Finite Group Databasebeta

← #10696537

Subgroups — #10696537

Order 768.

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

normal subgroups only

This group has 628 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#83180C211yesmaximal, characteristic, complemented
2#83163C211yesmaximal, characteristic
3#79318C211yesmaximal, characteristic, complemented
4C3×#1555C211yesmaximal, characteristic, complemented
5#70832C211yesmaximal, characteristic, complemented
6#70812C211yesmaximal, characteristic, complemented
7#67748C211yesmaximal, characteristic, complemented
8#10089—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×#932—22no—
11#5477C2211yescharacteristic
12C2×#870C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×#432C2211yescharacteristic
15#5222—22no—
16#5222—22no—
17#5221—21no—
18#5221—21no—
19#5076—21no—
20#5074—22no—
21#5073—21no—
22#4932C2211yescharacteristic
23C4×#208C2211yescharacteristic
24#4847—22no—
25#4736—21no—
26#4735—21no—
27#4713—21no—
28#4702—21no—
29C3×#331—22no—
30#4470—22no—
31#4465—22no—
32#3234—31no—
33#3151—31no—
34#2641—31no—
35#1555D311yescharacteristic, complemented
36#1528—31no—
37#1505—31no—
38#1300—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D412yes—
42C2×#235—21no—
43C3×#127—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#217—24no—
49C2×#217—24no—
50C2×#214—24no—
51#932—48no—
52C2×#213—24no—
53#930—21no—
54#930—21no—
55C2×C4⋊Dic3D411yescharacteristic
56C2×C4⋊Dic3D411yescharacteristic
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×C4×Dic3—21no—
60C2×#208—21no—
61#875—21no—
62#874D411yescharacteristic
63C4×D12—21no—
64C4×D12—21no—
65#870—21no—
66#870—21no—
67#870—21no—
68#870—21no—
69#869D411yescharacteristic
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×#97—21no—
75C3×#97—21no—
76C3⋊#95—21no—
77C3⋊#95D412yes—
78#821—22no—
79#821—22no—
80#818—22no—
81#817—22no—
82#803D412yes—
83C4×C3⋊C8—21no—
84C2×AGL(1,ℤ/8)—62no—
85#532—31no—
86C2×#127C2×D311yescharacteristic
87C2×#125—31no—
88#501—61no—
89#496—62no—
90#493—61no—
91#492—61no—
92#490—61no—
93#481—61no—
94#480—61no—
95#478—62no—
96#432C2×D311yescharacteristic
97#432C2×D311yescharacteristic
98C4×M16—31no—
99#394—62no—
100#394—62no—
101#390—61no—
102#390—61no—
103#376—31no—
104#331—22no—
105#330—62no—
106#327—62no—
107C24×C3—22no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—44no—
111C2×C3×D4—44no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—82no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C2×#67—41no—
121C22×Dic3—22no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124#235—44no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×C4×D3—41no—
129C2×Dic6—21no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136#217—84no—
137#217—84no—
138#214—48no—
139#213—48no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148Dic3⋊C4—41no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151Dic3⋊C4—41no—
152C4×Dic3—42no—
153#208—24no—
154C2×C3⋊C8—22no—
155C2×#41—61no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4D1212yes—
159AGL(1,ℤ/8)—128no—
160C2×SD16—64no—
161C2×D8—64no—
162C2×M16—61no—
163C4⋊Q8—31no—
164#127—21no—
165#127—21no—
166#127—21no—
167#127—21no—
168#127—31no—
169#125—61no—
170#125—61no—
171#125—61no—
172#125—61no—
173#124—61no—
174#121—61no—
175#121—61no—
176C4×D4—61no—
177C4×D4—61no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C22⋊C4—64no—
183C2×C22⋊C4—64no—
184C2×C42—61no—
185C2×C42C22×D311yescharacteristic
186#107—62no—
187#106—62no—
188C4⋊C8—21no—
189C4⋊C8#6712yes—
190#102—62no—
191#102—62no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194#97—21no—
195#97—21no—
196#97—32no—
197C4×C8—61no—
198#95—32no—
199#95—61no—
200C23×C3—44no—
201C23×C3—44no—
202C23×C3—42no—
203C23×C3—44no—
204C23×C3#12011yescharacteristic
205C22×D3—41no—
206C3×D4—44no—
207C3×D4—82no—
208C3×D4—82no—
209C3×D4—82no—
210C3×D4—82no—
211C3×D4—82no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12711yescharacteristic
221#67—82no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230D12—42no—
231D12—42no—
232D12—81no—
233C4×D3—82no—
234Dic6—42no—
235C3×C8—42no—
236C3×C8—81no—
237C3⋊C8—24no—
238C24—22no—
239#41—124no—
240C2×Q8—61no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—82no—
253C2×D4—121no—
254C22×C4—62no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—121no—
258C22×C4C2×D1211yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261SD16—128no—
262D8—128no—
263M16—64no—
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—62no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C22⋊C4—244no—
280C22⋊C4—244no—
281C42—122no—
282C42#23511yescharacteristic
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C42C2×#6711yescharacteristic
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—88no—
289C22×C3—82no—
290C22×C3—82no—
291C22×C3—82no—
292C22×C3#39211yescharacteristic
293C22×C3#45312yes—
294C2×D3—81no—
295C2×D3—81no—
296C2×D3—81no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4#45412yes—
303Dic3—82no—
304Dic3—81no—
305Dic3—81no—
306Dic3—81no—
307C23—44no—
308C23—44no—
309C23—42no—
310C23—44no—
311C23—121no—
312C23#88211yescharacteristic
313Q8—122no—
314D4—44no—
315D4—82no—
316D4—82no—
317D4—82no—
318D4—82no—
319D4—82no—
320D4—122no—
321D4—122no—
322D4—241no—
323D4—242no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—242no—
336C2×C4#88211yescharacteristic
337C2×C4#88411yescharacteristic
338C2×C4#88411yescharacteristic
339C2×C4#93011yescharacteristic
340C2×C4#93711yescharacteristic
341C2×C4#94011yescharacteristic
342C8—42no—
343C8—64no—
344C8—81no—
345C2×C3—22no—
346C2×C3—162no—
347C2×C3#200211yescharacteristic
348C2×C3#200311yescharacteristic
349C2×C3#200811yescharacteristic
350D3—161no—
351C22—21no—
352C22—21no—
353C22—88no—
354C22—82no—
355C22—82no—
356C22—82no—
357C22—241no—
358C22—241no—
359C22—241no—
360C22#474712yes—
361C22#496011yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—242no—
368C4—241no—
369C4—241no—
370C4—241no—
371C4#520112yes—
372C3#1008911yescharacteristic, complemented, Sylow
373C2—22no—
374C2—162no—
375C2—481no—
376C2#7008211yescharacteristic
377C2#7032611yescharacteristic
378C2#7033111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1555C2
socleC22×C3#392
prime radical#83163C2
hypercenter#1555D3
hypocenterC3#10089
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1555

Complements

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

NH ≅ G/Ntype
1. #83180C2semidirect
3. #79318C2semidirect
4. C3×#1555C2semidirect
5. #70832C2semidirect
6. #70812C2semidirect
7. #67748C2semidirect
35. #1555D3semidirect
372. C3#10089semidirect

Maximal subgroups

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

subgroupindex
#831802
#831632
#793182
C3×#15552
#708322
#708122
#677482
#100893