Finite Group Databasebeta

← #10697245

Subgroups — #10697245

Order 768.

496 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 103 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
2C2×#5007C211yesmaximal, characteristic
3#79272C211yesmaximal, characteristic, complemented
4C3×#1460C211yesmaximal, characteristic
5#70779C211yesmaximal, characteristic, complemented
6#70768C211yesmaximal, characteristic, complemented
7#67745C211yesmaximal, characteristic, complemented
8#10190—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5215—21no—
15C2×#832—21no—
16#5009—22no—
17#5008—21no—
18#5007—22no—
19#5007—21no—
20#5007—22no—
21#4964—21no—
22C2×#807—21no—
23#4924C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4850—22no—
26#4722—22no—
27#4682—21no—
28#4681—21no—
29C3×#323—22no—
30#4481—22no—
31#4456—22no—
32#3230—31no—
33#3077—31no—
34C2×#474—31no—
35#1548—31no—
36#1481—31no—
37#1460D311yescharacteristic
38#1301—31no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C22×Dic6—21no—
42C3×C4×D4—21no—
43C3×C4×D4—21no—
44C3×C4×D4—21no—
45C3×C4×D4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49#926—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52#924—21no—
53#924—21no—
54C2×#208—21no—
55C22×C3⋊C8—21no—
56#870—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62#869D412yes—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8D412yes—
69C3×C4×C8—21no—
70C3×C4×C8—21no—
71#832—24no—
72#832—22no—
73#832—22no—
74#818—24no—
75#816—24no—
76#808—22no—
77#807—24no—
78#807—22no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D412yes—
82C4⋊(C3⋊C8)D412yes—
83#533—31no—
84C2×C4⋊Q8—31no—
85C2×C4×D4C2×D311yescharacteristic
86#500—61no—
87#487—61no—
88#484—62no—
89#478—62no—
90#477—62no—
91#476—61no—
92#474—62no—
93#474—61no—
94#474—62no—
95#431C2×D311yescharacteristic
96#431C2×D311yescharacteristic
97#428—61no—
98C2×#107—61no—
99#423—31no—
100C2×C4⋊C8—31no—
101#418—61no—
102C2×#102—61no—
103#333—62no—
104Q8⋊C8—62no—
105#323—22no—
106C24×C3—21no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4Q1612yes—
113C22×C3×C4—21no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—82no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×Dic3—41no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×Dic6—22no—
124C2×Dic6—41no—
125C2×Dic6—82no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42C2×D411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136#217—41no—
137C4⋊Dic3—22no—
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—22no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145Dic3⋊C4—41no—
146Dic3⋊C4—41no—
147C4×Dic3—41no—
148#208—41no—
149#208—41no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—41no—
153C2×C3⋊C8—41no—
154C22×Q8—61no—
155C22×D4D1211yescharacteristic
156C23×C4—21no—
157C2×M16—61no—
158C22×C8—61no—
159C4⋊Q8—32no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165#125—61no—
166C22⋊Q8—61no—
167C22⋊Q8—61no—
168C4×Q8—61no—
169C4×Q8—61no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—21no—
174#117—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4D1211yescharacteristic
178C2×C22⋊C4—21no—
179C2×C42C22×D311yescharacteristic
180#107—64no—
181#107—62no—
182#107—64no—
183#106—62no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—32no—
187C4⋊C8—32no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190#103—64no—
191#102—64no—
192#102—62no—
193#102—62no—
194C22⋊C8—21no—
195C22⋊C8#6712yes—
196C4×C8—21no—
197C4×C8—21no—
198C23×C3—22no—
199C23×C3—41no—
200C23×C3—41no—
201C23×C3—81no—
202C23×C3—81no—
203C23×C3#12111yescharacteristic
204C3×D4—22no—
205C3×D4—22no—
206C3×D4—41no—
207C3×D4—41no—
208C2×C3×C4—22no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—82no—
212C2×C3×C4—82no—
213C2×C3×C4—81no—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12711yescharacteristic
220C2×C3×C4C2×Q1611yescharacteristic
221C2×C3×C4#13511yescharacteristic
222C2×C3×C4#13711yescharacteristic
223C2×C3×C4#13711yescharacteristic
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—42no—
227C2×Dic3—82no—
228Dic6—42no—
229Dic6—42no—
230Dic6—82no—
231Dic6—82no—
232C3×C8—42no—
233C3×C8—81no—
234C3⋊C8—42no—
235C3⋊C8—42no—
236C24—21no—
237C2×Q8—61no—
238C2×Q8—62no—
239C2×Q8—61no—
240C2×Q8—121no—
241C2×Q8—242no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4Dic1212yes—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—82no—
252C22×C4—121no—
253C22×C4C2×D1211yescharacteristic
254C22×C4#23511yescharacteristic
255C22×C4C2×#6711yescharacteristic
256M16—121no—
257M16—121no—
258C2×C8—22no—
259C2×C8—41no—
260C2×C8—62no—
261C2×C8—62no—
262C2×C8—121no—
263C2×C8—121no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—61no—
267C4⋊C4—62no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—62no—
271C4⋊C4—61no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—121no—
279C42—121no—
280C42D3×D411yescharacteristic
281C42D3×D411yescharacteristic
282C42D3×D411yescharacteristic
283C42D3×D411yescharacteristic
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—42no—
287C22×C3—82no—
288C22×C3—82no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3#39211yescharacteristic
292C22×C3#46711yescharacteristic
293C22×C3#47011yescharacteristic
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—81no—
298C3×C4#45111yescharacteristic
299C3×C4#45111yescharacteristic
300C3×C4#45211yescharacteristic
301C3×C4#45211yescharacteristic
302C3×C4#47212yes—
303Dic3—81no—
304Dic3—81no—
305Dic3—82no—
306C23—22no—
307C23—41no—
308C23—41no—
309C23—81no—
310C23—81no—
311C23#93011yescharacteristic
312Q8—122no—
313Q8—122no—
314Q8—242no—
315Q8—242no—
316D4—22no—
317D4—22no—
318D4—41no—
319D4—41no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—122no—
329C2×C4—242no—
330C2×C4#88211yescharacteristic
331C2×C4#88211yescharacteristic
332C2×C4#88411yescharacteristic
333C2×C4#88411yescharacteristic
334C2×C4#90411yescharacteristic
335C2×C4C2×Dic1211yescharacteristic
336C2×C4#90911yescharacteristic
337C2×C4#90911yescharacteristic
338C2×C4#93711yescharacteristic
339C2×C4#94011yescharacteristic
340C8—42no—
341C8—81no—
342C8—122no—
343C8—122no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—82no—
347C2×C3#200111yescharacteristic
348C2×C3#200911yescharacteristic
349C2×C3#201811yescharacteristic
350C22—21no—
351C22—21no—
352C22—42no—
353C22—82no—
354C22—82no—
355C22—81no—
356C22—81no—
357C22#496011yescharacteristic
358C22#511611yescharacteristic
359C22#513711yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—81no—
364C4—241no—
365C4—242no—
366C4—241no—
367C4#473311yescharacteristic
368C4#473311yescharacteristic
369C4#473911yescharacteristic
370C4#473911yescharacteristic
371C4#474912yes—
372C3#1019011yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—82no—
376C2#6996411yescharacteristic
377C2#7005311yescharacteristic
378C2#7009011yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1460C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#1460D3
hypocenterC3#10190
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
3. #79272C2semidirect
5. #70779C2semidirect
6. #70768C2semidirect
7. #67745C2semidirect
372. C3#10190semidirect

Maximal subgroups

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

subgroupindex
#822962
C2×#50072
#792722
C3×#14602
#707792
#707682
#677452
#101903