Finite Group Databasebeta

← #10691098

Subgroups — #10691098

Order 768.

490 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 113 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#83367C211yesmaximal, characteristic
2#74157C211yesmaximal, characteristic
3C3×#1518C211yesmaximal, characteristic
4#71835C211yesmaximal, characteristic
5#70767C211yesmaximal, characteristic
6#67746C211yesmaximal, characteristic
7#67631C211yesmaximal, characteristic
8#9176—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#887—21no—
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#382C2211yescharacteristic
14#5094C412yescomplemented
15#5091C412yescomplemented
16#5063—22no—
17#5004—21no—
18#4995—21no—
19#4972—21no—
20C4×C4⋊Dic3C2211yescharacteristic
21#4925C2211yescharacteristic
22#4924C2211yescharacteristic
23#4650—21no—
24#4649—21no—
25#4647—22no—
26C3×#328—22no—
27#4538—21no—
28#4538—21no—
29#4456—22no—
30#3078—31no—
31#2351—31no—
32#1518D311yescharacteristic
33#1503—31no—
34#1479—31no—
35#1303—31no—
36#1286—31no—
37C22×Dic6D411yescharacteristic
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×C4⋊Q8C2×C411yescharacteristic
40C3×#125—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×#117—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3D411yescharacteristic
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—22no—
52C2×Dic3⋊C4—22no—
53C2×C4×Dic3—21no—
54C2×C4×Dic3—22no—
55C2×C4×Dic3—21no—
56C2×#208—21no—
57#887—42no—
58#887—42no—
59C4×Dic6—21no—
60C4×Dic6—21no—
61C4×Dic6C2×C412yes—
62C4×Dic6D412yes—
63C3×C4⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×C4×C8—21no—
66C3×#95—21no—
67C3×#95D412yes—
68C3⋊#95—22no—
69C3⋊#95D411yescharacteristic
70C3⋊#95D411yescharacteristic
71#830—21no—
72#810—22no—
73#810—22no—
74#808—22no—
75#807—22no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)C2×C412yes—
79C4×C3⋊C8—21no—
80#533C2×D311yescharacteristic
81C2×C4×Q8—31no—
82C2×C4×Q8—61no—
83#479—62no—
84#477—32no—
85#475—32no—
86#431—31no—
87#431C2×D311yescharacteristic
88#428—61no—
89#423—31no—
90#419—61no—
91#389—61no—
92#389—62no—
93#382—61no—
94#382C2×D311yescharacteristic
95C4×C4⋊C4—31no—
96C4×C4⋊C4—61no—
97#328—22no—
98#328—61no—
99#328—61no—
100Q8⋊C8—62no—
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—
110C22×Dic3—21no—
111C2×Dic6—21no—
112C2×Dic6—21no—
113C2×Dic6—21no—
114C2×Dic6C22⋊C412yes—
115C2×Dic6SD1612yes—
116C2×Dic6Q1612yes—
117C2×C3×C8—41no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—21no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C42—41no—
128C3×C42C22×C411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131C3×C42#4111yescharacteristic
132C4⋊Dic3—21no—
133C4⋊Dic3—42no—
134C4⋊Dic3C22⋊C412yes—
135Dic3⋊C4—22no—
136Dic3⋊C4—22no—
137Dic3⋊C4—44no—
138Dic3⋊C4—44no—
139C4×Dic3—22no—
140C4×Dic3—22no—
141C4×Dic3—44no—
142C4×Dic3—41no—
143C4×Dic3—41no—
144#208—41no—
145#208—41no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148C22×Q8—31no—
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×Q8—32no—
158C4×Q8—32no—
159C4×Q8—61no—
160C4×Q8—61no—
161C4×Q8—122no—
162C4×Q8—122no—
163#117—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—62no—
168C2×C4⋊C4—62no—
169C2×C4⋊C4—61no—
170C2×C42—61no—
171C2×C42—62no—
172C2×C42—61no—
173C2×C42C22×D311yescharacteristic
174#107—62no—
175#106—62no—
176C4⋊C8—21no—
177C4⋊C8—32no—
178C4⋊C8—61no—
179C4⋊C8—61no—
180#103—62no—
181#103—62no—
182C22⋊C8—21no—
183C22⋊C8—61no—
184C4×C8—21no—
185C4×C8—61no—
186#95—21no—
187#95—31no—
188#95—31no—
189#95—62no—
190#95#6712yes—
191C23×C3#12011yescharacteristic
192C3×Q8—81no—
193C3×Q8—81no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4C2×C22⋊C411yescharacteristic
202C2×C3×C4C4×D411yescharacteristic
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12311yescharacteristic
207C2×C3×C4C2×SD1611yescharacteristic
208C2×C3×C4C2×Q1611yescharacteristic
209C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
210C2×C3×C4#13711yescharacteristic
211C2×Dic3—22no—
212C2×Dic3—22no—
213C2×Dic3—22no—
214C2×Dic3—22no—
215C2×Dic3—22no—
216C2×Dic3—42no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222Dic6—22no—
223Dic6—22no—
224Dic6—22no—
225Dic6#10314yes—
226C3×C8—81no—
227C3⋊C8—42no—
228C3⋊C8—81no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—32no—
232C2×Q8—32no—
233C2×Q8—32no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8—61no—
237C22×C4—22no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4D3×D411yescharacteristic
243C22×C4D3×D411yescharacteristic
244C22×C4C2×#6711yescharacteristic
245M16—121no—
246M16—121no—
247C2×C8—41no—
248C2×C8—62no—
249C2×C8—121no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—32no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—61no—
260C4⋊C4—62no—
261C4⋊C4—62no—
262C4⋊C4—124no—
263C4⋊C4—124no—
264C4⋊C4—122no—
265C22⋊C4—41no—
266C42—41no—
267C42—62no—
268C42—62no—
269C42—124no—
270C42—121no—
271C42—121no—
272C42C2×C4×D311yescharacteristic
273C42#23511yescharacteristic
274C42D3×D411yescharacteristic
275C42#23711yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3#38611yescharacteristic
279C22×C3#45011yescharacteristic
280C22×C3#45111yescharacteristic
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—82no—
286C3×C4—81no—
287C3×C4—81no—
288C3×C4C2×#10311yescharacteristic
289C3×C4#41711yescharacteristic
290C3×C4#48311yescharacteristic
291C3×C4#48411yescharacteristic
292Dic3—24no—
293Dic3—42no—
294Dic3—42no—
295Dic3—81no—
296C23#93711yescharacteristic
297Q8—34no—
298Q8—62no—
299Q8—62no—
300Q8—62no—
301Q8—81no—
302Q8—81no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—62no—
313C2×C4—62no—
314C2×C4—62no—
315C2×C4—122no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4D3×C22⋊C411yescharacteristic
322C2×C4#67⋊C411yescharacteristic
323C2×C4#88311yescharacteristic
324C2×C4#88411yescharacteristic
325C2×C4D3×SD1611yescharacteristic
326C2×C4#91411yescharacteristic
327C2×C4D3×Q1611yescharacteristic
328C2×C4#91811yescharacteristic
329C2×C4C4×#6711yescharacteristic
330C2×C4#93811yescharacteristic
331C8—81no—
332C8—122no—
333C8—241no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3#187211yescharacteristic
337C2×C3#190511yescharacteristic
338C2×C3#190611yescharacteristic
339C22—21no—
340C22—21no—
341C22#495611yescharacteristic
342C22#516911yescharacteristic
343C22#518611yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—64no—
349C4—82no—
350C4—81no—
351C4—81no—
352C4—122no—
353C4—122no—
354C4—241no—
355C4D3×#10311yescharacteristic
356C4#480311yescharacteristic
357C4#480511yescharacteristic
358C4#481111yescharacteristic
359C3#917611yescharacteristic, complemented, Sylow
360C2—21no—
361C2—21no—
362C2#7005511yescharacteristic
363C2#7026811yescharacteristic
364C2#7041011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1518C2
socleC22×C3#386
prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1518D3
hypocenterC3#9176
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
14. #5094C4semidirect
15. #5091C4semidirect
359. C3#9176semidirect

Maximal subgroups

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

subgroupindex
#833672
#741572
C3×#15182
#718352
#707672
#677462
#676312
#91763