Finite Group Databasebeta

← #10697855

Subgroups — #10697855

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×#2862C211yesmaximal, characteristic, complemented
2#83216C211yesmaximal, characteristic
3#83215C211yesmaximal, characteristic
4#71775C211yesmaximal, characteristic, complemented
5#71773C211yesmaximal, characteristic, complemented
6#69395C211yesmaximal, characteristic
7#67611C211yesmaximal, characteristic, complemented
8#10286—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#391—22no—
13C3×#391—22no—
14C3×#382—22no—
15C3×C43C2211yescharacteristic
16#5088—21no—
17#5088—21no—
18#5087—22no—
19#5084—22no—
20#5080—21no—
21#5078—21no—
22#4968—22no—
23#4967—22no—
24C2×#805—22no—
25#4925C2211yescharacteristic
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4547—22no—
29#4547—22no—
30#3227—31no—
31#3226—31no—
32#2862D311yescharacteristic, complemented
33#1962—31no—
34#1549—31no—
35#1546—31no—
36#1276—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C2×#235—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D412yes—
51C2×C3×C4⋊C4D412yes—
52C2×C3×C42—21no—
53C2×C3×C42—21no—
54C2×C3×C42D412yes—
55C2×C3×C42C2311yescharacteristic
56#930—21no—
57#930—21no—
58C2×#208—21no—
59#875—21no—
60#874D411yescharacteristic
61C4×D12—21no—
62C4×D12—21no—
63#869D411yescharacteristic
64C3×#95—28no—
65#830—21no—
66#830—21no—
67#809—24no—
68#809—24no—
69#808—24no—
70#807—24no—
71#805—44no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D412yes—
75C4⋊(C3⋊C8)D412yes—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78#532—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×C4⋊Q8C2×D311yescharacteristic
81#500—61no—
82#498—61no—
83#486—61no—
84#486—61no—
85#481—62no—
86#480—62no—
87#476—62no—
88#474—62no—
89#431—31no—
90#431—31no—
91#423—31no—
92C2×#104—62no—
93#391—22no—
94#391—22no—
95#382—22no—
96C43C2×D311yescharacteristic
97#332—62no—
98#332—62no—
99C2×C3×Q8—44no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—42no—
105C2×C3×Q8—42no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C2×#67—41no—
113#235—42no—
114#235—42no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×C4×D3—41no—
119C2×Dic6—21no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C4⋊C4—41no—
128C3×C42—24no—
129C3×C42—22no—
130C3×C42—22no—
131C3×C42—42no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136(C2×D3)⋊C4—41no—
137(C2×D3)⋊C4—41no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140#208—42no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—41no—
144C2×#41—61no—
145C22×Q8—21no—
146C22×Q8—21no—
147C2×M16—61no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—31no—
157#127—31no—
158#124—61no—
159#121—61no—
160#121—61no—
161C4×D4—61no—
162C4×D4—61no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4#6712yes—
166C2×C4⋊C4#6712yes—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42#6712yes—
170C2×C42C22×D311yescharacteristic
171#107—64no—
172#106—64no—
173C4⋊C8—32no—
174C4⋊C8—32no—
175C4⋊C8—61no—
176C4⋊C8—61no—
177#104—124no—
178#102—64no—
179#102—64no—
180C22⋊C8—61no—
181C22⋊C8—61no—
182C4×C8—61no—
183C4×C8—61no—
184#95—28no—
185C23×C3#12011yescharacteristic
186C22×D3—41no—
187C3×Q8—82no—
188C3×Q8—82no—
189C3×Q8—82no—
190C3×Q8—81no—
191C3×Q8—81no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—24no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—44no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—44no—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12711yescharacteristic
207C2×C3×C4#13512yes—
208C2×C3×C4AGL(1,ℤ/8)12yes—
209#67—82no—
210C2×Dic3—41no—
211D12—42no—
212D12—42no—
213D12—42no—
214C4×D3—81no—
215C4×D3—81no—
216Dic6—42no—
217C3⋊C8—42no—
218C3⋊C8—42no—
219C3⋊C8—81no—
220#41—122no—
221#41—122no—
222C2×Q8—44no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—121no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—121no—
238C22×C4C2×#6711yescharacteristic
239C22×C4C2×#6711yescharacteristic
240C22×C4C2×#6711yescharacteristic
241M16—122no—
242C2×C8—62no—
243C2×C8—62no—
244C2×C8—121no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C22⋊C4—121no—
256C22⋊C4—121no—
257C42—24no—
258C42—22no—
259C42—22no—
260C42—42no—
261C42D3×D411yescharacteristic
262C42D3×D411yescharacteristic
263C42D3×D411yescharacteristic
264C42#23711yescharacteristic
265C22×C3—21no—
266C22×C3—21no—
267C22×C3#39211yescharacteristic
268C22×C3#44912yes—
269C2×D3—81no—
270C2×D3—81no—
271C2×D3—81no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—44no—
275C3×C4—82no—
276C3×C4—82no—
277C3×C4#45412yes—
278C3×C4#45612yes—
279C3×C4#46312yes—
280C3×C4#49511yescharacteristic
281C3×C4#49611yescharacteristic
282Dic3—81no—
283C23—121no—
284C23#95311yescharacteristic
285Q8—82no—
286Q8—82no—
287Q8—81no—
288Q8—81no—
289Q8—82no—
290Q8—122no—
291D4—122no—
292D4—122no—
293D4—122no—
294D4—242no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—121no—
305C2×C4—241no—
306C2×C4—241no—
307C2×C4#93711yescharacteristic
308C2×C4#93711yescharacteristic
309C2×C4#93811yescharacteristic
310C2×C4#93911yescharacteristic
311C2×C4#93911yescharacteristic
312C2×C4#94011yescharacteristic
313C2×C4#94912yes—
314C2×C4#95012yes—
315C8—122no—
316C8—122no—
317C8—241no—
318C2×C3—22no—
319C2×C3#200311yescharacteristic
320C2×C3#200511yescharacteristic
321C2×C3#201211yescharacteristic
322D3—161no—
323C22—21no—
324C22—21no—
325C22—241no—
326C22—241no—
327C22—241no—
328C22#522311yescharacteristic
329C22#523812yes—
330C4—21no—
331C4—21no—
332C4—44no—
333C4—82no—
334C4—82no—
335C4—241no—
336C4#505712yes—
337C4#505811yescharacteristic
338C4#505911yescharacteristic
339C4#506212yes—
340C4#509512yes—
341C3#1028611yescharacteristic, complemented, Sylow
342C2—22no—
343C2—481no—
344C2#7172811yescharacteristic
345C2#7173011yescharacteristic
346C2#7179511yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2862C2
socleC22×C3#392
prime radical#5477C22
hypercenter#2862D3
hypocenterC3#10286
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
1. C3×#2862C2semidirect
4. #71775C2semidirect
5. #71773C2semidirect
7. #67611C2semidirect
32. #2862D3semidirect
341. C3#10286semidirect

Maximal subgroups

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

subgroupindex
C3×#28622
#832162
#832152
#717752
#717732
#693952
#676112
#102863