Finite Group Databasebeta

← #10698950

Subgroups — #10698950

Order 768.

398 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 95 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#82296C211yesmaximal, characteristic
2C3×#1460C211yesmaximal, characteristic
3#71605C211yesmaximal, characteristic
4#70793C211yesmaximal, characteristic, complemented
5#70776C211yesmaximal, characteristic, complemented
6#70749C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#10468—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#431C2211yescharacteristic
13#5215—21no—
14C2×#832—21no—
15#5009—22no—
16#5008—21no—
17#5007—21no—
18#4964—21no—
19#4936C2211yescharacteristic
20#4924C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×#323—22no—
23#4529—21no—
24#4482—21no—
25#4481—21no—
26#4467—22no—
27#4457—22no—
28#3077—31no—
29#1804—31no—
30#1570—31no—
31#1542—31no—
32#1518—31no—
33#1460D311yescharacteristic
34#1286—31no—
35C22×C3×D4D411yescharacteristic
36C23×C3×C4—21no—
37C3×C4×D4—21no—
38C3×C4×D4—21no—
39C3×C4×D4—21no—
40C3×C4×D4—21no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4—21no—
43C2×C3×C42C2311yescharacteristic
44#926—21no—
45C2×C4⋊Dic3—21no—
46#924—21no—
47#924—21no—
48C2×#208—21no—
49C22×C3⋊C8—21no—
50#870—21no—
51#869D411yescharacteristic
52#869D411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6—21no—
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×C22⋊C8—21no—
58C3×C22⋊C8D412yes—
59C3×C4×C8—21no—
60C3×C4×C8—21no—
61#832—22no—
62#832—22no—
63#832—42no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66#808—22no—
67#807—22no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)D412yes—
72#533—31no—
73C2×C4×D4C2×D311yescharacteristic
74#478—62no—
75#476—61no—
76#474—61no—
77#431C2×D311yescharacteristic
78#431C2×D311yescharacteristic
79#428—61no—
80#423—31no—
81C2×C4⋊C8—31no—
82#418—61no—
83C2×#102—61no—
84#382—31no—
85#340—61no—
86#334—62no—
87#333—61no—
88#332—61no—
89#328—62no—
90#323—22no—
91C24×C3—21no—
92C2×C3×D4—21no—
93C2×C3×D4—21no—
94C2×C3×D4—21no—
95C2×C3×D4—21no—
96C2×C3×D4—21no—
97C2×C3×D4SD1612yes—
98C22×C3×C4—21no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C22×C3×C4—82no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×Dic3—41no—
106C2×Dic6—21no—
107C2×Dic6—21no—
108C2×C3×C8—22no—
109C2×C3×C8—41no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C22⋊C4—41no—
113C3×C22⋊C4—41no—
114C3×C42C2×D411yescharacteristic
115C3×C42C2×D411yescharacteristic
116C3×C42#4111yescharacteristic
117C3×C42#4111yescharacteristic
118#217—41no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—41no—
124C4⋊Dic3—41no—
125Dic3⋊C4—41no—
126Dic3⋊C4—41no—
127C4×Dic3—41no—
128#208—41no—
129#208—41no—
130C2×C3⋊C8—22no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134C22×D4D1211yescharacteristic
135C23×C4—21no—
136C2×M16—61no—
137C22×C8—61no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140#125—61no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
143C4×Q8—61no—
144C4×Q8—61no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—21no—
149#117—61no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4D1211yescharacteristic
152C2×C22⋊C4—21no—
153C2×C42C22×D311yescharacteristic
154#107—62no—
155#106—62no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—32no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161C4⋊C8—61no—
162#102—62no—
163#102—62no—
164#102—122no—
165C22⋊C8—21no—
166C22⋊C8#6712yes—
167C4×C8—21no—
168C4×C8—21no—
169#95—61no—
170#95—61no—
171C23×C3—22no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3—81no—
175C23×C3—81no—
176C23×C3#12111yescharacteristic
177C3×D4—22no—
178C3×D4—22no—
179C3×D4—41no—
180C3×D4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—82no—
185C2×C3×C4—82no—
186C2×C3×C4—81no—
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12311yescharacteristic
191C2×C3×C4#12311yescharacteristic
192C2×C3×C4#12411yescharacteristic
193C2×C3×C4C2×SD1611yescharacteristic
194C2×C3×C4#13511yescharacteristic
195C2×C3×C4#13711yescharacteristic
196C2×C3×C4#13711yescharacteristic
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—81no—
200C2×Dic3—81no—
201C2×Dic3—81no—
202Dic6—81no—
203Dic6—81no—
204C3×C8—42no—
205C3×C8—81no—
206C3⋊C8—42no—
207C3⋊C8—81no—
208C24—21no—
209C2×Q8—61no—
210C2×Q8—61no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—21no—
215C2×D4—21no—
216C2×D4#20412yes—
217C22×C4—21no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—82no—
221C22×C4—121no—
222C22×C4C2×D1211yescharacteristic
223C22×C4#23511yescharacteristic
224C22×C4C2×#6711yescharacteristic
225M16—121no—
226M16—121no—
227C2×C8—22no—
228C2×C8—41no—
229C2×C8—62no—
230C2×C8—121no—
231C2×C8—121no—
232C2×C8—121no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—121no—
246C42—121no—
247C42D3×D411yescharacteristic
248C42D3×D411yescharacteristic
249C42#23711yescharacteristic
250C42#23711yescharacteristic
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—42no—
254C22×C3—82no—
255C22×C3—82no—
256C22×C3—81no—
257C22×C3—81no—
258C22×C3#39411yescharacteristic
259C22×C3#46511yescharacteristic
260C22×C3#47011yescharacteristic
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—81no—
266C3×C4#44811yescharacteristic
267C3×C4#45211yescharacteristic
268C3×C4#48311yescharacteristic
269C3×C4#48511yescharacteristic
270Dic3—81no—
271Dic3—81no—
272Dic3—161no—
273C23—22no—
274C23—41no—
275C23—41no—
276C23—81no—
277C23—81no—
278C23#93011yescharacteristic
279Q8—241no—
280Q8—241no—
281D4—22no—
282D4—22no—
283D4—41no—
284D4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—81no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—241no—
294C2×C4—241no—
295C2×C4—241no—
296C2×C4#88211yescharacteristic
297C2×C4#88311yescharacteristic
298C2×C4#88511yescharacteristic
299C2×C4#88611yescharacteristic
300C2×C4C2×#20411yescharacteristic
301C2×C4#90411yescharacteristic
302C2×C4#90911yescharacteristic
303C2×C4#90911yescharacteristic
304C2×C4#93811yescharacteristic
305C2×C4#93911yescharacteristic
306C8—42no—
307C8—81no—
308C8—122no—
309C8—241no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—82no—
313C2×C3#203511yescharacteristic
314C2×C3#204811yescharacteristic
315C2×C3#205311yescharacteristic
316C22—21no—
317C22—21no—
318C22—42no—
319C22—82no—
320C22—82no—
321C22—81no—
322C22—81no—
323C22#496111yescharacteristic
324C22#511411yescharacteristic
325C22#513711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—81no—
331C4—241no—
332C4—241no—
333C4—481no—
334C4#472111yescharacteristic
335C4#472311yescharacteristic
336C4#473311yescharacteristic
337C4#473811yescharacteristic
338C3#1046811yescharacteristic, complemented, Sylow
339C2—21no—
340C2—21no—
341C2—82no—
342C2#6993611yescharacteristic
343C2#7004311yescharacteristic
344C2#7007611yescharacteristic
subgroupG/H
centerC22#4961
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1460C2
socleC22×C3#394
prime radicalC22×C3×D4D4
hypercenter#1460D3
hypocenterC3#10468
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
4. #70793C2semidirect
5. #70776C2semidirect
6. #70749C2semidirect
7. #67738C2semidirect
338. C3#10468semidirect

Maximal subgroups

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

subgroupindex
#822962
C3×#14602
#716052
#707932
#707762
#707492
#677382
#104683