Finite Group Databasebeta

← #10696538

Subgroups — #10696538

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#83067C211yesmaximal, characteristic, complemented
2#83019C211yesmaximal, characteristic, complemented
3#78530C211yesmaximal, characteristic
4C3×#1528C211yesmaximal, characteristic, complemented
5#70832C211yesmaximal, characteristic, complemented
6#70818C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#10089—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10C2×#908—22no—
11#5477C2211yescharacteristic
12C2×#874C2211yescharacteristic
13C3×#432C2211yescharacteristic
14C3×C4×M16C2211yescharacteristic
15#5067—21no—
16#5066—22no—
17#5064—21no—
18#4991—22no—
19#4991—22no—
20#4988—21no—
21#4988—21no—
22#4968—21no—
23#4967—21no—
24#4932C2211yescharacteristic
25#4926C2211yescharacteristic
26#4712—22no—
27#4695—21no—
28#4693—21no—
29C3×#330—22no—
30#4485—22no—
31#4470—22no—
32#3234—31no—
33#3151—31no—
34#2641—31no—
35#1555—31no—
36#1528D311yescharacteristic, complemented
37#1505—31no—
38#1300—31no—
39C2×#235—21no—
40C22×D12—21no—
41C22×D12—21no—
42C22×D12D412yes—
43C2×C3×M16—21no—
44C3×#125—21no—
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C42C2311yescharacteristic
53#930—21no—
54#930—21no—
55C2×(C2×D3)⋊C4—24no—
56C2×(C2×D3)⋊C4—24no—
57C2×C4×Dic3—21no—
58#908—48no—
59C2×D24—24no—
60C2×#204—24no—
61#875—21no—
62#874—21no—
63#874—21no—
64#874—21no—
65#874—21no—
66#874D411yescharacteristic
67C4×D12—21no—
68C4×D12—21no—
69#869D411yescharacteristic
70C3×C4⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#97—21no—
73C3×#97D412yes—
74C3×C4×C8—21no—
75C3⋊#95—21no—
76C3⋊#95D412yes—
77#830—21no—
78#809—22no—
79#809—22no—
80#808—22no—
81#807—22no—
82C4⋊(C3⋊C8)D412yes—
83#803—21no—
84C2×AGL(1,ℤ/8)—62no—
85#532—31no—
86C2×#127—31no—
87C2×#125C2×D311yescharacteristic
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#432—31no—
97#432C2×D311yescharacteristic
98C4×M16C2×D311yescharacteristic
99#394—62no—
100#394—62no—
101#390—61no—
102#390—61no—
103#376—31no—
104#331—62no—
105#330—22no—
106#327—62no—
107C23×D3—22no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C2×#67—41no—
114C22×Dic3—22no—
115#235—44no—
116C2×D12—21no—
117C2×D12—22no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—22no—
121C2×D12—44no—
122C2×D12—44no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—82no—
128C2×C4×D3—41no—
129C2×Dic6—21no—
130C3×M16—24no—
131C2×C3×C8—22no—
132C2×C3×C8—41no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—84no—
148(C2×D3)⋊C4—84no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4×Dic3—42no—
152C2×C3⋊C8—22no—
153D24—48no—
154#204—48no—
155C2×#41—61no—
156C22×D4—32no—
157C22×D4—61no—
158C22×D4—61no—
159AGL(1,ℤ/8)—128no—
160C2×SD16—64no—
161C2×D8—64no—
162C2×M16—21no—
163C4⋊Q8—31no—
164#127—31no—
165#127—61no—
166#127—61no—
167#127—61no—
168#127—61no—
169#125—21no—
170#125—21no—
171#125—21no—
172#125—21no—
173#124—61no—
174#121—61no—
175#121—61no—
176C4×D4—61no—
177C4×D4—61no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4D1211yescharacteristic
181C2×C4⋊C4D1211yescharacteristic
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—32no—
190#102—62no—
191#102—62no—
192C22⋊C8—21no—
193C22⋊C8—61no—
194#97—21no—
195#97—61no—
196#97#6712yes—
197C4×C8—21no—
198#95—32no—
199#95—61no—
200C23×C3#12011yescharacteristic
201C22×D3—44no—
202C22×D3—44no—
203C22×D3—41no—
204C22×D3—42no—
205C22×D3—44no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12711yescharacteristic
219#67—82no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—41no—
223C2×Dic3—42no—
224D12—44no—
225D12—42no—
226D12—42no—
227D12—81no—
228D12—82no—
229D12—82no—
230D12—82no—
231D12—82no—
232D12—82no—
233C4×D3—82no—
234Dic6—42no—
235C3×C8—24no—
236C3×C8—81no—
237C3⋊C8—42no—
238C24—62no—
239#41—124no—
240C2×Q8—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—62no—
244C2×D4—61no—
245C2×D4—62no—
246C2×D4—124no—
247C2×D4—124no—
248C2×D4—121no—
249C2×D4—121no—
250C2×D4—121no—
251C2×D4—121no—
252C2×D4—121no—
253C2×D4—242no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—62no—
257C22×C4—121no—
258C22×C4C2×D1211yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261SD16—128no—
262D8—128no—
263M16—24no—
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—62no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—61no—
276C4⋊C4—61no—
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#39211yescharacteristic
289C22×C3#45312yes—
290C2×D3—88no—
291C2×D3—81no—
292C2×D3—82no—
293C2×D3—81no—
294C2×D3—82no—
295C2×D3—82no—
296C2×D3—81no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—81no—
303C3×C4—81no—
304C3×C4#45412yes—
305Dic3—82no—
306Dic3—81no—
307C23—124no—
308C23—124no—
309C23—121no—
310C23—122no—
311C23—124no—
312C23#88211yescharacteristic
313Q8—122no—
314D4—122no—
315D4—124no—
316D4—122no—
317D4—242no—
318D4—241no—
319D4—242no—
320D4—242no—
321D4—242no—
322D4—242no—
323D4—242no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—121no—
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—24no—
343C8—81no—
344C8—122no—
345C2×C3—22no—
346C2×C3#200211yescharacteristic
347C2×C3#200311yescharacteristic
348C2×C3#200811yescharacteristic
349D3—161no—
350D3—162no—
351C22—21no—
352C22—21no—
353C22—248no—
354C22—242no—
355C22—241no—
356C22—241no—
357C22—242no—
358C22—241no—
359C22—242no—
360C22#482212yes—
361C22#496011yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—81no—
368C4—81no—
369C4—242no—
370C4—241no—
371C4#475312yes—
372C3#1008911yescharacteristic, complemented, Sylow
373C2—22no—
374C2—482no—
375C2—481no—
376C2#7011611yescharacteristic
377C2#7038511yescharacteristic
378C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1528C2
socleC22×C3#392
prime radical#78530C2
hypercenter#1528D3
hypocenterC3#10089
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1528

Complements

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

NH ≅ G/Ntype
1. #83067C2semidirect
2. #83019C2semidirect
4. C3×#1528C2semidirect
5. #70832C2semidirect
6. #70818C2semidirect
7. #67665C2semidirect
36. #1528D3semidirect
372. C3#10089semidirect

Maximal subgroups

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

subgroupindex
#830672
#830192
#785302
C3×#15282
#708322
#708182
#676652
#100893