Finite Group Databasebeta

← #10696750

Subgroups — #10696750

Order 768.

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

normal subgroups only

This group has 578 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×#3153C211yesmaximal, characteristic, complemented
2#83308C211yesmaximal, characteristic, complemented
3#82947C211yesmaximal, characteristic, complemented
4#71803C211yesmaximal, characteristic, complemented
5#71720C211yesmaximal, characteristic, complemented
6#69853C211yesmaximal, characteristic, complemented
7#67670C211yesmaximal, characteristic, complemented
8#10120—31nomaximal, Sylow
9C2×C3×#137—21no—
10C2×C3×AGL(1,ℤ/8)—21no—
11C3×#532C2211yescharacteristic
12C2×C3×#124C2211yescharacteristic
13#5477C2211yescharacteristic
14C3×#497—21no—
15C3×#496—21no—
16C3×#495—22no—
17C3×C4×M16C2211yescharacteristic
18#5227—21no—
19#5217—21no—
20#5089—21no—
21#5074—21no—
22#5058—21no—
23#5057—22no—
24#5055—21no—
25#5037—22no—
26#5019—21no—
27#5015—21no—
28#4926C2211yescharacteristic
29#4926C2211yescharacteristic
30#4925C2211yescharacteristic
31D4⋊(C3⋊C8)—22no—
32#4525—22no—
33#4459—22no—
34#3214—31no—
35#3213—31no—
36#3153D311yescharacteristic, complemented
37#1496—31no—
38#1484—31no—
39#1469—31no—
40#1291—31no—
41C2×C3×#41—21no—
42C22×C3×Q8D411yescharacteristic
43C22×C3×D4D411yescharacteristic
44C2×#235—21no—
45C3×#137—44no—
46C3×AGL(1,ℤ/8)—44no—
47C2×C3×Q16—22no—
48C2×C3×SD16—22no—
49C2×C3×SD16—22no—
50C2×C3×D8—22no—
51C2×C3×M16—21no—
52C3×C4⋊Q8D411yescharacteristic
53C3×#127D411yescharacteristic
54C3×#124—21no—
55C3×#124—21no—
56C3×#124—21no—
57C3×#124—21no—
58C3×#124—21no—
59C3×#121—21no—
60C3×#121—21no—
61C3×C4×D4—21no—
62C3×C4×D4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C42C2311yescharacteristic
66C2×#215—22no—
67C2×#214—22no—
68C2×#213—22no—
69C2×#213—22no—
70#930—21no—
71#930—21no—
72#875—21no—
73#874D411yescharacteristic
74C4×D12—21no—
75C4×D12—21no—
76#869D411yescharacteristic
77C3×#97—21no—
78C3×C4×C8D412yes—
79#835—22no—
80#832—22no—
81#830—21no—
82#830—21no—
83#830D412yes—
84#810—22no—
85#809—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)D412yes—
89#803—21no—
90#803—21no—
91C4×C3⋊C8—21no—
92C2×#137—21no—
93C2×AGL(1,ℤ/8)—21no—
94#532—31no—
95#532C2×D311yescharacteristic
96C2×#124C2×D311yescharacteristic
97#497—21no—
98#496—21no—
99#496—61no—
100#496—61no—
101#495—22no—
102#489—61no—
103#489—61no—
104#463—62no—
105#461—61no—
106#459—61no—
107#449—62no—
108#448—61no—
109#447—61no—
110#432—31no—
111#432—31no—
112#431—31no—
113C4×M16C2×D311yescharacteristic
114D4⋊C8—62no—
115D4⋊C8—62no—
116#323—62no—
117C24×C3—21no—
118C3×#41—42no—
119C3×#41—42no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—42no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—42no—
130C2×C3×D4—41no—
131C22×C3×C4—21no—
132C22×C3×C4—41no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C2×#67—41no—
137#235—82no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12—21no—
141C2×C4×D3—41no—
142C2×Dic6—21no—
143C3×Q16—44no—
144C3×SD16—44no—
145C3×SD16—44no—
146C3×D8—44no—
147C3×M16—24no—
148C2×C3×C8—22no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161#215—82no—
162#214—82no—
163#213—82no—
164#213—82no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—41no—
172C2×#41—21no—
173C2×#41—61no—
174C22×Q8#6711yescharacteristic
175C22×D4#6711yescharacteristic
176#137—44no—
177AGL(1,ℤ/8)—44no—
178C2×Q16—22no—
179C2×SD16—22no—
180C2×SD16—22no—
181C2×SD16—62no—
182C2×SD16—62no—
183C2×D8—22no—
184C2×D8—62no—
185C2×D8—62no—
186C2×M16—21no—
187C4⋊Q8—31no—
188C4⋊Q8#6711yescharacteristic
189#127—31no—
190#127#6711yescharacteristic
191#124—21no—
192#124—21no—
193#124—21no—
194#124—21no—
195#124—21no—
196#124—61no—
197#121—21no—
198#121—21no—
199#121—61no—
200#121—61no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—61no—
204C4×D4—61no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42C22×D311yescharacteristic
208C4⋊C8—32no—
209C4⋊C8—61no—
210C4⋊C8—61no—
211#103—62no—
212#103—62no—
213#102—62no—
214#102—62no—
215C22⋊C8—32no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218#97—21no—
219#97—61no—
220#97—61no—
221C4×C8—61no—
222C4×C8#6712yes—
223C23×C3—42no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—81no—
227C23×C3—81no—
228C23×C3#12111yescharacteristic
229C22×D3—41no—
230C3×Q8—41no—
231C3×Q8—42no—
232C3×Q8—41no—
233C3×Q8—42no—
234C3×D4—41no—
235C3×D4—41no—
236C3×D4—41no—
237C3×D4—42no—
238C3×D4—41no—
239C3×D4—42no—
240C3×D4—42no—
241C3×D4—82no—
242C2×C3×C4—22no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—82no—
248C2×C3×C4—81no—
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12711yescharacteristic
255#67—82no—
256C2×Dic3—41no—
257D12—42no—
258D12—42no—
259D12—42no—
260C4×D3—82no—
261Dic6—42no—
262C3×C8—24no—
263C3⋊C8—42no—
264C3⋊C8—42no—
265C3⋊C8—81no—
266C24—21no—
267#41—42no—
268#41—42no—
269#41—242no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—42no—
274C2×Q8—61no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—121no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—121no—
289C22×C4D3×D411yescharacteristic
290C22×C4#23711yescharacteristic
291C22×C4C2×#6711yescharacteristic
292Q16—44no—
293SD16—44no—
294SD16—44no—
295SD16—242no—
296SD16—242no—
297D8—44no—
298D8—242no—
299D8—242no—
300M16—24no—
301C2×C8—22no—
302C2×C8—62no—
303C2×C8—62no—
304C2×C8—121no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C42D3×D411yescharacteristic
318C42D3×D411yescharacteristic
319C42C2×#6711yescharacteristic
320C42C2×#6711yescharacteristic
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—162no—
331C22×C3#39211yescharacteristic
332C2×D3—81no—
333C2×D3—81no—
334C2×D3—81no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—81no—
341C3×C4—81no—
342C3×C4#45312yes—
343Dic3—81no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—81no—
348C23—81no—
349C23—121no—
350C23#93911yescharacteristic
351Q8—41no—
352Q8—42no—
353Q8—41no—
354Q8—42no—
355Q8—122no—
356D4—41no—
357D4—42no—
358D4—41no—
359D4—41no—
360D4—41no—
361D4—42no—
362D4—42no—
363D4—82no—
364D4—122no—
365D4—122no—
366D4—122no—
367D4—242no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—121no—
376C2×C4—242no—
377C2×C4#93711yescharacteristic
378C2×C4#93711yescharacteristic
379C2×C4#93811yescharacteristic
380C2×C4#93911yescharacteristic
381C2×C4#94011yescharacteristic
382C2×C4#95311yescharacteristic
383C8—24no—
384C8—122no—
385C8—122no—
386C8—241no—
387C2×C3—41no—
388C2×C3—161no—
389C2×C3—161no—
390C2×C3#200211yescharacteristic
391C2×C3#200811yescharacteristic
392C2×C3#201711yescharacteristic
393D3—161no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—162no—
404C22—241no—
405C22—241no—
406C22—241no—
407C22#522311yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—81no—
414C4—81no—
415C4—241no—
416C4#507712yes—
417C3#1012011yescharacteristic, complemented, Sylow
418C2—41no—
419C2—161no—
420C2—161no—
421C2—481no—
422C2#7176911yescharacteristic
423C2#7211911yescharacteristic
424C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3153C2
socleC22×C3#392
hypercenter#3153D3
hypocenterC3#10120
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3153

Complements

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

NH ≅ G/Ntype
1. C3×#3153C2semidirect
2. #83308C2semidirect
3. #82947C2semidirect
4. #71803C2semidirect
5. #71720C2semidirect
6. #69853C2semidirect
7. #67670C2semidirect
36. #3153D3semidirect
417. C3#10120semidirect

Maximal subgroups

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

subgroupindex
C3×#31532
#833082
#829472
#718032
#717202
#698532
#676702
#101203