Finite Group Databasebeta

← #10698811

Subgroups — #10698811

Order 768.

408 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#4846C211yesmaximal, characteristic
2C3×#1570C211yesmaximal, characteristic, complemented
3#70780C211yesmaximal, characteristic, complemented
4#70756C211yesmaximal, characteristic
5#70527C211yesmaximal, characteristic
6#70507C211yesmaximal, characteristic, complemented
7#67707C211yesmaximal, characteristic, complemented
8#10448—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×#823—21no—
15C2×#821—21no—
16C2×#818—21no—
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19#4846—22no—
20#4846—22no—
21C3×#334—22no—
22#4554—21no—
23#4483—22no—
24#4482—22no—
25#4481—22no—
26C2×#474—31no—
27#1800—31no—
28#1570D311yescharacteristic, complemented
29#1549—31no—
30#1548—31no—
31#1478—31no—
32#1301—31no—
33C22×D12D411yescharacteristic
34C22×C4×D3—21no—
35C22×Dic6—21no—
36C2×C3×M16—21no—
37C22×C3×C8—21no—
38C2×C3×C42C2311yescharacteristic
39C2×(C2×D3)⋊C4—21no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3D411yescharacteristic
43C2×#208—21no—
44C4×D12—21no—
45C4×D12—21no—
46C4×D12—21no—
47C4×D12—21no—
48#869—21no—
49#869—21no—
50#869—21no—
51#869D412yes—
52C3×C4⋊C8—21no—
53C3×C4⋊C8—21no—
54C3×C4⋊C8—21no—
55C3×C4⋊C8D412yes—
56C3×C4⋊C8D412yes—
57C3×C22⋊C8—21no—
58C3×C4×C8—21no—
59#830—21no—
60#823—42no—
61#821—24no—
62#818—24no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)—21no—
65C4⋊(C3⋊C8)—21no—
66C4×C3⋊C8—21no—
67C2×C4⋊Q8—31no—
68C2×C4×D4—31no—
69#474—62no—
70#474—62no—
71#431—31no—
72#431C2×D311yescharacteristic
73C2×#107—61no—
74#423—31no—
75#423C2×D311yescharacteristic
76C2×C4⋊C8C2×D311yescharacteristic
77C2×#102—61no—
78C2×#100—61no—
79#343—61no—
80#334—22no—
81#333—62no—
82#332—62no—
83D4⋊C8—62no—
84C23×D3—21no—
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4#4111yescharacteristic
87C22×C3×C4#4111yescharacteristic
88C22×Dic3—21no—
89C22×Dic3—41no—
90C2×D12—21no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12—21no—
95C2×D12SD1612yes—
96C2×C4×D3—41no—
97C2×C4×D3—41no—
98C2×C4×D3—82no—
99C2×Dic6—22no—
100C2×Dic6—41no—
101C2×Dic6—82no—
102C3×M16—41no—
103C3×M16—41no—
104C2×C3×C8—22no—
105C2×C3×C8—22no—
106C2×C3×C8—41no—
107C2×C3×C8—41no—
108C2×C3×C8—41no—
109C3×C42C2×D411yescharacteristic
110C3×C42C2×D411yescharacteristic
111C3×C42C2×D411yescharacteristic
112C3×C42#4111yescharacteristic
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4—41no—
115C4⋊Dic3—21no—
116C4⋊Dic3—22no—
117C4⋊Dic3—22no—
118C4⋊Dic3—21no—
119C4⋊Dic3—41no—
120C4⋊Dic3—41no—
121#208—41no—
122#208—41no—
123C2×C3⋊C8—41no—
124C2×C3⋊C8—41no—
125C22×Q8—61no—
126C22×D4—31no—
127C23×C4—61no—
128C2×M16—21no—
129C2×M16—61no—
130C22×C8—21no—
131C4⋊Q8—32no—
132C4⋊Q8—61no—
133C4⋊Q8—61no—
134C4⋊Q8—61no—
135C4×D4—61no—
136C4×D4—61no—
137C4×D4—61no—
138C4×D4—61no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—61no—
141C2×C4⋊C4—61no—
142C2×C22⋊C4—61no—
143C2×C42C22×D311yescharacteristic
144#107—64no—
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—61no—
149C4⋊C8—61no—
150C4⋊C8—61no—
151C4⋊C8D1212yes—
152C4⋊C8#6712yes—
153#102—64no—
154#100—122no—
155C22⋊C8—21no—
156C22⋊C8—61no—
157C4×C8—21no—
158C4×C8—61no—
159C23×C3#12311yescharacteristic
160C22×D3—22no—
161C22×D3—41no—
162C22×D3—41no—
163C22×D3—81no—
164C22×D3—81no—
165C2×C3×C4—22no—
166C2×C3×C4—41no—
167C2×C3×C4#12011yescharacteristic
168C2×C3×C4#12111yescharacteristic
169C2×C3×C4#12111yescharacteristic
170C2×C3×C4#12111yescharacteristic
171C2×C3×C4#12311yescharacteristic
172C2×C3×C4#12411yescharacteristic
173C2×C3×C4C2×SD1611yescharacteristic
174C2×C3×C4#13511yescharacteristic
175C2×C3×C4#13711yescharacteristic
176C2×C3×C4#13711yescharacteristic
177C2×Dic3—42no—
178C2×Dic3—41no—
179C2×Dic3—82no—
180C2×Dic3—81no—
181D12—22no—
182D12—22no—
183D12—41no—
184D12—41no—
185C4×D3—82no—
186C4×D3—82no—
187Dic6—82no—
188Dic6—82no—
189C3×C8—42no—
190C3×C8—42no—
191C3×C8—81no—
192C3⋊C8—81no—
193C3⋊C8—81no—
194C24—61no—
195C2×Q8—62no—
196C2×Q8—121no—
197C2×Q8—242no—
198C2×D4—32no—
199C2×D4—61no—
200C2×D4—61no—
201C2×D4—61no—
202C2×D4—61no—
203C2×D4—61no—
204C22×C4—61no—
205C22×C4—121no—
206C22×C4—121no—
207C22×C4—121no—
208C22×C4—242no—
209C22×C4#23511yescharacteristic
210C22×C4D3×D411yescharacteristic
211C22×C4#23911yescharacteristic
212M16—41no—
213M16—41no—
214M16—121no—
215M16—121no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—121no—
222C2×C8—121no—
223C4⋊C4—61no—
224C4⋊C4—62no—
225C4⋊C4—62no—
226C4⋊C4—61no—
227C4⋊C4—121no—
228C4⋊C4—121no—
229C22⋊C4—121no—
230C22⋊C4—121no—
231C42C2×D1211yescharacteristic
232C42D3×D411yescharacteristic
233C42#23911yescharacteristic
234C42C2×#6711yescharacteristic
235C22×C3—21no—
236C22×C3—21no—
237C22×C3#39411yescharacteristic
238C22×C3#48311yescharacteristic
239C22×C3#48511yescharacteristic
240C2×D3—42no—
241C2×D3—82no—
242C2×D3—81no—
243C2×D3—81no—
244C2×D3—82no—
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—21no—
248C3×C4#44811yescharacteristic
249C3×C4#45211yescharacteristic
250C3×C4#46511yescharacteristic
251C3×C4#47011yescharacteristic
252C3×C4#47212yes—
253Dic3—82no—
254Dic3—81no—
255C23—62no—
256C23—121no—
257C23—121no—
258C23—241no—
259C23—241no—
260C23#89411yescharacteristic
261Q8—242no—
262Q8—242no—
263D4—62no—
264D4—62no—
265D4—121no—
266D4—121no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—121no—
270C2×C4—122no—
271C2×C4—242no—
272C2×C4—242no—
273C2×C4—242no—
274C2×C4—241no—
275C2×C4#89411yescharacteristic
276C2×C4#89511yescharacteristic
277C2×C4#89511yescharacteristic
278C2×C4D3×SD1611yescharacteristic
279C2×C4#91511yescharacteristic
280C2×C4#91511yescharacteristic
281C2×C4#91611yescharacteristic
282C2×C4#93011yescharacteristic
283C2×C4#93711yescharacteristic
284C2×C4#94711yescharacteristic
285C8—42no—
286C8—42no—
287C8—81no—
288C8—241no—
289C8—241no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3#203511yescharacteristic
293C2×C3#204511yescharacteristic
294C2×C3#204911yescharacteristic
295D3—82no—
296C22—21no—
297C22—21no—
298C22—122no—
299C22—242no—
300C22—242no—
301C22—241no—
302C22—241no—
303C22#486211yescharacteristic
304C22#486811yescharacteristic
305C22#499111yescharacteristic
306C4—21no—
307C4—21no—
308C4—21no—
309C4—241no—
310C4—242no—
311C4#486411yescharacteristic
312C4#486711yescharacteristic
313C4#514212yes—
314C4#517011yescharacteristic
315C4#517311yescharacteristic
316C3#1044811yescharacteristic, complemented, Sylow
317C2—21no—
318C2—21no—
319C2—242no—
320C2#7054011yescharacteristic
321C2#7058211yescharacteristic
322C2#7073711yescharacteristic
subgroupG/H
centerC22#4991
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1570C2
socleC22×C3#394
prime radicalC22×D12D4
hypercenter#1570D3
hypocenterC3#10448
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
2. C3×#1570C2semidirect
3. #70780C2semidirect
6. #70507C2semidirect
7. #67707C2semidirect
28. #1570D3semidirect
316. C3#10448semidirect

Maximal subgroups

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

subgroupindex
C2×#48462
C3×#15702
#707802
#707562
#705272
#705072
#677072
#104483