Finite Group Databasebeta

← #10691883

Subgroups — #10691883

Order 768.

594 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 107 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#83355C211yesmaximal, characteristic
2#74153C211yesmaximal, characteristic
3C3×#1518C211yesmaximal, characteristic, complemented
4#71840C211yesmaximal, characteristic, complemented
5#70774C211yesmaximal, characteristic
6#67737C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#9295—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#893—21no—
11C2×#874C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#382C2211yescharacteristic
14#5088C412yescomplemented
15#5085C412yescomplemented
16#5046—22no—
17#5027—22no—
18#4988—21no—
19#4932C2211yescharacteristic
20#4925C2211yescharacteristic
21#4924C2211yescharacteristic
22#4665—22no—
23#4665—22no—
24C3×#328—22no—
25#4532—21no—
26#4532—21no—
27#4485—22no—
28#3208—31no—
29#2328—31no—
30#1995—31no—
31#1545—31no—
32#1518D311yescharacteristic, complemented
33#1286—31no—
34#1274—31no—
35C22×D12—21no—
36C22×D12—21no—
37C22×D12D412yes—
38C22×C4×D3—22no—
39C3×C4⋊Q8C2×C411yescharacteristic, complemented
40C3×C4⋊Q8C2×C411yescharacteristic, complemented
41C3×#125—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×#117—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×#215—24no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×C4×Dic3D411yescharacteristic
54C2×C4×Dic3D411yescharacteristic
55C2×#208—21no—
56#893—42no—
57#893—42no—
58#874—21no—
59#874—21no—
60#874C2×C412yes—
61#874D412yes—
62C3×C4⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×C4×C8—21no—
65C3×#95—21no—
66C3×#95D412yes—
67C3⋊#95—21no—
68C3⋊#95—21no—
69#830—21no—
70#809—24no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8C2×C412yescomplemented
75#533C2×D311yescharacteristic
76C2×#127—31no—
77C2×C4×D4—61no—
78#492—32no—
79#486—32no—
80#460—62no—
81#450—62no—
82#431—31no—
83#431C2×D311yescharacteristic
84#423—31no—
85#390—61no—
86#386—62no—
87#386—62no—
88#382C2×D311yescharacteristic
89#376—31no—
90#339—61no—
91#339—61no—
92#331—62no—
93#328—22no—
94C23×D3—22no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C22×C3×C4—22no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×Dic3—21no—
102C22×Dic3—21no—
103C2×D12—22no—
104C2×D12—22no—
105C2×D12—22no—
106C2×D12—22no—
107C2×D12—22no—
108C2×D12—41no—
109C2×D12—41no—
110C2×D12—41no—
111C2×D12—41no—
112C2×D12—82no—
113C2×D12—82no—
114C2×D12C22⋊C414yes—
115C2×C4×D3—44no—
116C2×C4×D3—82no—
117C2×C4×D3—82no—
118C2×C3×C8—41no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C42—41no—
129C3×C42C22×C411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132C3×C42#4111yescharacteristic
133#215—84no—
134(C2×D3)⋊C4—44no—
135(C2×D3)⋊C4—82no—
136(C2×D3)⋊C4—82no—
137C4×Dic3—21no—
138C4×Dic3—21no—
139C4×Dic3—21no—
140C4×Dic3SD1612yescomplemented
141#208—41no—
142#208—41no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145C22×D4—32no—
146C22×D4—61no—
147C22×D4—61no—
148C23×C4—62no—
149C2×SD16—64no—
150C2×M16—61no—
151C4⋊Q8C4×D311yescharacteristic, complemented
152C4⋊Q8C4×D311yescharacteristic, complemented
153#127—32no—
154#127—32no—
155#127—61no—
156#127—61no—
157#125—21no—
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160C4×Q8—21no—
161C4×Q8—21no—
162C4×D4—122no—
163C4×D4—122no—
164#117—21no—
165C2×C4⋊C4—21no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—62no—
168C2×C22⋊C4—62no—
169C2×C42—31no—
170C2×C42—31no—
171C2×C42C22×D311yescharacteristic
172C4⋊C8—21no—
173C4⋊C8—61no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176#102—64no—
177C22⋊C8—21no—
178C22⋊C8—61no—
179C4×C8—21no—
180C4×C8—32no—
181#95—21no—
182#95—61no—
183#95—61no—
184#95#6712yes—
185C23×C3#12011yescharacteristic
186C22×D3—24no—
187C22×D3—42no—
188C22×D3—42no—
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—42no—
193C3×Q8—42no—
194C3×Q8—81no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4C2×C22⋊C411yescharacteristic
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12311yescharacteristic
208C2×C3×C4C2×SD1611yescharacteristic
209C2×C3×C4#13511yescharacteristic
210C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
211C2×C3×C4#13711yescharacteristic
212C2×Dic3—22no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218D12—44no—
219D12—42no—
220D12—42no—
221D12—42no—
222D12—42no—
223D12—82no—
224D12—82no—
225D12—82no—
226C4×D3—88no—
227C3×C8—81no—
228C3⋊C8—42no—
229C3⋊C8—81no—
230C24—62no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×D4—34no—
234C2×D4—62no—
235C2×D4—62no—
236C2×D4—62no—
237C2×D4—62no—
238C2×D4—62no—
239C2×D4—121no—
240C2×D4—121no—
241C2×D4—121no—
242C2×D4—121no—
243C2×D4—242no—
244C2×D4—242no—
245C22×C4—22no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—124no—
249C22×C4—242no—
250C22×C4—242no—
251C22×C4D3×D411yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4C2×#6711yescharacteristic
254SD16—244no—
255M16—121no—
256M16—121no—
257C2×C8—41no—
258C2×C8—62no—
259C2×C8—121no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—124no—
270C22⋊C4—242no—
271C22⋊C4—242no—
272C42—32no—
273C42—41no—
274C42—61no—
275C42—61no—
276C42—61no—
277C42C2×C4×D311yescharacteristic
278C42#23511yescharacteristic
279C42D3×D411yescharacteristic
280C42#23711yescharacteristic
281C22×C3—21no—
282C22×C3—21no—
283C22×C3#38611yescharacteristic
284C22×C3#45011yescharacteristic
285C22×C3#45211yescharacteristic
286C2×D3—44no—
287C2×D3—44no—
288C2×D3—44no—
289C2×D3—82no—
290C2×D3—82no—
291C2×D3—82no—
292C2×D3—82no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—82no—
298C3×C4—81no—
299C3×C4—81no—
300C3×C4C4×SD1611yescharacteristic
301C3×C4#44011yescharacteristic
302C3×C4#46111yescharacteristic
303C3×C4#46311yescharacteristic
304Dic3—42no—
305Dic3—81no—
306C23—64no—
307C23—122no—
308C23—122no—
309C23—122no—
310C23—122no—
311C23—122no—
312C23—122no—
313C23#93711yescharacteristic
314Q8—42no—
315Q8—81no—
316D4—124no—
317D4—122no—
318D4—122no—
319D4—122no—
320D4—122no—
321D4—242no—
322D4—242no—
323D4—242no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—62no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—248no—
338C2×C4D3×C22⋊C411yescharacteristic
339C2×C4#67⋊C411yescharacteristic
340C2×C4#88311yescharacteristic
341C2×C4#88411yescharacteristic
342C2×C4D3×SD1611yescharacteristic
343C2×C4#91411yescharacteristic
344C2×C4#91811yescharacteristic
345C2×C4#91911yescharacteristic
346C2×C4C4×#6711yescharacteristic
347C2×C4#93811yescharacteristic
348C8—81no—
349C8—122no—
350C8—241no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3#187711yescharacteristic
354C2×C3#190411yescharacteristic
355C2×C3#191111yescharacteristic
356D3—84no—
357C22—21no—
358C22—21no—
359C22—124no—
360C22—124no—
361C22—124no—
362C22—242no—
363C22—242no—
364C22—242no—
365C22—242no—
366C22#495611yescharacteristic
367C22#516911yescharacteristic
368C22#518711yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—82no—
374C4—81no—
375C4—81no—
376C4—122no—
377C4—241no—
378C4#478811yescharacteristic
379C4#481711yescharacteristic
380C4#481811yescharacteristic
381C4#481911yescharacteristic
382C3#929511yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—244no—
386C2#7004911yescharacteristic
387C2#7027611yescharacteristic
388C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1518C2
socleC22×C3#386
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1518D3
hypocenterC3#9295
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
3. C3×#1518C2semidirect
4. #71840C2semidirect
6. #67737C2semidirect
7. #67631C2semidirect
14. #5088C4semidirect
15. #5085C4semidirect
32. #1518D3semidirect
39. C3×C4⋊Q8C2×C4semidirect
40. C3×C4⋊Q8C2×C4semidirect
74. C4×C3⋊C8C2×C4semidirect
140. C4×Dic3SD16semidirect
151. C4⋊Q8C4×D3semidirect
152. C4⋊Q8C4×D3semidirect
382. C3#9295semidirect

Maximal subgroups

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

subgroupindex
#833552
#741532
C3×#15182
#718402
#707742
#677372
#676312
#92953