Finite Group Databasebeta

← #10698946

Subgroups — #10698946

Order 768.

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

normal subgroups only

This group has 398 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
1C3×#3077C211yesmaximal, characteristic
2#71838C211yesmaximal, characteristic, complemented
3#71828C211yesmaximal, characteristic, complemented
4#71605C211yesmaximal, characteristic
5#71601C211yesmaximal, characteristic
6#70493C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#10468—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C3×#478—22no—
12C3×#476—21no—
13C3×#474—21no—
14C3×#428—21no—
15C3×#423C2211yescharacteristic
16C3×#418—21no—
17C2×#832—21no—
18#4936C2211yescharacteristic
19#4925C2211yescharacteristic
20#4925C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22#4547—21no—
23#4546—21no—
24#4538—22no—
25#4529—21no—
26#4525—22no—
27#4480—22no—
28#3077D311yescharacteristic
29#1804—31no—
30#1570—31no—
31#1542—31no—
32#1518—31no—
33#1460—31no—
34#1286—31no—
35C22×C3×D4D411yescharacteristic
36C23×C3×C4—21no—
37C2×C3×M16—21no—
38C3×C4⋊Q8D411yescharacteristic
39C3×C4⋊Q8D411yescharacteristic
40C3×#125—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×#117—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C42C2311yescharacteristic
53C2×C4⋊Dic3—21no—
54C22×C3⋊C8—21no—
55C3×#107—22no—
56C3×#106—22no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8D412yes—
59C3×#102—22no—
60C3×#102—22no—
61#832—42no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64#830—21no—
65#830D412yes—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4×C3⋊C8—21no—
71C4×C3⋊C8—21no—
72#533C2×D311yescharacteristic
73C2×C4×D4C2×D311yescharacteristic
74#478—22no—
75#476—21no—
76#474—21no—
77#431—31no—
78#431—31no—
79#428—21no—
80#423C2×D311yescharacteristic
81C2×C4⋊C8—31no—
82#418—21no—
83C2×#102—61no—
84#382—31no—
85#340—61no—
86#334—62no—
87#333—61no—
88#332—61no—
89#328—62no—
90#323—62no—
91C24×C3—21no—
92C2×C3×Q8—21no—
93C2×C3×Q8—21no—
94C2×C3×D4—21no—
95C2×C3×D4—21no—
96C2×C3×D4—21no—
97C2×C3×D4—21no—
98C2×C3×D4—21no—
99C2×C3×D4SD1612yes—
100C22×C3×C4—21no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×C3×C4—82no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4111yescharacteristic
107C22×Dic3—41no—
108C3×M16—41no—
109C3×M16—41no—
110C2×C3×C8—22no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—41no—
122C3×C42—41no—
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42#4111yescharacteristic
126C3×C42#4111yescharacteristic
127C4⋊Dic3—41no—
128C4⋊Dic3—41no—
129C2×C3⋊C8—22no—
130C2×C3⋊C8—41no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—41no—
133C2×C3⋊C8—41no—
134C22×D4#6711yescharacteristic
135C23×C4—21no—
136C2×M16—21no—
137C22×C8—61no—
138C4⋊Q8#6711yescharacteristic
139C4⋊Q8#6711yescharacteristic
140#125—21no—
141C22⋊Q8—21no—
142C22⋊Q8—21no—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—21no—
149#117—21no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4#6711yescharacteristic
152C2×C22⋊C4—21no—
153C2×C42C22×D311yescharacteristic
154#107—22no—
155#106—22no—
156C4⋊C8—21no—
157C4⋊C8—61no—
158C4⋊C8—61no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161C4⋊C8#6712yes—
162#102—22no—
163#102—22no—
164#102—122no—
165C22⋊C8—32no—
166C22⋊C8—61no—
167C4×C8—61no—
168C4×C8—61no—
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×Q8—81no—
178C3×Q8—81no—
179C3×D4—22no—
180C3×D4—22no—
181C3×D4—41no—
182C3×D4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—82no—
189C2×C3×C4—82no—
190C2×C3×C4—81no—
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12311yescharacteristic
195C2×C3×C4#12311yescharacteristic
196C2×C3×C4#12411yescharacteristic
197C2×C3×C4C2×SD1611yescharacteristic
198C2×C3×C4#13511yescharacteristic
199C2×C3×C4#13711yescharacteristic
200C2×C3×C4#13711yescharacteristic
201C2×Dic3—81no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204C3×C8—42no—
205C3⋊C8—42no—
206C3⋊C8—81no—
207C3⋊C8—81no—
208C24—21no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—21no—
215C2×D4—21no—
216C2×D4#21512yes—
217C22×C4—21no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—82no—
221C22×C4—121no—
222C22×C4D3×D411yescharacteristic
223C22×C4#23711yescharacteristic
224C22×C4C2×#6711yescharacteristic
225M16—41no—
226M16—41no—
227C2×C8—22no—
228C2×C8—62no—
229C2×C8—121no—
230C2×C8—121no—
231C2×C8—121no—
232C2×C8—121no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C42—41no—
247C42#23711yescharacteristic
248C42#23711yescharacteristic
249C42C2×#6711yescharacteristic
250C42C2×#6711yescharacteristic
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—81no—
267C3×C4—81no—
268C3×C4#44811yescharacteristic
269C3×C4#45211yescharacteristic
270C3×C4#48311yescharacteristic
271C3×C4#48511yescharacteristic
272Dic3—161no—
273C23—22no—
274C23—41no—
275C23—41no—
276C23—81no—
277C23—81no—
278C23#93911yescharacteristic
279Q8—81no—
280Q8—81no—
281D4—22no—
282D4—22no—
283D4—41no—
284D4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—241no—
294C2×C4—241no—
295C2×C4—241no—
296C2×C4#93511yescharacteristic
297C2×C4#93511yescharacteristic
298C2×C4#93611yescharacteristic
299C2×C4#93811yescharacteristic
300C2×C4#93911yescharacteristic
301C2×C4C2×#21511yescharacteristic
302C2×C4#94211yescharacteristic
303C2×C4#95011yescharacteristic
304C2×C4#95111yescharacteristic
305C2×C4#95311yescharacteristic
306C8—42no—
307C8—122no—
308C8—241no—
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#504811yescharacteristic
324C22#505211yescharacteristic
325C22#522211yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—481no—
334C4#504311yescharacteristic
335C4#504511yescharacteristic
336C4#522711yescharacteristic
337C4#523911yescharacteristic
338C3#1046811yescharacteristic, complemented, Sylow
339C2—21no—
340C2—21no—
341C2—82no—
342C2#7169411yescharacteristic
343C2#7209211yescharacteristic
344C2#7211511yescharacteristic
subgroupG/H
centerC22#5222
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3077C2
socleC22×C3#394
prime radicalC22×C3×D4D4
hypercenter#3077D3
hypocenterC3#10468
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #71838C2semidirect
3. #71828C2semidirect
6. #70493C2semidirect
7. #67713C2semidirect
338. C3#10468semidirect

Maximal subgroups

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

subgroupindex
C3×#30772
#718382
#718282
#716052
#716012
#704932
#677132
#104683