Finite Group Databasebeta

← #10697601

Subgroups — #10697601

Order 768.

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

normal subgroups only

This group has 740 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
1C3×#3208C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic, complemented
3#83188C211yesmaximal, characteristic
4#71835C211yesmaximal, characteristic, complemented
5#71759C211yesmaximal, characteristic, complemented
6#70756C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#10246—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#127C2211yescharacteristic
11C2×#940—21no—
12C2×#938—22no—
13C2×C4×D12C2211yescharacteristic
14C3×#492—21no—
15C3×#486—21no—
16C3×#460—22no—
17C3×#450—22no—
18C3×#431C2211yescharacteristic
19#5223—22no—
20#5221—21no—
21#5087—21no—
22#5084—21no—
23#5066—22no—
24#5001—21no—
25#4995—21no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4544—22no—
30#4538—22no—
31#4483—22no—
32#3208D311yescharacteristic, complemented
33#3077—31no—
34#2541—31no—
35#1545—31no—
36#1503—31no—
37#1478—31no—
38#1303—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D412yes—
42C22×#67—24no—
43C22×D12D411yescharacteristic
44C22×C4×D3—21no—
45C2×C3×SD16—24no—
46C3×C4⋊Q8D411yescharacteristic, complemented
47C3×C4⋊Q8D411yescharacteristic, complemented
48C3×#127—21no—
49C3×#127—21no—
50C3×#127—21no—
51C3×#127—21no—
52C3×#125—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×Q8—21no—
57C3×#117—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#217—24no—
60#940—42no—
61#940—42no—
62#938—48no—
63C2×(C2×D3)⋊C4—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3D411yescharacteristic
66C2×C4×Dic3—21no—
67C2×#208—21no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12—21no—
71C4×D12—21no—
72C3×C4⋊C8D412yescomplemented
73C3×#102—24no—
74C3×C22⋊C8—21no—
75C3×C4×C8—21no—
76C3⋊#95D412yes—
77#830—21no—
78#809—22no—
79#809—22no—
80#808—22no—
81#807—22no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)D412yescomplemented
85C4×C3⋊C8—21no—
86#533C2×D311yescharacteristic
87C2×#127—61no—
88C2×#127C2×D311yescharacteristic
89C2×#121—62no—
90C2×C4×D4—31no—
91#492—21no—
92#486—21no—
93#478—62no—
94#476—61no—
95#474—61no—
96#460—22no—
97#450—22no—
98#431—31no—
99#431C2×D311yescharacteristic
100#428—61no—
101#423—31no—
102#418—61no—
103#392—62no—
104#390—61no—
105C4×C4⋊C4—31no—
106#331—62no—
107#328—62no—
108D4⋊C8—62no—
109C24×C3—22no—
110C23×D3—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×D4—22no—
114C2×C3×D4—22no—
115C2×C3×D4—44no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—82no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C2×#67—48no—
129C2×#67—48no—
130C2×#67—84no—
131C22×Dic3—22no—
132C22×Dic3—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12SD1612yescomplemented
139C2×C4×D3—44no—
140C2×C4×D3—41no—
141C2×C4×D3—41no—
142C3×SD16—84no—
143C2×C3×C8—22no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C42—41no—
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156#217—48no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—41no—
159C4⋊Dic3—24no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4×Dic3—41no—
163C4×Dic3—41no—
164#208—41no—
165#208—41no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—41no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—31no—
171C22×D4—64no—
172C22×D4#6712yes—
173C23×C4—61no—
174C2×SD16—24no—
175C2×M16—61no—
176C4⋊Q8#6711yescharacteristic, complemented
177C4⋊Q8#6711yescharacteristic, complemented
178#127—21no—
179#127—21no—
180#127—21no—
181#127—21no—
182#127—122no—
183#127—122no—
184#125—21no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187#121—128no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×D4—61no—
191C4×D4—61no—
192C4×D4—61no—
193C4×D4—61no—
194#117—21no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—61no—
197C2×C22⋊C4—64no—
198C2×C22⋊C4—61no—
199C2×C42—61no—
200C2×C42C22×D311yescharacteristic
201#107—62no—
202#106—62no—
203C4⋊C8—32no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206C4⋊C8#6712yescomplemented
207#102—24no—
208#102—62no—
209#102—62no—
210C22⋊C8—21no—
211C22⋊C8—61no—
212C4×C8—21no—
213C4×C8—61no—
214#95—32no—
215C23×C3—44no—
216C23×C3—44no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3#12011yescharacteristic
221C22×D3—22no—
222C22×D3—42no—
223C22×D3—42no—
224C22×D3—41no—
225C22×D3—41no—
226C3×Q8—42no—
227C3×Q8—81no—
228C3×D4—44no—
229C3×D4—82no—
230C3×D4—82no—
231C3×D4—82no—
232C3×D4—81no—
233C3×D4—81no—
234C3×D4—81no—
235C3×D4—81no—
236C2×C3×C4—22no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12711yescharacteristic
246C2×C3×C4C2×SD1611yescharacteristic
247C2×C3×C4#13511yescharacteristic
248C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
249C2×C3×C4#13711yescharacteristic
250#67—816no—
251C2×Dic3—24no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—41no—
256D12—22no—
257D12—22no—
258D12—41no—
259D12—41no—
260C4×D3—82no—
261C4×D3—82no—
262C3×C8—42no—
263C3⋊C8—42no—
264C3⋊C8—81no—
265C24—22no—
266C24—61no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—32no—
272C2×D4—44no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—61no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—82no—
287C2×D4—128no—
288C2×D4—128no—
289C2×D4—244no—
290C22×C4—62no—
291C22×C4—61no—
292C22×C4—124no—
293C22×C4—121no—
294C22×C4—121no—
295C22×C4D3×D411yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4C2×#6711yescharacteristic
298SD16—84no—
299M16—121no—
300M16—121no—
301C2×C8—22no—
302C2×C8—62no—
303C2×C8—121no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—64no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C22⋊C4—41no—
314C22⋊C4—128no—
315C22⋊C4—121no—
316C22⋊C4—121no—
317C42—41no—
318C42—121no—
319C42—121no—
320C42D3×D411yescharacteristic
321C42#23711yescharacteristic
322C42C2×#6711yescharacteristic
323C42C2×#6711yescharacteristic
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—84no—
327C22×C3—84no—
328C22×C3—82no—
329C22×C3—82no—
330C22×C3—82no—
331C22×C3#39211yescharacteristic
332C22×C3#45011yescharacteristic
333C22×C3#45211yescharacteristic
334C2×D3—44no—
335C2×D3—42no—
336C2×D3—82no—
337C2×D3—81no—
338C2×D3—81no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—81no—
343C3×C4—81no—
344C3×C4#44811yescharacteristic
345C3×C4#44911yescharacteristic
346C3×C4#45312yes—
347C3×C4#46511yescharacteristic
348C3×C4#46811yescharacteristic
349Dic3—44no—
350Dic3—81no—
351C23—44no—
352C23—44no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—62no—
357C23—122no—
358C23—122no—
359C23—121no—
360C23—121no—
361C23#93711yescharacteristic
362Q8—42no—
363Q8—81no—
364D4—44no—
365D4—62no—
366D4—62no—
367D4—82no—
368D4—82no—
369D4—82no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4—81no—
374D4—121no—
375D4—121no—
376D4—2416no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—64no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—121no—
386C2×C4—242no—
387C2×C4—242no—
388C2×C4D3×SD1611yescharacteristic
389C2×C4#91411yescharacteristic
390C2×C4#91511yescharacteristic
391C2×C4#91611yescharacteristic
392C2×C4#93711yescharacteristic
393C2×C4#93811yescharacteristic
394C2×C4#93911yescharacteristic
395C2×C4#93911yescharacteristic
396C2×C4#94011yescharacteristic
397C2×C4#95311yescharacteristic
398C8—42no—
399C8—122no—
400C8—241no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—162no—
404C2×C3#200011yescharacteristic
405C2×C3#200211yescharacteristic
406C2×C3#200811yescharacteristic
407D3—82no—
408C22—21no—
409C22—21no—
410C22—84no—
411C22—84no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—124no—
416C22—122no—
417C22—242no—
418C22—241no—
419C22—241no—
420C22#516911yescharacteristic
421C22#517311yescharacteristic
422C22#522311yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—81no—
427C4—81no—
428C4—124no—
429C4—241no—
430C4#517011yescharacteristic
431C4#517111yescharacteristic
432C4#517211yescharacteristic
433C4#517411yescharacteristic
434C4#524112yes—
435C3#1024611yescharacteristic, complemented, Sylow
436C2—21no—
437C2—21no—
438C2—162no—
439C2—242no—
440C2#7185611yescharacteristic
441C2#7205411yescharacteristic
442C2#7209911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3208C2
socleC22×C3#392
prime radical#83188C2
hypercenter#3208D3
hypocenterC3#10246
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. C3×#3208C2semidirect
2. #83350C2semidirect
4. #71835C2semidirect
5. #71759C2semidirect
6. #70756C2semidirect
7. #67746C2semidirect
32. #3208D3semidirect
46. C3×C4⋊Q8D4semidirect
47. C3×C4⋊Q8D4semidirect
72. C3×C4⋊C8D4semidirect
84. C4⋊(C3⋊C8)D4semidirect
138. C2×D12SD16semidirect
176. C4⋊Q8#67semidirect
177. C4⋊Q8#67semidirect
206. C4⋊C8#67semidirect
435. C3#10246semidirect

Maximal subgroups

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

subgroupindex
C3×#32082
#833502
#831882
#718352
#717592
#707562
#677462
#102463