Finite Group Databasebeta

← #10698951

Subgroups — #10698951

Order 768.

398 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 93 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#79059C211yesmaximal, characteristic
2C3×#1570C211yesmaximal, characteristic, complemented
3#70793C211yesmaximal, characteristic, complemented
4#70783C211yesmaximal, characteristic
5#70480C211yesmaximal, characteristic
6#70437C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#10468—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5131—21no—
15#5121—21no—
16C2×#821—21no—
17#4936C2211yescharacteristic
18#4925C2211yescharacteristic
19#4847—22no—
20#4846—21no—
21#4842—21no—
22C3×#334—22no—
23#4551—21no—
24#4486—21no—
25#4481—21no—
26#4468—22no—
27#4467—22no—
28#3077—31no—
29#1804—31no—
30#1570D311yescharacteristic, complemented
31#1542—31no—
32#1518—31no—
33#1460—31no—
34#1286—31no—
35C22×D12D411yescharacteristic
36C22×C4×D3—21no—
37C2×C3×M16—21no—
38C22×C3×C8—21no—
39C2×C3×C42C2311yescharacteristic
40C2×(C2×D3)⋊C4—21no—
41#926—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3D411yescharacteristic
44#924—21no—
45#924—21no—
46C4×D12—21no—
47C4×D12—21no—
48C4×D12—21no—
49C4×D12—21no—
50#870—21no—
51#869D411yescharacteristic
52#869D411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6—21no—
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C4⋊C8D412yes—
60C3×C22⋊C8D412yes—
61C3×C4×C8—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64#830—21no—
65#821—22no—
66#821—22no—
67#821—42no—
68#818—22no—
69#817—22no—
70C4⋊(C3⋊C8)—21no—
71C4×C3⋊C8—21no—
72#533—31no—
73C2×C4×D4—31no—
74#478—62no—
75#476—61no—
76#474—61no—
77#431—31no—
78#431C2×D311yescharacteristic
79#428—61no—
80#423C2×D311yescharacteristic
81C2×C4⋊C8C2×D311yescharacteristic
82#418—61no—
83C2×#102—61no—
84#382—31no—
85#340—61no—
86#334—22no—
87#333—61no—
88#332—61no—
89#328—62no—
90#323—62no—
91C23×D3—21no—
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4#4111yescharacteristic
95C22×Dic3—21no—
96C22×Dic3—41no—
97C2×D12—21no—
98C2×D12—21no—
99C2×D12—21no—
100C2×D12—21no—
101C2×D12—21no—
102C2×D12SD1612yes—
103C2×C4×D3—41no—
104C2×C4×D3—41no—
105C2×C4×D3—82no—
106C2×Dic6—21no—
107C2×Dic6—21no—
108C3×M16—41no—
109C3×M16—41no—
110C2×C3×C8—22no—
111C2×C3×C8—22no—
112C2×C3×C8—41no—
113C2×C3×C8—41no—
114C2×C3×C8—41no—
115C3×C42C2×D411yescharacteristic
116C3×C42C2×D411yescharacteristic
117C3×C42#4111yescharacteristic
118C3×C42#4111yescharacteristic
119#217—41no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—41no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—41no—
129C4⋊Dic3—41no—
130Dic3⋊C4—41no—
131Dic3⋊C4—41no—
132C4×Dic3—41no—
133C2×C3⋊C8—41no—
134C22×D4—31no—
135C23×C4—61no—
136C2×M16—21no—
137C22×C8—21no—
138C4⋊Q8—31no—
139C4⋊Q8—31no—
140#125—61no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
143C4×Q8—61no—
144C4×Q8—61no—
145C4×D4—61no—
146C4×D4—61no—
147C4×D4—61no—
148C4×D4—61no—
149#117—61no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C42C22×D311yescharacteristic
154#107—62no—
155#106—62no—
156C4⋊C8—21no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—61no—
161C4⋊C8#6712yes—
162#102—62no—
163#102—62no—
164#102—122no—
165C22⋊C8—61no—
166C22⋊C8D1212yes—
167C4×C8—21no—
168C4×C8—61no—
169#95—61no—
170#95—61no—
171C23×C3#12111yescharacteristic
172C22×D3—22no—
173C22×D3—41no—
174C22×D3—41no—
175C22×D3—81no—
176C22×D3—81no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4#12011yescharacteristic
180C2×C3×C4#12111yescharacteristic
181C2×C3×C4#12111yescharacteristic
182C2×C3×C4#12311yescharacteristic
183C2×C3×C4#12311yescharacteristic
184C2×C3×C4#12411yescharacteristic
185C2×C3×C4C2×SD1611yescharacteristic
186C2×C3×C4#13511yescharacteristic
187C2×C3×C4#13711yescharacteristic
188C2×C3×C4#13711yescharacteristic
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—81no—
193C2×Dic3—81no—
194C2×Dic3—81no—
195C2×Dic3—81no—
196D12—22no—
197D12—22no—
198D12—41no—
199D12—41no—
200C4×D3—82no—
201C4×D3—82no—
202Dic6—81no—
203Dic6—81no—
204C3×C8—42no—
205C3×C8—42no—
206C3×C8—81no—
207C3⋊C8—81no—
208C24—61no—
209C2×Q8—61no—
210C2×Q8—61no—
211C2×D4—32no—
212C2×D4—61no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4—61no—
217C22×C4—61no—
218C22×C4—121no—
219C22×C4—121no—
220C22×C4—121no—
221C22×C4—242no—
222C22×C4C2×D1211yescharacteristic
223C22×C4D3×D411yescharacteristic
224C22×C4#23911yescharacteristic
225M16—41no—
226M16—41no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—121no—
233C4⋊C4—61no—
234C4⋊C4—61no—
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—121no—
244C22⋊C4—121no—
245C22⋊C4—121no—
246C42—121no—
247C42#23511yescharacteristic
248C42D3×D411yescharacteristic
249C42#23911yescharacteristic
250C42C2×#6711yescharacteristic
251C22×C3—21no—
252C22×C3—21no—
253C22×C3#39411yescharacteristic
254C22×C3#46511yescharacteristic
255C22×C3#47011yescharacteristic
256C2×D3—42no—
257C2×D3—82no—
258C2×D3—81no—
259C2×D3—81no—
260C2×D3—82no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4#44811yescharacteristic
266C3×C4#45211yescharacteristic
267C3×C4#48311yescharacteristic
268C3×C4#48511yescharacteristic
269Dic3—81no—
270Dic3—81no—
271Dic3—81no—
272Dic3—161no—
273C23—62no—
274C23—121no—
275C23—121no—
276C23—241no—
277C23—241no—
278C23#89511yescharacteristic
279Q8—241no—
280Q8—241no—
281D4—62no—
282D4—62no—
283D4—121no—
284D4—121no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—242no—
291C2×C4—242no—
292C2×C4—241no—
293C2×C4—241no—
294C2×C4—241no—
295C2×C4—241no—
296C2×C4#89411yescharacteristic
297C2×C4#89411yescharacteristic
298C2×C4#89511yescharacteristic
299C2×C4D3×SD1611yescharacteristic
300C2×C4#91511yescharacteristic
301C2×C4#91511yescharacteristic
302C2×C4#91611yescharacteristic
303C2×C4#93011yescharacteristic
304C2×C4#93711yescharacteristic
305C2×C4#94711yescharacteristic
306C8—42no—
307C8—42no—
308C8—81no—
309C8—241no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3#203511yescharacteristic
313C2×C3#204811yescharacteristic
314C2×C3#205311yescharacteristic
315D3—82no—
316C22—21no—
317C22—21no—
318C22—122no—
319C22—242no—
320C22—242no—
321C22—241no—
322C22—241no—
323C22#486411yescharacteristic
324C22#486711yescharacteristic
325C22#499111yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—241no—
331C4—241no—
332C4—241no—
333C4—481no—
334C4#486211yescharacteristic
335C4#486811yescharacteristic
336C4#517011yescharacteristic
337C4#517311yescharacteristic
338C3#1046811yescharacteristic, complemented, Sylow
339C2—21no—
340C2—21no—
341C2—242no—
342C2#7054711yescharacteristic
343C2#7058211yescharacteristic
344C2#7073611yescharacteristic
subgroupG/H
centerC22#4991
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1570C2
socleC22×C3#394
prime radicalC22×D12D4
hypercenter#1570D3
hypocenterC3#10468
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. #70793C2semidirect
6. #70437C2semidirect
7. #67713C2semidirect
30. #1570D3semidirect
338. C3#10468semidirect

Maximal subgroups

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

subgroupindex
#790592
C3×#15702
#707932
#707832
#704802
#704372
#677132
#104683