C4⋊Dic3D411yescharacteristic
52C2×C4⋊Dic3D411yescharacteristic
53C2×C4⋊Dic3D411yescharacteristic
54C2×Dic3⋊C4—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C22×C3⋊C8—21no—
59C22×C3⋊C8—21no—
60#870—21no—
61#870—21no—
62#870—21no—
63#870—21no—
64C3×#97—21no—
65C3×#97—21no—
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74#808—42no—
75#807—42no—
76#803—21no—
77#803—21no—
78#803—21no—
79#803D412yes—
80C2×#125—31no—
81C2×#125C2×D311yescharacteristic
82C2×#107—61no—
83C2×#106—61no—
84C2×#97—31no—
85C2×#97—31no—
86C2×#97C2×D311yescharacteristic
87#400—61no—
88#400—61no—
89#398—61no—
90#398—61no—
91#398—61no—
92#398—61no—
93#384C2×D311yescharacteristic
94#382—31no—
95#340—21no—
96#340—21no—
97#340—61no—
98#340—61no—
99#330—62no—
100#330—62no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—41no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C2×C3×C8—22no—
112C2×C3×C8—41no—
113C2×C3×C8—41no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—42no—
121C3×C4⋊C4—42no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42#4111yescharacteristic
128C3×C42#4111yescharacteristic
129C3×C42#4111yescharacteristic
130C4⋊Dic3—21no—
131C4⋊Dic3—21no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136Dic3⋊C4—41no—
137Dic3⋊C4—41no—
138Dic3⋊C4—41no—
139Dic3⋊C4—41no—
140Dic3⋊C4—81no—
141Dic3⋊C4—81no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—41no—
147C2×C3⋊C8—41no—
148C22×C8—21no—
149C22×C8—61no—
150C22×C8—61no—
151#125—21no—
152#125—21no—
153#125—21no—
154#125—21no—
155#125—61no—
156#125—61no—
157#125—61no—
158#125—61no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4#6711yescharacteristic
173C2×C4⋊C4#6711yescharacteristic
174C2×C42C22×D311yescharacteristic
175#107—122no—
176#106—122no—
177#97—21no—
178#97—21no—
179#97—32no—
180#97—61no—
181#97—61no—
182#97—61no—
183#95—61no—
184#95—61no—
185#95—61no—
186#95—61no—
187#95—61no—
188#95—61no—
189#95—61no—
190#95—61no—
191C23×C3#12011yescharacteristic
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12311yescharacteristic
211C2×C3×C4#12311yescharacteristic
212C2×C3×C4#12411yescharacteristic
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C3×C8—42no—
226C3⋊C8—24no—
227C3⋊C8—42no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—41no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4D3×D411yescharacteristic
236C22×C4D3×D411yescharacteristic
237C22×C4C2×#6711yescharacteristic
238C2×C8—22no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—62no—
242C2×C8—62no—
243C2×C8—62no—
244C2×C8—62no—
245C2×C8—121no—
246C2×C8—121no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—241no—
270C4⋊C4—241no—
271C42#23511yescharacteristic
272C42#23511yescharacteristic
273C42D3×D411yescharacteristic
274C42#23711yescharacteristic
275C22×C3#39411yescharacteristic
276C22×C3#45311yescharacteristic
277C22×C3#45411yescharacteristic
278C22×C3#45411yescharacteristic
279C22×C3#45611yescharacteristic
280C22×C3#45611yescharacteristic
281C22×C3#45611yescharacteristic
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—81no—
289C3×C4—81no—
290C3×C4—161no—
291Dic3—81no—
292Dic3—81no—
293Dic3—81no—
294Dic3—81no—
295C23#93711yescharacteristic
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—121no—
312C2×C4—121no—
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—121no—
323C2×C4#88311yescharacteristic
324C2×C4#88411yescharacteristic
325C2×C4#88411yescharacteristic
326C2×C4#88511yescharacteristic
327C2×C4#92911yescharacteristic
328C2×C4#93911yescharacteristic
329C8—42no—
330C8—64no—
331C8—122no—
332C2×C3#165711yescharacteristic
333C2×C3#166011yescharacteristic
334C2×C3#166311yescharacteristic
335C2×C3#166311yescharacteristic
336C2×C3#203811yescharacteristic
337C2×C3#203811yescharacteristic
338C2×C3#203911yescharacteristic
339C22#495911yescharacteristic
340C22#506211yescharacteristic
341C22#509511yescharacteristic
342C22#519811yescharacteristic
343C22#520111yescharacteristic
344C22#520511yescharacteristic
345C22#520511yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—81no—
353C4—81no—
354C4—161no—
355C4—241no—
356C4—241no—
357C4—241no—
358C4—241no—
359C3#1042911yescharacteristic, complemented, Sylow
360C2#6969511yescharacteristic
361C2#7030911yescharacteristic
362C2#7031111yescharacteristic
363C2#7124911yescharacteristic
364C2#7125211yescharacteristic
365C2#7139111yescharacteristic
366C2#7139311yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1992C2
socle, prime radicalC23×C3#120
hypercenter#1992D3
hypocenterC3#10429
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
359. C3#10429semidirect

Maximal subgroups

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

subgroupindex
#830552
C3×#19922
C2×#45432
C2×#44652
#695832
#695782
#676592
#104293