Finite Group Databasebeta

← #10696077

Subgroups — #10696077

Order 768.

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

normal subgroups only

This group has 598 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×#2632C211yesmaximal, characteristic, complemented
2#82943C211yesmaximal, characteristic
3#82444C211yesmaximal, characteristic, complemented
4#71719C211yesmaximal, characteristic, complemented
5#71623C211yesmaximal, characteristic, complemented
6#69564C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#10020—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×#942—21no—
12C2×#932—21no—
13#5477C2211yescharacteristic
14C3×#399—21no—
15C3×#395—21no—
16C3×#394—21no—
17C3×#393—21no—
18C3×#388—22no—
19C3×#383C2211yescharacteristic
20#5227—21no—
21#5060—21no—
22#5059—22no—
23#5058—21no—
24#5027—21no—
25#5026—22no—
26#5021—21no—
27#4967—21no—
28#4926C2211yescharacteristic
29#4926C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4540—22no—
32#4535—22no—
33#4451—22no—
34#3153—31no—
35#3112—31no—
36#2632D311yescharacteristic, complemented
37#1533—31no—
38#1514—31no—
39#1491—31no—
40#1288—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×Q8D411yescharacteristic
43C22×C3×D4D411yescharacteristic
44C2×#235—21no—
45C3×#124—21no—
46C3×#124—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#216—22no—
63#942—44no—
64C2×#215—22no—
65C2×#215—22no—
66C2×#214—22no—
67#932—44no—
68C2×#213—22no—
69#930—21no—
70#930—21no—
71C2×#208—21no—
72#875—21no—
73#874D411yescharacteristic
74C4×D12—21no—
75C4×D12—21no—
76#869D411yescharacteristic
77C3×#95—22no—
78C3×#95—21no—
79C3×#95—21no—
80C3×#95D412yes—
81#835—22no—
82#830—21no—
83#830—21no—
84#809—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89#803—21no—
90#803D412yes—
91C4×C3⋊C8D412yes—
92C2×#137—61no—
93C2×AGL(1,ℤ/8)—61no—
94#532—31no—
95C2×#124C2×D311yescharacteristic
96C2×C4×Q8C2×D311yescharacteristic
97#497—61no—
98#496—61no—
99#495—62no—
100#480—61no—
101#477—61no—
102#460—61no—
103#448—61no—
104#440—62no—
105#432—31no—
106#432—31no—
107C4×M16—31no—
108#399—21no—
109#395—21no—
110#394—21no—
111#393—21no—
112#388—22no—
113#383C2×D311yescharacteristic
114#329—62no—
115#327—62no—
116Q8⋊C8—62no—
117C24×C3—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—42no—
123C2×C3×Q8—42no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—42no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C2×#67—41no—
135#235—42no—
136#235—42no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×C4×D3—41no—
141C2×Dic6—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—82no—
147C3×C22⋊C4—44no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—82no—
153C3×C42—41no—
154C3×C42—41no—
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42#4111yescharacteristic
159#216—44no—
160#215—44no—
161#215—44no—
162#214—44no—
163#213—44no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168#208—24no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—41no—
172C2×#41—61no—
173C22×Q8#6711yescharacteristic
174C22×Q8#6711yescharacteristic
175C22×D4#6711yescharacteristic
176#137—124no—
177AGL(1,ℤ/8)—124no—
178C2×Q16—62no—
179C2×SD16—62no—
180C2×SD16—62no—
181C2×SD16—62no—
182C2×D8—62no—
183C2×M16—61no—
184C4⋊Q8—31no—
185#127—31no—
186#124—21no—
187#124—21no—
188#124—21no—
189#124—21no—
190#124—61no—
191#121—61no—
192#121—61no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×D4—61no—
198C4×D4—61no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4#6711yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C42—21no—
207C2×C42C22×D311yescharacteristic
208#107—62no—
209C4⋊C8—61no—
210C4⋊C8—61no—
211#103—62no—
212#102—62no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215#97—32no—
216#97—61no—
217#97—61no—
218C4×C8—32no—
219#95—22no—
220#95—21no—
221#95—21no—
222#95#6712yes—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—41no—
226C23×C3—42no—
227C23×C3#12011yescharacteristic
228C22×D3—41no—
229C3×Q8—22no—
230C3×Q8—22no—
231C3×Q8—41no—
232C3×Q8—41no—
233C3×Q8—42no—
234C3×Q8—81no—
235C3×D4—42no—
236C3×D4—81no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12711yescharacteristic
256#67—82no—
257C2×Dic3—41no—
258D12—41no—
259D12—41no—
260D12—42no—
261D12—81no—
262C4×D3—82no—
263Dic6—41no—
264Dic6—41no—
265C3⋊C8—24no—
266C3⋊C8—42no—
267C3⋊C8—81no—
268C24—21no—
269#41—122no—
270#41—122no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—61no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—42no—
281C2×D4—61no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—121no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—121no—
290C22×C4C2×#6711yescharacteristic
291C22×C4C2×#6711yescharacteristic
292C22×C4C2×#6711yescharacteristic
293Q16—124no—
294SD16—124no—
295SD16—124no—
296SD16—124no—
297D8—124no—
298M16—64no—
299C2×C8—62no—
300C2×C8—62no—
301C2×C8—121no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—82no—
309C22⋊C4—44no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—82no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C42—41no—
318C42—41no—
319C42D3×D411yescharacteristic
320C42D3×D411yescharacteristic
321C42D3×D411yescharacteristic
322C42#23711yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—84no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3#39211yescharacteristic
330C22×C3#45312yes—
331C2×D3—81no—
332C2×D3—81no—
333C2×D3—81no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—42no—
341C3×C4—82no—
342C3×C4—81no—
343C3×C4—81no—
344Dic3—81no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23—42no—
349C23—121no—
350C23#95311yescharacteristic
351Q8—22no—
352Q8—22no—
353Q8—41no—
354Q8—41no—
355Q8—42no—
356Q8—81no—
357Q8—121no—
358Q8—121no—
359D4—42no—
360D4—81no—
361D4—121no—
362D4—121no—
363D4—122no—
364D4—241no—
365D4—242no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—121no—
380C2×C4—242no—
381C2×C4#93711yescharacteristic
382C2×C4#93711yescharacteristic
383C2×C4#93811yescharacteristic
384C2×C4#93911yescharacteristic
385C2×C4#93911yescharacteristic
386C2×C4#94011yescharacteristic
387C8—64no—
388C8—122no—
389C8—241no—
390C2×C3—22no—
391C2×C3—161no—
392C2×C3#200211yescharacteristic
393C2×C3#200911yescharacteristic
394C2×C3#201011yescharacteristic
395D3—161no—
396C22—21no—
397C22—21no—
398C22—84no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—241no—
403C22—241no—
404C22—241no—
405C22#522311yescharacteristic
406C22#524112yes—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—82no—
415C4—81no—
416C4—81no—
417C4—241no—
418C3#1002011yescharacteristic, complemented, Sylow
419C2—22no—
420C2—161no—
421C2—481no—
422C2#7171511yescharacteristic
423C2#7185611yescharacteristic
424C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2632C2
socleC22×C3#392
prime radical#82943C2
hypercenter#2632D3
hypocenterC3#10020
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2632

Complements

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

NH ≅ G/Ntype
1. C3×#2632C2semidirect
3. #82444C2semidirect
4. #71719C2semidirect
5. #71623C2semidirect
6. #69564C2semidirect
7. #67633C2semidirect
36. #2632D3semidirect
418. C3#10020semidirect

Maximal subgroups

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

subgroupindex
C3×#26322
#829432
#824442
#717192
#716232
#695642
#676332
#100203