Finite Group Databasebeta

← #10697643

Subgroups — #10697643

Order 768.

500 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C3×#3209C211yesmaximal, characteristic
2#83367C211yesmaximal, characteristic
3#83244C211yesmaximal, characteristic
4#71835C211yesmaximal, characteristic
5#71774C211yesmaximal, characteristic
6#70766C211yesmaximal, characteristic
7#67746C211yesmaximal, characteristic
8#10252—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#494—21no—
13C3×#487—21no—
14C3×#462—22no—
15C3×#451—22no—
16C3×#431C2211yescharacteristic
17#5229—22no—
18#5094—21no—
19#5091—21no—
20#5063—22no—
21#5004—21no—
22#4995—21no—
23#4985—22no—
24#4977—21no—
25#4973—21no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4546—22no—
30#4538—22no—
31#4484—22no—
32#3209D311yescharacteristic
33#3078—31no—
34#2568—31no—
35#1548—31no—
36#1503—31no—
37#1479—31no—
38#1303—31no—
39C22×C3×Q8D412yes—
40C22×Dic6D411yescharacteristic
41C2×C3×Q16—24no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D411yescharacteristic
47C3×C4⋊Q8D411yescharacteristic
48C3×#125—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C42C2311yescharacteristic
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×Dic3⋊C4—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3—21no—
62C2×#208—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8D412yes—
68C3×#103—24no—
69C3×C22⋊C8—21no—
70C3×C4×C8—21no—
71C3⋊#95—24no—
72C3⋊#95—24no—
73C3⋊#95D412yes—
74#830—21no—
75#810—22no—
76#810—22no—
77#808—22no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yes—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×C4×Q8—31no—
86#494—21no—
87#487—21no—
88#479—62no—
89#477—61no—
90#475—61no—
91#462—22no—
92#451—22no—
93#431—31no—
94#431C2×D311yescharacteristic
95#428—61no—
96#423—31no—
97#419—61no—
98#400—62no—
99#391—61no—
100#391—62no—
101#382—61no—
102C4×C4⋊C4—31no—
103#333—62no—
104#328—62no—
105Q8⋊C8—62no—
106C2×C3×Q8—22no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—22no—
110C2×C3×Q8—44no—
111C22×C3×C4—22no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—22no—
117C22×Dic3—21no—
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6Q1612yescomplemented
124C3×Q16—84no—
125C2×C3×C8—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—42no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147C4×Dic3—41no—
148C4×Dic3—41no—
149C4×Dic3—41no—
150C4×Dic3—41no—
151#208—41no—
152#208—41no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C22×Q8—31no—
156C22×Q8#6712yes—
157C2×Q16—24no—
158C2×M16—61no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8#6711yescharacteristic
164C4⋊Q8#6711yescharacteristic
165#125—21no—
166C22⋊Q8—21no—
167C22⋊Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—61no—
171C4×Q8—61no—
172C4×Q8—61no—
173C4×Q8—61no—
174#117—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C42—61no—
181C2×C42—61no—
182C2×C42C22×D311yescharacteristic
183#107—62no—
184#106—62no—
185C4⋊C8—32no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8#6712yes—
189#103—24no—
190#103—62no—
191#103—62no—
192C22⋊C8—21no—
193C22⋊C8—61no—
194C4×C8—21no—
195C4×C8—61no—
196#95—32no—
197#95—64no—
198#95—64no—
199C23×C3#12011yescharacteristic
200C3×Q8—42no—
201C3×Q8—44no—
202C3×Q8—81no—
203C3×Q8—82no—
204C2×C3×C4—22no—
205C2×C3×C4—41no—
206C2×C3×C4—44no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12711yescharacteristic
216C2×C3×C4C2×Q1611yescharacteristic
217C2×C3×C4#13511yescharacteristic
218C2×C3×C4#13711yescharacteristic
219C2×C3×C4#13711yescharacteristic
220C2×Dic3—22no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—42no—
228Dic6—22no—
229Dic6—22no—
230Dic6—41no—
231Dic6—41no—
232C3×C8—42no—
233C3⋊C8—42no—
234C3⋊C8—81no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—21no—
239C2×Q8—32no—
240C2×Q8—44no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—61no—
246C22×C4—22no—
247C22×C4—61no—
248C22×C4—62no—
249C22×C4—61no—
250C22×C4D3×D411yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4C2×#6711yescharacteristic
253Q16—84no—
254M16—121no—
255M16—121no—
256C2×C8—22no—
257C2×C8—62no—
258C2×C8—121no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—122no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C22⋊C4—41no—
275C42—41no—
276C42—121no—
277C42—121no—
278C42—121no—
279C42—121no—
280C42D3×D411yescharacteristic
281C42#23711yescharacteristic
282C42C2×#6711yescharacteristic
283C42C2×#6711yescharacteristic
284C22×C3—21no—
285C22×C3—21no—
286C22×C3#39211yescharacteristic
287C22×C3#45111yescharacteristic
288C22×C3#45211yescharacteristic
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—82no—
293C3×C4—81no—
294C3×C4—81no—
295C3×C4#45111yescharacteristic
296C3×C4#45211yescharacteristic
297C3×C4#45512yes—
298C3×C4#46711yescharacteristic
299C3×C4#47011yescharacteristic
300Dic3—42no—
301Dic3—82no—
302Dic3—81no—
303C23#93711yescharacteristic
304Q8—42no—
305Q8—44no—
306Q8—62no—
307Q8—62no—
308Q8—82no—
309Q8—81no—
310Q8—121no—
311Q8—121no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—44no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—62no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—121no—
326C2×C4D3×Q1611yescharacteristic
327C2×C4#91811yescharacteristic
328C2×C4#91811yescharacteristic
329C2×C4#91911yescharacteristic
330C2×C4#93711yescharacteristic
331C2×C4#93811yescharacteristic
332C2×C4#93911yescharacteristic
333C2×C4#93911yescharacteristic
334C2×C4#94011yescharacteristic
335C2×C4#95311yescharacteristic
336C8—42no—
337C8—122no—
338C8—241no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3#200111yescharacteristic
342C2×C3#200411yescharacteristic
343C2×C3#200911yescharacteristic
344C22—21no—
345C22—21no—
346C22#518611yescharacteristic
347C22#518711yescharacteristic
348C22#522311yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—81no—
353C4—82no—
354C4—81no—
355C4—122no—
356C4—242no—
357C4—241no—
358C4#518611yescharacteristic
359C4#518711yescharacteristic
360C4#518811yescharacteristic
361C4#518911yescharacteristic
362C4#524212yes—
363C3#1025211yescharacteristic, complemented, Sylow
364C2—21no—
365C2—21no—
366C2#7185711yescharacteristic
367C2#7207111yescharacteristic
368C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3209C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#3209D3
hypocenterC3#10252
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
123. C2×Dic6Q16semidirect
363. C3#10252semidirect

Maximal subgroups

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

subgroupindex
C3×#32092
#833672
#832442
#718352
#717742
#707662
#677462
#102523