Finite Group Databasebeta

← #10698949

Subgroups — #10698949

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#82296C211yesmaximal, characteristic
2C3×#1804C211yesmaximal, characteristic
3#71601C211yesmaximal, characteristic
4#70437C211yesmaximal, characteristic, complemented
5#70421C211yesmaximal, characteristic, complemented
6#69546C211yesmaximal, characteristic, complemented
7#67631C211yesmaximal, characteristic, complemented
8#10468—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C2×C3×#102—21no—
13C3×#382C2211yescharacteristic
14#5215—21no—
15#5009—22no—
16#5008—21no—
17#5007—21no—
18#4964—21no—
19#4925C2211yescharacteristic
20#4925C2211yescharacteristic
21#4924C2211yescharacteristic
22C3×#340—21no—
23#4525—22no—
24#4486—21no—
25#4481—21no—
26#4480—22no—
27#4452—22no—
28#3077—31no—
29#1804D311yescharacteristic
30#1570—31no—
31#1542—31no—
32#1518—31no—
33#1460—31no—
34#1286—31no—
35C22×C3×D4D411yescharacteristic
36C23×C3×C4—21no—
37C22×C3×C8—21no—
38C3×C4×D4—21no—
39C3×C4×D4—21no—
40C3×C4×D4—21no—
41C3×C4×D4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C22⋊C4—21no—
45C2×C3×C42C2311yescharacteristic
46#926—21no—
47#924—21no—
48#924—21no—
49C2×#208—21no—
50#870—21no—
51#869D411yescharacteristic
52#869D411yescharacteristic
53C4×Dic6—21no—
54C4×Dic6—21no—
55C3×C4⋊C8—21no—
56C3×C4⋊C8—21no—
57C3×#102—42no—
58C3×#95—21no—
59C3×#95—21no—
60#832—22no—
61#832—22no—
62#830—21no—
63#830D412yes—
64#808—22no—
65#807—22no—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)D412yes—
70C4×C3⋊C8—21no—
71C4×C3⋊C8—21no—
72#533—31no—
73C2×C4×D4C2×D311yescharacteristic
74#478—62no—
75#476—61no—
76#474—61no—
77#431—31no—
78#431—31no—
79#428—61no—
80#423—31no—
81C2×C4⋊C8C2×D311yescharacteristic
82#418—61no—
83C2×#102—21no—
84#382C2×D311yescharacteristic
85#340—21no—
86#334—62no—
87#333—61no—
88#332—61no—
89#328—62no—
90#323—62no—
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—41no—
102C22×C3×C4—82no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4111yescharacteristic
106C2×Dic6—21no—
107C2×Dic6—21no—
108C2×C3×C8—41no—
109C2×C3×C8—41no—
110C2×C3×C8—41no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C42C2×D411yescharacteristic
118C3×C42C2×D411yescharacteristic
119C3×C42#4111yescharacteristic
120C3×C42#4111yescharacteristic
121#217—41no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—21no—
125C4⋊Dic3—21no—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128C4×Dic3—41no—
129#208—41no—
130#208—41no—
131C2×C3⋊C8—22no—
132C2×C3⋊C8—22no—
133C2×C3⋊C8—41no—
134C22×D4#6711yescharacteristic
135C23×C4—21no—
136C2×M16—61no—
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—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—21no—
149#117—61no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4#6711yescharacteristic
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—42no—
163#102—62no—
164#102—62no—
165C22⋊C8—32no—
166C22⋊C8—61no—
167C4×C8—61no—
168C4×C8—61no—
169#95—21no—
170#95—21no—
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—81no—
188C2×C3×C4—81no—
189C2×C3×C4—81no—
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12111yescharacteristic
193C2×C3×C4#12311yescharacteristic
194C2×C3×C4#12311yescharacteristic
195C2×C3×C4#12411yescharacteristic
196C2×C3×C4C2×SD1611yescharacteristic
197C2×C3×C4#13511yescharacteristic
198C2×C3×C4#13711yescharacteristic
199C2×C3×C4#13711yescharacteristic
200C2×Dic3—41no—
201C2×Dic3—41no—
202Dic6—81no—
203Dic6—81no—
204C3×C8—81no—
205C3⋊C8—42no—
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#21412yes—
217C22×C4—21no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—82no—
222C22×C4D3×D411yescharacteristic
223C22×C4#23711yescharacteristic
224C22×C4C2×#6711yescharacteristic
225M16—121no—
226M16—121no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—62no—
231C2×C8—62no—
232C2×C8—121no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—121no—
242C4⋊C4—121no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—121no—
246C42—121no—
247C42#23511yescharacteristic
248C42#23511yescharacteristic
249C42D3×D411yescharacteristic
250C42D3×D411yescharacteristic
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—161no—
267C3×C4#44811yescharacteristic
268C3×C4#45211yescharacteristic
269C3×C4#48311yescharacteristic
270C3×C4#48511yescharacteristic
271Dic3—81no—
272Dic3—81no—
273C23—22no—
274C23—41no—
275C23—41no—
276C23—81no—
277C23—81no—
278C23#93911yescharacteristic
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—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—121no—
295C2×C4—121no—
296C2×C4#88311yescharacteristic
297C2×C4#88411yescharacteristic
298C2×C4#88411yescharacteristic
299C2×C4#88511yescharacteristic
300C2×C4#92911yescharacteristic
301C2×C4C2×#21411yescharacteristic
302C2×C4#93711yescharacteristic
303C2×C4#94211yescharacteristic
304C2×C4#95011yescharacteristic
305C2×C4#95111yescharacteristic
306C8—81no—
307C8—122no—
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#495911yescharacteristic
324C22#504111yescharacteristic
325C22#505211yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—81no—
331C4—161no—
332C4—241no—
333C4—241no—
334C4#497111yescharacteristic
335C4#499611yescharacteristic
336C4#504011yescharacteristic
337C4#504711yescharacteristic
338C3#1046811yescharacteristic, complemented, Sylow
339C2—21no—
340C2—21no—
341C2—82no—
342C2#6952611yescharacteristic
343C2#7029611yescharacteristic
344C2#7033211yescharacteristic
subgroupG/H
centerC22#4959
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1804C2
socleC22×C3#394
prime radicalC22×C3×D4D4
hypercenter#1804D3
hypocenterC3#10468
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
4. #70437C2semidirect
5. #70421C2semidirect
6. #69546C2semidirect
7. #67631C2semidirect
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×#18042
#716012
#704372
#704212
#695462
#676312
#104683