Finite Group Databasebeta

← #10697563

Subgroups — #10697563

Order 768.

658 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 101 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#83259C211yesmaximal, characteristic, complemented
2#83213C211yesmaximal, characteristic, complemented
3#79055C211yesmaximal, characteristic
4C3×#1548C211yesmaximal, characteristic, complemented
5#70789C211yesmaximal, characteristic, complemented
6#70757C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#10240—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#946—22no—
11C2×#908—21no—
12#5477C2211yescharacteristic
13C2×C4×D12C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×#423C2211yescharacteristic
16#5227—22no—
17#5133—21no—
18#5085—21no—
19#5083—22no—
20#5079—21no—
21#4993—22no—
22#4988—21no—
23#4975—21no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26#4844—22no—
27#4843—21no—
28#4841—21no—
29C3×#333—22no—
30#4470—22no—
31#4459—22no—
32#3206—31no—
33#3036—31no—
34#2542—31no—
35#1548D311yescharacteristic, complemented
36#1502—31no—
37#1478—31no—
38#1303—31no—
39C22×C3×Q8D412yes—
40C2×#235—21no—
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C2×C3×M16—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51#946—48no—
52C2×#215—24no—
53#930—21no—
54#930—21no—
55C2×(C2×D3)⋊C4—24no—
56C2×(C2×D3)⋊C4—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×Dic3⋊C4—24no—
60C2×C4×Dic3—21no—
61#908—42no—
62#908—42no—
63C2×D24—22no—
64C2×#204—22no—
65#875—21no—
66#874D411yescharacteristic
67C4×D12—21no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12—21no—
71C4×D12—21no—
72C4×D12—21no—
73#869D411yescharacteristic
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8D412yes—
77C3×C22⋊C8—21no—
78C3×C4×C8—21no—
79#835—24no—
80C3⋊#95D412yes—
81#830—21no—
82#821—22no—
83#816—22no—
84C4⋊(C3⋊C8)D412yes—
85C4×C3⋊C8—21no—
86C2×AGL(1,ℤ/8)—61no—
87#532—31no—
88C2×C4⋊Q8C2×D311yescharacteristic
89C2×C22⋊Q8—62no—
90C2×C4×D4—31no—
91#492—61no—
92#487—61no—
93#463—62no—
94#461—61no—
95#461—62no—
96#459—61no—
97#448—62no—
98#431—31no—
99#431C2×D311yescharacteristic
100#423C2×D311yescharacteristic
101#417—61no—
102#399—62no—
103#390—61no—
104#384—61no—
105C4×C4⋊C4—31no—
106#333—22no—
107#327—62no—
108D4⋊C8—62no—
109C23×D3—21no—
110C2×C3×Q8—22no—
111C2×C3×Q8—22no—
112C2×C3×Q8—44no—
113C22×C3×C4—22no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C2×#67—41no—
118C22×Dic3—22no—
119C22×Dic3—21no—
120#235—41no—
121#235—41no—
122#235—41no—
123#235—41no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12SD1612yescomplemented
133C2×C4×D3—44no—
134C2×C4×D3—41no—
135C2×C4×D3—41no—
136C2×C4×D3—41no—
137C2×Dic6—21no—
138C3×M16—41no—
139C3×M16—41no—
140C2×C3×C8—22no—
141C2×C3×C8—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150#215—84no—
151(C2×D3)⋊C4—48no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—24no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161Dic3⋊C4—48no—
162C4×Dic3—41no—
163C4×Dic3—41no—
164C2×C3⋊C8—22no—
165D24—44no—
166#204—44no—
167C2×#41—61no—
168C22×Q8D1212yes—
169C22×D4—31no—
170C23×C4—61no—
171AGL(1,ℤ/8)—122no—
172AGL(1,ℤ/8)—122no—
173C2×SD16—62no—
174C2×SD16—64no—
175C2×D8—62no—
176C2×M16—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—31no—
182#127—31no—
183#124—61no—
184C22⋊Q8—128no—
185#121—61no—
186#121—61no—
187C4×D4—61no—
188C4×D4—61no—
189C4×D4—61no—
190C4×D4—61no—
191C4×D4—61no—
192C4×D4—61no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—64no—
198C2×C22⋊C4—64no—
199C2×C22⋊C4—61no—
200C2×C42—61no—
201C2×C42C22×D311yescharacteristic
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—32no—
205C4⋊C8#6712yes—
206#103—62no—
207#103—64no—
208#102—62no—
209C22⋊C8—21no—
210C22⋊C8—61no—
211C4×C8—21no—
212C4×C8—61no—
213#95—32no—
214C23×C3#12011yescharacteristic
215C22×D3—22no—
216C22×D3—42no—
217C22×D3—42no—
218C22×D3—41no—
219C22×D3—41no—
220C22×D3—41no—
221C3×Q8—44no—
222C3×Q8—82no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—44no—
226C2×C3×C4—42no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12711yescharacteristic
233C2×C3×C4C2×SD1611yescharacteristic
234C2×C3×C4#13511yescharacteristic
235C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
236C2×C3×C4#13711yescharacteristic
237#67—81no—
238#67—81no—
239C2×Dic3—24no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245D12—22no—
246D12—22no—
247D12—41no—
248D12—41no—
249D12—41no—
250D12—42no—
251D12—41no—
252D12—81no—
253C4×D3—82no—
254C4×D3—82no—
255C4×D3—81no—
256C4×D3—81no—
257Dic6—41no—
258Dic6—41no—
259C3×C8—42no—
260C3×C8—81no—
261C3⋊C8—42no—
262C24—61no—
263#41—121no—
264#41—121no—
265#41—121no—
266#41—121no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—44no—
270C2×Q8—61no—
271C2×D4—32no—
272C2×D4—61no—
273C2×D4—61no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—61no—
280C2×D4—121no—
281C22×C4—22no—
282C22×C4—62no—
283C22×C4—61no—
284C22×C4—124no—
285C22×C4—121no—
286C22×C4—121no—
287C22×C4—121no—
288C22×C4C2×D1211yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291SD16—124no—
292SD16—244no—
293D8—124no—
294M16—41no—
295M16—41no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—62no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—64no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—128no—
309C22⋊C4—128no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C42—121no—
315C42—121no—
316C42#23511yescharacteristic
317C42D3×D411yescharacteristic
318C42D3×D411yescharacteristic
319C42C2×#6711yescharacteristic
320C22×C3—21no—
321C22×C3—21no—
322C22×C3#39211yescharacteristic
323C22×C3#44811yescharacteristic
324C22×C3#44911yescharacteristic
325C2×D3—44no—
326C2×D3—42no—
327C2×D3—82no—
328C2×D3—81no—
329C2×D3—81no—
330C2×D3—81no—
331C2×D3—81no—
332C2×D3—81no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—82no—
337C3×C4#45011yescharacteristic
338C3×C4#45211yescharacteristic
339C3×C4#45312yes—
340C3×C4#46511yescharacteristic
341C3×C4#46811yescharacteristic
342Dic3—44no—
343Dic3—81no—
344Dic3—81no—
345C23—62no—
346C23—122no—
347C23—122no—
348C23—121no—
349C23—121no—
350C23—121no—
351C23#88211yescharacteristic
352Q8—44no—
353Q8—82no—
354Q8—121no—
355Q8—121no—
356D4—62no—
357D4—62no—
358D4—121no—
359D4—121no—
360D4—121no—
361D4—122no—
362D4—121no—
363D4—241no—
364D4—241no—
365D4—241no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—64no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—242no—
377C2×C4—242no—
378C2×C4—241no—
379C2×C4—241no—
380C2×C4#88211yescharacteristic
381C2×C4#88411yescharacteristic
382C2×C4#88411yescharacteristic
383C2×C4D3×SD1611yescharacteristic
384C2×C4#91411yescharacteristic
385C2×C4#91811yescharacteristic
386C2×C4#91911yescharacteristic
387C2×C4#93011yescharacteristic
388C2×C4#93711yescharacteristic
389C2×C4#94011yescharacteristic
390C8—42no—
391C8—81no—
392C8—122no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3#200011yescharacteristic
396C2×C3#200211yescharacteristic
397C2×C3#200911yescharacteristic
398D3—82no—
399D3—161no—
400C22—21no—
401C22—21no—
402C22—124no—
403C22—122no—
404C22—242no—
405C22—241no—
406C22—241no—
407C22—241no—
408C22—241no—
409C22—241no—
410C22#480611yescharacteristic
411C22#481011yescharacteristic
412C22#496011yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—82no—
417C4—124no—
418C4—241no—
419C4—241no—
420C4#474712yes—
421C4#480711yescharacteristic
422C4#481211yescharacteristic
423C4#516911yescharacteristic
424C4#518711yescharacteristic
425C3#1024011yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—242no—
429C2—481no—
430C2#7008211yescharacteristic
431C2#7027211yescharacteristic
432C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1548C2
socleC22×C3#392
prime radical#79055C2
hypercenter#1548D3
hypocenterC3#10240
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. #83259C2semidirect
2. #83213C2semidirect
4. C3×#1548C2semidirect
5. #70789C2semidirect
6. #70757C2semidirect
7. #67715C2semidirect
35. #1548D3semidirect
132. C2×D12SD16semidirect
425. C3#10240semidirect

Maximal subgroups

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

subgroupindex
#832592
#832132
#790552
C3×#15482
#707892
#707572
#677152
#102403