Finite Group Databasebeta

← #10696091

Subgroups — #10696091

Order 768.

576 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 81 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×#2631C211yesmaximal, characteristic
2#82939C211yesmaximal, characteristic, complemented
3#82315C211yesmaximal, characteristic, complemented
4#71719C211yesmaximal, characteristic, complemented
5#71599C211yesmaximal, characteristic, complemented
6#69566C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#10022—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#397—21no—
13C3×#393—21no—
14C3×#392—21no—
15C3×#392—21no—
16C3×#386—22no—
17C3×#383C2211yescharacteristic
18#5225—21no—
19#5218—21no—
20#5215—21no—
21#5056—22no—
22#5055—21no—
23#5055—21no—
24#5019—21no—
25#5018—22no—
26#5007—21no—
27#4970—21no—
28#4926C2211yescharacteristic
29#4926C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4540—22no—
32D4⋊(C3⋊C8)—22no—
33#4452—22no—
34#3140—31no—
35#3110—31no—
36#2631D311yescharacteristic
37#1533—31no—
38#1515—31no—
39#1490—31no—
40#1288—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4—22no—
43C22×C3×D4D411yescharacteristic
44C22×C3×D4D411yescharacteristic
45C23×C3×C4—21no—
46C3×#124—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C42C2311yescharacteristic
61C2×#214—22no—
62#926—21no—
63#924—21no—
64#924—21no—
65C2×#208—21no—
66#870—21no—
67#869D411yescharacteristic, complemented
68#869D411yescharacteristic, complemented
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×#95—22no—
72C3×#95—21no—
73C3×#95—21no—
74C3×#95D412yes—
75#835—22no—
76#835—22no—
77#832—22no—
78#832—22no—
79#832—22no—
80#830—21no—
81#830—21no—
82#810—22no—
83#807—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86#803—21no—
87#803—21no—
88#803D412yescomplemented
89C4×C3⋊C8D412yescomplemented
90#533—31no—
91C2×#124C2×D311yescharacteristic
92C2×C4×D4C2×D311yescharacteristic
93#489—61no—
94#489—61no—
95#488—62no—
96#484—61no—
97#474—61no—
98#461—61no—
99#450—61no—
100#440—62no—
101#432—31no—
102#432—31no—
103#419—61no—
104#418—61no—
105C4×M16—31no—
106#397—21no—
107#393—21no—
108#392—21no—
109#392—21no—
110#386—22no—
111#383C2×D311yescharacteristic
112#329—62no—
113#328—62no—
114D4⋊C8—62no—
115C24×C3—21no—
116C24×C3—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—42no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—42no—
125C2×C3×D4—42no—
126C2×C3×D4—44no—
127C2×C3×D4—82no—
128C2×C3×D4—82no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—82no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C2×Dic6—21no—
139C2×Dic6—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—82no—
149C3×C22⋊C4—82no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154#217—41no—
155#214—44no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162C4×Dic3—41no—
163#208—42no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167C22×Q8#6711yescharacteristic
168C22×D4—22no—
169C22×D4#6711yescharacteristic
170C22×D4#6711yescharacteristic
171C23×C4—21no—
172C2×SD16—62no—
173C2×M16—61no—
174C4⋊Q8—31no—
175C4⋊Q8—31no—
176#125—61no—
177#124—21no—
178#124—21no—
179#124—21no—
180#124—21no—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189#117—61no—
190C2×C4⋊C4#6711yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C42C22×D311yescharacteristic
197#107—62no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200#103—62no—
201#103—62no—
202#103—62no—
203#102—62no—
204#102—62no—
205#102—62no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208#97—32no—
209#97—61no—
210#97—61no—
211C4×C8—32no—
212#95—22no—
213#95—21no—
214#95—21no—
215#95#6712yes—
216C23×C3—22no—
217C23×C3—42no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—42no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3—41no—
225C23×C3—42no—
226C23×C3#12011yescharacteristic
227C3×Q8—42no—
228C3×Q8—81no—
229C3×D4—22no—
230C3×D4—22no—
231C3×D4—41no—
232C3×D4—41no—
233C3×D4—42no—
234C3×D4—81no—
235C3×D4—88no—
236C2×C3×C4—22no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—82no—
247C2×C3×C4—82no—
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12711yescharacteristic
254C2×Dic3—41no—
255C2×Dic3—41no—
256Dic6—42no—
257Dic6—81no—
258C3⋊C8—42no—
259C3⋊C8—42no—
260C3⋊C8—81no—
261C24—21no—
262C24—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—42no—
266C2×Q8—61no—
267C2×Q8—61no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—44no—
275C2×D4—82no—
276C2×D4—82no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—82no—
283C22×C4C2×#6711yescharacteristic
284C22×C4C2×#6711yescharacteristic
285C22×C4C2×#6711yescharacteristic
286SD16—124no—
287M16—122no—
288C2×C8—62no—
289C2×C8—62no—
290C2×C8—121no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—121no—
308C42—121no—
309C42D3×D411yescharacteristic
310C42D3×D411yescharacteristic
311C42D3×D411yescharacteristic
312C42#23711yescharacteristic
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—44no—
316C22×C3—42no—
317C22×C3—82no—
318C22×C3—82no—
319C22×C3—82no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3#39211yescharacteristic
326C22×C3#45512yes—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—82no—
334C3×C4—81no—
335C3×C4—81no—
336Dic3—81no—
337Dic3—81no—
338C23—22no—
339C23—42no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—42no—
346C23—41no—
347C23—42no—
348C23#95311yescharacteristic
349Q8—42no—
350Q8—81no—
351Q8—122no—
352Q8—241no—
353D4—22no—
354D4—22no—
355D4—41no—
356D4—41no—
357D4—42no—
358D4—81no—
359D4—88no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4#93711yescharacteristic
375C2×C4#93711yescharacteristic
376C2×C4#93811yescharacteristic
377C2×C4#93911yescharacteristic
378C2×C4#93911yescharacteristic
379C2×C4#94011yescharacteristic
380C8—122no—
381C8—122no—
382C8—241no—
383C2×C3—22no—
384C2×C3—82no—
385C2×C3—161no—
386C2×C3#200411yescharacteristic
387C2×C3#200811yescharacteristic
388C2×C3#201011yescharacteristic
389C22—21no—
390C22—21no—
391C22—44no—
392C22—42no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22#522311yescharacteristic
402C22#524212yes—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—82no—
410C4—81no—
411C4—81no—
412C4—241no—
413C4—241no—
414C3#1002211yescharacteristic, complemented, Sylow
415C2—22no—
416C2—82no—
417C2—161no—
418C2#7171511yescharacteristic
419C2#7185711yescharacteristic
420C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2631C2
socleC22×C3#392
hypercenter#2631D3
hypocenterC3#10022
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2631

Complements

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

NH ≅ G/Ntype
2. #82939C2semidirect
3. #82315C2semidirect
4. #71719C2semidirect
5. #71599C2semidirect
6. #69566C2semidirect
7. #67633C2semidirect
67. #869D4semidirect
68. #869D4semidirect
88. #803D4semidirect
89. C4×C3⋊C8D4semidirect
414. C3#10022semidirect

Maximal subgroups

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

subgroupindex
C3×#26312
#829392
#823152
#717192
#715992
#695662
#676332
#100223