Finite Group Databasebeta

← #10697852

Subgroups — #10697852

Order 768.

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

normal subgroups only

This group has 542 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
1C3×#3226C211yesmaximal, characteristic
2#83271C211yesmaximal, characteristic
3#83252C211yesmaximal, characteristic, complemented
4#71815C211yesmaximal, characteristic
5#71773C211yesmaximal, characteristic, complemented
6#70780C211yesmaximal, characteristic, complemented
7#67736C211yesmaximal, characteristic, complemented
8#10286—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#500—21no—
13C3×#486—21no—
14C3×#480—22no—
15C3×#474—22no—
16C3×#431C2211yescharacteristic
17C2×#837—22no—
18#5229—22no—
19#5080—21no—
20#5070—21no—
21#5034—22no—
22#5008—22no—
23#4977—22no—
24#4973—22no—
25C42×Dic3C2211yescharacteristic
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4547—22no—
29#4482—22no—
30#3227—31no—
31#3226D311yescharacteristic
32#2862—31no—
33#1962—31no—
34#1549—31no—
35#1546—31no—
36#1276—31no—
37C2×C3×#41—21no—
38C22×C3×Q8—21no—
39C22×Dic6—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×#127D411yescharacteristic
46C3×#124—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D412yes—
53C2×C3×C42C2311yescharacteristic
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3D412yes—
56C2×C4×Dic3—21no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3D412yes—
59C2×#208—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64C3×#107—24no—
65C3×C4⋊C8D412yes—
66C3×#102—24no—
67C3×C22⋊C8—21no—
68C3×C4×C8—21no—
69#837—44no—
70#832—24no—
71C3⋊#95—28no—
72#830—21no—
73#808—24no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77C4×C3⋊C8—21no—
78#532C2×D311yescharacteristic
79C2×C4⋊Q8—31no—
80C2×C4⋊Q8C2×D311yescharacteristic
81#500—21no—
82#498—61no—
83#486—21no—
84#486—61no—
85#481—62no—
86#480—22no—
87#476—62no—
88#474—22no—
89#431—31no—
90#431C2×D311yescharacteristic
91#423—31no—
92C2×#104—62no—
93#391—62no—
94#391—62no—
95#382—62no—
96C43—31no—
97#332—62no—
98#332—62no—
99C3×#41—42no—
100C3×#41—42no—
101C2×C3×Q8—21no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—42no—
105C2×C3×Q8—42no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—41no—
110C22×C3×C4—22no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—22no—
116C22×Dic3—22no—
117C2×Dic6—44no—
118C2×Dic6—41no—
119C2×Dic6—41no—
120C2×C3×C8—22no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C4⋊Dic3—22no—
134C4⋊Dic3—22no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137C4×Dic3—24no—
138C4×Dic3—22no—
139C4×Dic3—22no—
140C4×Dic3—42no—
141#208—42no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—41no—
144C2×#41—21no—
145C22×Q8—21no—
146C22×Q8—61no—
147C2×M16—61no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8#6711yescharacteristic
157#127#6711yescharacteristic
158#124—21no—
159#121—21no—
160#121—21no—
161C4×D4—21no—
162C4×D4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—32no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4#6712yes—
167C2×C42—32no—
168C2×C42—61no—
169C2×C42—61no—
170C2×C42C22×D311yescharacteristic
171#107—24no—
172#106—64no—
173C4⋊C8—32no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176C4⋊C8#6712yes—
177#104—124no—
178#102—24no—
179#102—64no—
180C22⋊C8—21no—
181C22⋊C8—61no—
182C4×C8—21no—
183C4×C8—61no—
184#95—68no—
185C23×C3—41no—
186C23×C3#12011yescharacteristic
187C3×Q8—42no—
188C3×Q8—82no—
189C3×Q8—81no—
190C3×Q8—81no—
191C3×D4—42no—
192C3×D4—42no—
193C3×D4—42no—
194C3×D4—82no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—41no—
198C2×C3×C4—42no—
199C2×C3×C4—44no—
200C2×C3×C4—81no—
201C2×C3×C4—81no—
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12711yescharacteristic
208C2×C3×C4#13512yes—
209C2×C3×C4AGL(1,ℤ/8)12yes—
210C2×Dic3—24no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—44no—
214C2×Dic3—42no—
215Dic6—82no—
216Dic6—82no—
217C3×C8—42no—
218C3⋊C8—42no—
219C3⋊C8—81no—
220#41—42no—
221#41—42no—
222C2×Q8—21no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—124no—
228C2×Q8—121no—
229C2×Q8—121no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—41no—
234C22×C4—22no—
235C22×C4—41no—
236C22×C4—62no—
237C22×C4—62no—
238C22×C4D3×D411yescharacteristic
239C22×C4D3×D411yescharacteristic
240C22×C4C2×#6711yescharacteristic
241M16—122no—
242C2×C8—22no—
243C2×C8—62no—
244C2×C8—121no—
245C4⋊C4—21no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—21no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C42—64no—
258C42—62no—
259C42—62no—
260C42—122no—
261C42D3×D411yescharacteristic
262C42#23711yescharacteristic
263C42C2×#6711yescharacteristic
264C42C2×#6711yescharacteristic
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—81no—
268C22×C3—81no—
269C22×C3—81no—
270C22×C3#39211yescharacteristic
271C22×C3#44912yes—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—82no—
275C3×C4—81no—
276C3×C4#45412yes—
277C3×C4#45612yes—
278C3×C4#46312yes—
279C3×C4#49511yescharacteristic
280C3×C4#49611yescharacteristic
281Dic3—44no—
282Dic3—82no—
283C23—41no—
284C23#93711yescharacteristic
285Q8—42no—
286Q8—82no—
287Q8—81no—
288Q8—81no—
289Q8—242no—
290Q8—242no—
291D4—42no—
292D4—42no—
293D4—42no—
294D4—82no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—64no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4—124no—
306C2×C4—122no—
307C2×C4#91112yes—
308C2×C4#91212yes—
309C2×C4#93711yescharacteristic
310C2×C4#93811yescharacteristic
311C2×C4#93911yescharacteristic
312C2×C4#93911yescharacteristic
313C2×C4#94011yescharacteristic
314C2×C4#95311yescharacteristic
315C8—42no—
316C8—122no—
317C8—241no—
318C2×C3—22no—
319C2×C3—161no—
320C2×C3#200311yescharacteristic
321C2×C3#200511yescharacteristic
322C2×C3#201211yescharacteristic
323C22—21no—
324C22—21no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22#515812yes—
329C22#522311yescharacteristic
330C4—21no—
331C4—21no—
332C4—82no—
333C4—81no—
334C4—124no—
335C4—242no—
336C4#515312yes—
337C4#515411yescharacteristic
338C4#515511yescharacteristic
339C4#520512yes—
340C4#524512yes—
341C3#1028611yescharacteristic, complemented, Sylow
342C2—22no—
343C2—161no—
344C2#7186711yescharacteristic
345C2#7204511yescharacteristic
346C2#7210811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3226C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3226D3
hypocenterC3#10286
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3226

Complements

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

NH ≅ G/Ntype
3. #83252C2semidirect
5. #71773C2semidirect
6. #70780C2semidirect
7. #67736C2semidirect
341. C3#10286semidirect

Maximal subgroups

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

subgroupindex
C3×#32262
#832712
#832522
#718152
#717732
#707802
#677362
#102863