Finite Group Databasebeta

← #10690510

Subgroups — #10690510

Order 768.

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

normal subgroups only

This group has 426 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
1#83281C211yesmaximal, characteristic
2#79267C211yesmaximal, characteristic, complemented
3C3×#1484C211yesmaximal, characteristic
4#71801C211yesmaximal, characteristic
5#69862C211yesmaximal, characteristic, complemented
6#67739C211yesmaximal, characteristic, complemented
7#67667C211yesmaximal, characteristic, complemented
8#9070—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5233—22no—
14#5079C411yescharacteristic
15#5079C411yescharacteristic
16#5070C411yescharacteristic
17#5070C411yescharacteristic
18#5056—22no—
19#4925C2211yescharacteristic
20C4×#208C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22#4831—21no—
23#4830—21no—
24#4720—22no—
25#4679—21no—
26C8×Dic6—21no—
27C3×D4⋊C8—21no—
28C3×D4⋊C8—21no—
29D4⋊(C3⋊C8)—21no—
30D4⋊(C3⋊C8)—21no—
31#4465—22no—
32#3131—31no—
33#2986—31no—
34#1526—31no—
35#1484—31no—
36#1484D311yescharacteristic
37#1278—31no—
38#1278—31no—
39C2×C3×#41—21no—
40C2×C3×M16—21no—
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×#127C2×C411yescharacteristic
43C3×#124—21no—
44C3×#121—21no—
45C3×#121—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C2×C3×C42C2311yescharacteristic
49#926D412yes—
50#924—21no—
51#924—21no—
52C2×#208—21no—
53C22×C3⋊C8—21no—
54#870D412yes—
55#869C2×C411yescharacteristic
56#869C2×C411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8D411yescharacteristic
60C3×C4⋊C8D411yescharacteristic
61C3×C22⋊C8—21no—
62C3×#97—21no—
63C3×C4×C8—21no—
64C3×C4×C8—21no—
65#835—22no—
66#835—22no—
67#832—22no—
68#832—22no—
69#830—21no—
70#815—22no—
71Dic3⋊C8—22no—
72Dic3⋊C8—22no—
73C8×Dic3—22no—
74C4⋊(C3⋊C8)D411yescharacteristic
75C4⋊(C3⋊C8)D411yescharacteristic
76#803—21no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80C4×C3⋊C8C2×C411yescharacteristic
81C4×C3⋊C8C2×C411yescharacteristic
82#533—31no—
83#532C2×D311yescharacteristic
84#488—62no—
85#487—31no—
86#487—31no—
87#486—31no—
88#486—31no—
89#446—61no—
90#446—61no—
91C8×Q8—61no—
92C8×Q8—61no—
93#433—62no—
94#431—31no—
95#431C2×D311yescharacteristic
96#416—62no—
97C4×M16—31no—
98C4×M16C2×D311yescharacteristic
99C2×C4×C8—31no—
100#330—62no—
101D4⋊C8—21no—
102D4⋊C8—21no—
103D4⋊C8—61no—
104D4⋊C8—61no—
105C3×#41—42no—
106C3×#41—42no—
107C2×C3×Q8—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—41no—
112C22×C3×C4—41no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4111yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C2×Dic6—21no—
117C2×Dic6—21no—
118C3×M16—22no—
119C3×M16—41no—
120C2×C3×C8—21no—
121C2×C3×C8—21no—
122C2×C3×C8—41no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C42C22×C411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131#217—22no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3C22⋊C412yes—
135C4⋊Dic3C22⋊C412yes—
136Dic3⋊C4—22no—
137Dic3⋊C4—22no—
138C4×Dic3—22no—
139#208—22no—
140#208—41no—
141C2×C3⋊C8—21no—
142C2×C3⋊C8—21no—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—21no—
145C2×C3⋊C8—41no—
146C2×C3⋊C8—41no—
147C2×C3⋊C8—41no—
148C2×#41—21no—
149C2×M16—21no—
150C2×M16—61no—
151C22×C8—61no—
152C4⋊Q8—31no—
153C4⋊Q8—31no—
154C4⋊Q8C4×D311yescharacteristic
155#127C4×D311yescharacteristic
156#125—32no—
157#124—21no—
158C22⋊Q8—61no—
159C22⋊Q8—61no—
160#121—21no—
161#121—21no—
162C4×Q8—61no—
163C4×Q8—61no—
164C4×D4—21no—
165C4×D4—21no—
166#117—32no—
167C2×C42C22×D311yescharacteristic
168C4⋊C8—31no—
169C4⋊C8—31no—
170C4⋊C8—62no—
171C4⋊C8—62no—
172C4⋊C8#6711yescharacteristic
173C4⋊C8#6711yescharacteristic
174#103—62no—
175#103—62no—
176#102—62no—
177#102—62no—
178C22⋊C8—21no—
179C22⋊C8—61no—
180#97—21no—
181#97—62no—
182#97—61no—
183C4×C8—21no—
184C4×C8—21no—
185C4×C8—31no—
186C4×C8—31no—
187C4×C8—62no—
188C4×C8—61no—
189C4×C8—61no—
190C4×C8—61no—
191C23×C3—41no—
192C23×C3#12311yescharacteristic
193C3×Q8—42no—
194C3×D4—42no—
195C3×D4—41no—
196C3×D4—41no—
197C3×D4—41no—
198C3×D4—41no—
199C3×D4—82no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—41no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4C2×C22⋊C411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×Dic3—22no—
216C2×Dic3—22no—
217Dic6—41no—
218Dic6—41no—
219Dic6—41no—
220Dic6—41no—
221C3×C8—42no—
222C3×C8—41no—
223C3×C8—41no—
224C3⋊C8—41no—
225C3⋊C8—41no—
226C3⋊C8—42no—
227C3⋊C8—41no—
228C3⋊C8—41no—
229#41—42no—
230#41—42no—
231C2×Q8—21no—
232C2×Q8—61no—
233C2×Q8—61no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—41no—
238C22×C4—41no—
239C22×C4#23511yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4#23711yescharacteristic
242M16—22no—
243M16—41no—
244M16—62no—
245M16—121no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—41no—
249C2×C8—61no—
250C2×C8—61no—
251C2×C8—61no—
252C2×C8—61no—
253C2×C8—121no—
254C2×C8—121no—
255C2×C8—121no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—32no—
259C4⋊C4—32no—
260C4⋊C4—61no—
261C4⋊C4—62no—
262C4⋊C4—61no—
263C4⋊C4—62no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—62no—
267C42—62no—
268C42C2×C4×D311yescharacteristic
269C42D3×D411yescharacteristic
270C42D3×D411yescharacteristic
271C42C2×#6711yescharacteristic
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—81no—
276C22×C3—81no—
277C22×C3—81no—
278C22×C3#38611yescharacteristic
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—81no—
283C3×C4#44311yescharacteristic
284C3×C4#44311yescharacteristic
285C3×C4#44411yescharacteristic
286C3×C4#44411yescharacteristic
287C3×C4#45411yescharacteristic
288C3×C4#45411yescharacteristic
289Dic3—42no—
290Dic3—42no—
291C23—41no—
292C23#88311yescharacteristic
293Q8—42no—
294Q8—121no—
295Q8—121no—
296Q8—121no—
297Q8—121no—
298D4—42no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—82no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—62no—
312C2×C4—62no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4D3×C22⋊C411yescharacteristic
316C2×C4#67⋊C411yescharacteristic
317C2×C4#88411yescharacteristic
318C2×C4C4×#6711yescharacteristic
319C2×C4#93711yescharacteristic
320C2×C4#93811yescharacteristic
321C8—42no—
322C8—41no—
323C8—41no—
324C8—121no—
325C8—121no—
326C8—122no—
327C8—121no—
328C8—121no—
329C2×C3—41no—
330C2×C3—161no—
331C2×C3#187611yescharacteristic
332C2×C3#187611yescharacteristic
333C2×C3#188811yescharacteristic
334C22—21no—
335C22—21no—
336C22—21no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22#495611yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—81no—
345C4—122no—
346C4—122no—
347C4#482311yescharacteristic
348C4#482411yescharacteristic
349C4#514411yescharacteristic
350C4#514511yescharacteristic
351C4#520111yescharacteristic
352C4#520111yescharacteristic
353C3#907011yescharacteristic, complemented, Sylow
354C2—41no—
355C2—161no—
356C2#7001111yescharacteristic
357C2#7032511yescharacteristic
358C2#7039011yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1484C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1484D3
hypocenterC3#9070
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1484

Complements

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

NH ≅ G/Ntype
2. #79267C2semidirect
5. #69862C2semidirect
6. #67739C2semidirect
7. #67667C2semidirect
353. C3#9070semidirect

Maximal subgroups

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

subgroupindex
#832812
#792672
C3×#14842
#718012
#698622
#677392
#676672
#90703