Finite Group Databasebeta

← #10692085

Subgroups — #10692085

Order 768.

612 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 97 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#83353C211yesmaximal, characteristic
2#73751C211yesmaximal, characteristic
3C3×#1986C211yesmaximal, characteristic, complemented
4#71836C211yesmaximal, characteristic, complemented
5#69855C211yesmaximal, characteristic
6#67665C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#9323—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#893—21no—
11C2×C4×D12C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5087C412yescomplemented
15#5083C412yescomplemented
16#5046—21no—
17#5044—21no—
18#5026—22no—
19Dic3×C4⋊C4—21no—
20C4×(C2×D3)⋊C4—22no—
21#4932C2211yescharacteristic
22#4926C2211yescharacteristic
23#4926C2211yescharacteristic
24#4666—21no—
25D3×#95—21no—
26C3×#339—22no—
27#4538—21no—
28#4538—21no—
29#4459—22no—
30#3110—31no—
31#2275—31no—
32#1986D311yescharacteristic, complemented
33#1506—31no—
34#1490—31no—
35#1300—31no—
36#1285—31no—
37C22×D12D411yescharacteristic
38C22×C4×D3—22no—
39C22×C4×D3—21no—
40C2×C3×M16—21no—
41C3×C4⋊Q8C2×C411yescharacteristic, complemented
42C3×C4⋊Q8C2×C411yescharacteristic, complemented
43C3×#125—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×#117—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42D412yes—
51C2×C3×C42C2311yescharacteristic
52C2×#215—22no—
53C2×(C2×D3)⋊C4—22no—
54C2×(C2×D3)⋊C4—21no—
55C2×C4⋊Dic3D411yescharacteristic
56C2×C4×Dic3—21no—
57C2×C4×Dic3—22no—
58#893—28no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12C2×C412yes—
62C4×D12D412yes—
63C3×#97—21no—
64C3×C4×C8—21no—
65C3×#95—21no—
66#835—22no—
67C3⋊#95—22no—
68C3⋊#95—21no—
69C3⋊#95D411yescharacteristic
70C3⋊#95D411yescharacteristic
71#830—21no—
72#830—21no—
73#809—22no—
74#807—22no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)C2×C412yescomplemented
77#803—21no—
78#803—21no—
79#533C2×D311yescharacteristic
80C2×C4×D4—31no—
81C2×C4×D4—61no—
82#484—61no—
83#474—32no—
84#461—32no—
85#450—61no—
86#440—62no—
87#432—31no—
88#432—31no—
89C4×M16C2×D311yescharacteristic
90C4×C4⋊C4—61no—
91C4×C4⋊C4C2×D311yescharacteristic
92C4×C22⋊C4—61no—
93C4×C22⋊C4—62no—
94#376—31no—
95C2×#95—61no—
96#339—22no—
97#328—61no—
98#328—61no—
99D4⋊C8—62no—
100C23×D3—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C22×C3×C4—22no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C2×D12—21no—
111C2×D12—24no—
112C2×D12C22⋊C412yes—
113C2×C4×D3—28no—
114C2×C4×D3—22no—
115C2×C4×D3—22no—
116C2×C4×D3—82no—
117C2×C4×D3—82no—
118C2×C4×D3—82no—
119C3×M16—22no—
120C3×M16—41no—
121C2×C3×C8—41no—
122C3×C4⋊C4—24no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—41no—
129C3×C22⋊C4—41no—
130C3×C42—42no—
131C3×C42—41no—
132C3×C42C22×C411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135C3×C42#4111yescharacteristic
136#215—44no—
137(C2×D3)⋊C4—28no—
138(C2×D3)⋊C4—22no—
139(C2×D3)⋊C4—22no—
140C4⋊Dic3—21no—
141C4⋊Dic3C22⋊C412yes—
142C4×Dic3—24no—
143C4×Dic3—82no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—41no—
146C22×D4—31no—
147C23×C4—62no—
148C23×C4—61no—
149C2×SD16—62no—
150C2×M16—21no—
151C4⋊Q8C4×D311yescharacteristic, complemented
152C4⋊Q8C4×D311yescharacteristic, complemented
153#125—21no—
154C22⋊Q8—21no—
155C22⋊Q8—21no—
156C4×Q8—21no—
157C4×Q8—21no—
158C4×D4—32no—
159C4×D4—32no—
160C4×D4—68no—
161C4×D4—61no—
162C4×D4—61no—
163#117—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—31no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—61no—
168C2×C42—61no—
169C2×C42—62no—
170C2×C42#6712yes—
171C2×C42C22×D311yescharacteristic
172#107—62no—
173C4⋊C8—32no—
174C4⋊C8—61no—
175#103—62no—
176#102—62no—
177C22⋊C8—61no—
178C22⋊C8—61no—
179#97—21no—
180#97—61no—
181#97—61no—
182C4×C8—21no—
183#95—21no—
184#95—31no—
185#95—31no—
186#95—62no—
187#95—61no—
188C23×C3#12011yescharacteristic
189C22×D3—24no—
190C22×D3—22no—
191C22×D3—42no—
192C22×D3—41no—
193C22×D3—41no—
194C3×Q8—42no—
195C3×Q8—81no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—22no—
199C2×C3×C4—24no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4C2×C22⋊C411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12311yescharacteristic
211C2×Dic3—22no—
212C2×Dic3—22no—
213C2×Dic3—22no—
214C2×Dic3—41no—
215C2×Dic3—42no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219D12—24no—
220D12—22no—
221D12#10414yes—
222C4×D3—82no—
223C4×D3—82no—
224C4×D3—88no—
225C3×C8—42no—
226C3⋊C8—42no—
227C3⋊C8—81no—
228C24—61no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×D4—32no—
232C2×D4—64no—
233C2×D4—61no—
234C22×C4—22no—
235C22×C4—68no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—62no—
240C22×C4—62no—
241C22×C4—242no—
242C22×C4—242no—
243C22×C4—242no—
244C22×C4D3×D411yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4C2×#6711yescharacteristic
247SD16—124no—
248M16—22no—
249M16—41no—
250C2×C8—41no—
251C2×C8—62no—
252C2×C8—121no—
253C4⋊C4—24no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—32no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—61no—
262C22⋊C4—41no—
263C22⋊C4—68no—
264C22⋊C4—62no—
265C22⋊C4—62no—
266C42—42no—
267C42—41no—
268C42—64no—
269C42—242no—
270C42C2×C4×D311yescharacteristic
271C42#23511yescharacteristic
272C42D3×D411yescharacteristic
273C42#23711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3#38611yescharacteristic
277C22×C3#45511yescharacteristic
278C22×C3#45611yescharacteristic
279C2×D3—24no—
280C2×D3—44no—
281C2×D3—44no—
282C2×D3—42no—
283C2×D3—42no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—44no—
290C3×C4—81no—
291C3×C4—81no—
292C3×C4C2×#10412yes—
293Dic3—42no—
294Dic3—42no—
295Dic3—42no—
296C23—64no—
297C23—62no—
298C23—122no—
299C23—121no—
300C23—121no—
301C23#93711yescharacteristic
302Q8—42no—
303Q8—81no—
304D4—34no—
305D4—64no—
306D4—62no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—62no—
317C2×C4—62no—
318C2×C4—62no—
319C2×C4—121no—
320C2×C4—122no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—242no—
325C2×C4—242no—
326C2×C4—248no—
327C2×C4D3×C22⋊C411yescharacteristic
328C2×C4#67⋊C411yescharacteristic
329C2×C4#88311yescharacteristic
330C2×C4#88411yescharacteristic
331C2×C4C4×#6711yescharacteristic
332C2×C4#93811yescharacteristic
333C8—42no—
334C8—122no—
335C8—241no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3#188211yescharacteristic
339C2×C3#189611yescharacteristic
340C2×C3#190411yescharacteristic
341D3—44no—
342C22—21no—
343C22—21no—
344C22—64no—
345C22—124no—
346C22—124no—
347C22—122no—
348C22—122no—
349C22#495611yescharacteristic
350C22#506111yescharacteristic
351C22#509511yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—44no—
358C4—81no—
359C4—81no—
360C4—122no—
361C4—122no—
362C4—122no—
363C4D3×#10412yes—
364C3#932311yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—124no—
368C2#6969311yescharacteristic
369C2#7038211yescharacteristic
370C2#7040711yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1986C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1986D3
hypocenterC3#9323
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
3. C3×#1986C2semidirect
4. #71836C2semidirect
6. #67665C2semidirect
7. #67621C2semidirect
14. #5087C4semidirect
15. #5083C4semidirect
32. #1986D3semidirect
41. C3×C4⋊Q8C2×C4semidirect
42. C3×C4⋊Q8C2×C4semidirect
76. C4⋊(C3⋊C8)C2×C4semidirect
151. C4⋊Q8C4×D3semidirect
152. C4⋊Q8C4×D3semidirect
364. C3#9323semidirect

Maximal subgroups

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

subgroupindex
#833532
#737512
C3×#19862
#718362
#698552
#676652
#676212
#93233