Finite Group Databasebeta

← #10691111

Subgroups — #10691111

Order 768.

494 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 105 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#83350C211yesmaximal, characteristic
2#73645C211yesmaximal, characteristic
3C3×#1503C211yesmaximal, characteristic, complemented
4#71838C211yesmaximal, characteristic, complemented
5#70756C211yesmaximal, characteristic
6#67738C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9177—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5087C412yescomplemented
14#5084C412yescomplemented
15#5066—22no—
16#5001—21no—
17#4995—21no—
18#4990—22no—
19#4939—21no—
20#4936C2211yescharacteristic
21C4×C4⋊Dic3—21no—
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4666—21no—
25#4664—21no—
26C3×#328—21no—
27C3×#328—21no—
28#4538—22no—
29#4483—22no—
30#3077—31no—
31#2357—31no—
32#1518—31no—
33#1503D311yescharacteristic, complemented
34#1478—31no—
35#1303—31no—
36#1286—31no—
37C22×D12D411yescharacteristic
38C22×C4×D3—21no—
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×C4⋊Q8C2×C411yescharacteristic
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×C42—21no—
49C2×C3×C42C2311yescharacteristic
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—22no—
52C2×(C2×D3)⋊C4—22no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3D411yescharacteristic
55C2×C4×Dic3—22no—
56C2×#208—21no—
57C4×D12—21no—
58C4×D12—21no—
59C4×D12C2×C412yes—
60C4×D12D412yes—
61C3×C4⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C4×C8—21no—
64C3×#95D411yescharacteristic
65C3×#95D411yescharacteristic
66C3⋊#95—22no—
67C3⋊#95—21no—
68C3⋊#95D412yes—
69#830—21no—
70#809—22no—
71#809—22no—
72#808—22no—
73#807—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)C2×C412yes—
77C4×C3⋊C8—21no—
78#533C2×D311yescharacteristic
79C2×C4×D4—31no—
80#478—62no—
81#476—32no—
82#474—32no—
83#431—31no—
84#431C2×D311yescharacteristic
85#428—61no—
86#423—31no—
87#418—61no—
88#390—61no—
89#388—62no—
90#382—31no—
91#381—61no—
92C4×C4⋊C4—61no—
93C4×C4⋊C4C2×D311yescharacteristic
94C4×C22⋊C4—61no—
95#328—21no—
96#328—21no—
97#328—62no—
98D4⋊C8—62no—
99C23×D3—21no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—22no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×D12—21no—
113C2×D12—21no—
114C2×D12C22⋊C412yes—
115C2×C4×D3—22no—
116C2×C4×D3—22no—
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—42no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C42—41no—
128C3×C42—41no—
129C3×C42—41no—
130C3×C42C22×C411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C3×C42#4111yescharacteristic
134(C2×D3)⋊C4—22no—
135(C2×D3)⋊C4—22no—
136(C2×D3)⋊C4—44no—
137(C2×D3)⋊C4—82no—
138C4⋊Dic3—21no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C4⋊Dic3C22⋊C412yes—
142C4×Dic3—82no—
143#208—41no—
144#208—41no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—41no—
147C22×D4—31no—
148C23×C4—61no—
149C2×M16—61no—
150C4⋊Q8C4×D311yescharacteristic
151C4⋊Q8C4×D311yescharacteristic
152#125—21no—
153C22⋊Q8—21no—
154C22⋊Q8—21no—
155C4×Q8—21no—
156C4×Q8—21no—
157C4×D4—32no—
158C4×D4—32no—
159C4×D4—61no—
160C4×D4—61no—
161#117—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—62no—
168C2×C42—21no—
169C2×C42—62no—
170C2×C42C22×D311yescharacteristic
171#107—62no—
172#106—62no—
173C4⋊C8—21no—
174C4⋊C8—32no—
175C4⋊C8—61no—
176C4⋊C8—61no—
177#102—62no—
178#102—62no—
179C22⋊C8—21no—
180C22⋊C8—61no—
181C4×C8—21no—
182C4×C8—61no—
183#95—32no—
184#95—62no—
185#95—61no—
186#95#6711yescharacteristic
187#95#6711yescharacteristic
188C23×C3#12011yescharacteristic
189C22×D3—22no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—41no—
193C22×D3—41no—
194C3×Q8—81no—
195C3×Q8—81no—
196C2×C3×C4—22no—
197C2×C3×C4—41no—
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—41no—
205C2×C3×C4—41no—
206C2×C3×C4C2×C22⋊C411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12311yescharacteristic
212C2×C3×C4#13511yescharacteristic
213C2×C3×C4#13511yescharacteristic
214C2×C3×C4#13711yescharacteristic
215C2×C3×C4#13711yescharacteristic
216C2×Dic3—22no—
217C2×Dic3—42no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221D12—22no—
222D12—22no—
223D12—41no—
224D12—41no—
225C4×D3—82no—
226C4×D3—82no—
227C3×C8—81no—
228C3⋊C8—42no—
229C3⋊C8—81no—
230C24—61no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×D4—32no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—61no—
242C22×C4—62no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—242no—
246C22×C4D3×D411yescharacteristic
247C22×C4D3×D411yescharacteristic
248C22×C4C2×#6711yescharacteristic
249M16—121no—
250M16—121no—
251C2×C8—41no—
252C2×C8—62no—
253C2×C8—121no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—32no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—61no—
263C4⋊C4—121no—
264C4⋊C4—121no—
265C22⋊C4—41no—
266C22⋊C4—62no—
267C22⋊C4—62no—
268C22⋊C4—124no—
269C22⋊C4—242no—
270C42—41no—
271C42—41no—
272C42—41no—
273C42—242no—
274C42C2×C4×D311yescharacteristic
275C42#23511yescharacteristic
276C42D3×D411yescharacteristic
277C42#23711yescharacteristic
278C22×C3—21no—
279C22×C3—21no—
280C22×C3#38611yescharacteristic
281C22×C3#45211yescharacteristic
282C22×C3#45211yescharacteristic
283C2×D3—42no—
284C2×D3—42no—
285C2×D3—42no—
286C2×D3—82no—
287C2×D3—82no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—42no—
293C3×C4—81no—
294C3×C4—81no—
295C3×C4—81no—
296C3×C4#41611yescharacteristic
297C3×C4#41911yescharacteristic
298C3×C4#48511yescharacteristic
299C3×C4#48511yescharacteristic
300Dic3—42no—
301Dic3—82no—
302C23—62no—
303C23—122no—
304C23—122no—
305C23—121no—
306C23—121no—
307C23#93711yescharacteristic
308Q8—81no—
309Q8—81no—
310D4—62no—
311D4—62no—
312D4—121no—
313D4—121no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—62no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—122no—
329C2×C4—242no—
330C2×C4—242no—
331C2×C4D3×C22⋊C411yescharacteristic
332C2×C4#67⋊C411yescharacteristic
333C2×C4#88311yescharacteristic
334C2×C4#88411yescharacteristic
335C2×C4#91511yescharacteristic
336C2×C4#91611yescharacteristic
337C2×C4#91811yescharacteristic
338C2×C4#91911yescharacteristic
339C2×C4C4×#6711yescharacteristic
340C2×C4#93811yescharacteristic
341C8—81no—
342C8—122no—
343C8—241no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3#187311yescharacteristic
347C2×C3#190411yescharacteristic
348C2×C3#190611yescharacteristic
349D3—82no—
350C22—21no—
351C22—21no—
352C22—122no—
353C22—122no—
354C22—122no—
355C22—242no—
356C22—242no—
357C22#495611yescharacteristic
358C22#517311yescharacteristic
359C22#518711yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—122no—
369C4—242no—
370C4#480211yescharacteristic
371C4#480411yescharacteristic
372C4#481411yescharacteristic
373C4#481511yescharacteristic
374C3#917711yescharacteristic, complemented, Sylow
375C2—21no—
376C2—21no—
377C2—242no—
378C2#7004911yescharacteristic
379C2#7026911yescharacteristic
380C2#7039111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1503D3
hypocenterC3#9177
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
3. C3×#1503C2semidirect
4. #71838C2semidirect
6. #67738C2semidirect
7. #67620C2semidirect
13. #5087C4semidirect
14. #5084C4semidirect
33. #1503D3semidirect
374. C3#9177semidirect

Maximal subgroups

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

subgroupindex
#833502
#736452
C3×#15032
#718382
#707562
#677382
#676202
#91773