Finite Group Databasebeta

← #10695932

Subgroups — #10695932

Order 768.

602 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 97 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#82501C211yesmaximal, characteristic, complemented
2#82442C211yesmaximal, characteristic, complemented
3#79055C211yesmaximal, characteristic
4C3×#1487C211yesmaximal, characteristic, complemented
5#70790C211yesmaximal, characteristic, complemented
6#70756C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9998—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#908—21no—
11#5477C2211yescharacteristic
12C2×C4×D12C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5231—21no—
16#5227—21no—
17#5133—21no—
18#5027—21no—
19C4×#215—22no—
20#5020—21no—
21#4993—21no—
22#4989—22no—
23#4968—21no—
24#4939—21no—
25#4936C2211yescharacteristic
26#4928—21no—
27#4925C2211yescharacteristic
28#4844—22no—
29#4843—21no—
30#4841—21no—
31C3×Q8⋊C8—22no—
32#4483—22no—
33#4470—22no—
34#3103—31no—
35#3036—31no—
36#2580—31no—
37#1517—31no—
38#1487D311yescharacteristic, complemented
39#1478—31no—
40#1286—31no—
41C22×C3×Q8D411yescharacteristic
42C2×#235—21no—
43C22×D12D411yescharacteristic
44C22×C4×D3—21no—
45C2×C3×M16—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C42—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#215—22no—
55#930—21no—
56#930—21no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—22no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3D411yescharacteristic
62C2×Dic3⋊C4—22no—
63#908—42no—
64#908—42no—
65C2×D24—22no—
66C2×#204—22no—
67#875—21no—
68#874D411yescharacteristic
69C4×D12—21no—
70C4×D12—21no—
71C4×D12—21no—
72C4×D12—21no—
73C4×D12—21no—
74C4×D12—21no—
75#869D411yescharacteristic
76C3×C4⋊C8—21no—
77C3×C4⋊C8—21no—
78C3×C4⋊C8D412yes—
79C3×C22⋊C8—21no—
80C3×C4×C8—21no—
81#835—22no—
82C3⋊#95—22no—
83C3⋊#95—21no—
84C3⋊#95D412yes—
85#830—21no—
86#821—22no—
87#816—22no—
88#809—22no—
89#808—22no—
90C4⋊(C3⋊C8)—21no—
91C4×C3⋊C8D412yes—
92C2×AGL(1,ℤ/8)—61no—
93#532—31no—
94C2×C4×Q8C2×D311yescharacteristic
95C2×C4×D4—31no—
96#481—61no—
97#475—61no—
98#463—62no—
99#461—61no—
100#460—61no—
101#459—61no—
102#448—61no—
103C4×SD16—62no—
104#431—31no—
105#431C2×D311yescharacteristic
106#423C2×D311yescharacteristic
107#417—61no—
108#399—61no—
109#393—61no—
110#390—61no—
111#389—61no—
112#385—62no—
113#382—31no—
114#327—62no—
115Q8⋊C8—22no—
116D4⋊C8—62no—
117C23×D3—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—42no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C2×#67—41no—
127C22×Dic3—22no—
128C22×Dic3—21no—
129#235—41no—
130#235—41no—
131#235—41no—
132#235—41no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12SD1612yescomplemented
142C2×C4×D3—41no—
143C2×C4×D3—41no—
144C2×C4×D3—41no—
145C2×C4×D3—82no—
146C2×Dic6—21no—
147C3×M16—41no—
148C3×M16—41no—
149C2×C3×C8—22no—
150C2×C3×C8—41no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C42—41no—
156C3×C42—41no—
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42#4111yescharacteristic
161#215—44no—
162(C2×D3)⋊C4—44no—
163(C2×D3)⋊C4—41no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167(C2×D3)⋊C4—82no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4⋊Dic3—41no—
173C4⋊Dic3—41no—
174Dic3⋊C4—82no—
175C2×C3⋊C8—22no—
176D24—44no—
177#204—44no—
178C2×#41—61no—
179C22×Q8D1211yescharacteristic
180C22×D4—31no—
181C23×C4—61no—
182AGL(1,ℤ/8)—122no—
183AGL(1,ℤ/8)—122no—
184C2×SD16—62no—
185C2×SD16—62no—
186C2×D8—62no—
187C2×M16—21no—
188C4⋊Q8—31no—
189#127—31no—
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—
199C4×D4—61no—
200C4×D4—61no—
201C4×D4—61no—
202C4×D4—61no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—62no—
207C2×C4⋊C4D1211yescharacteristic
208C2×C22⋊C4—62no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—62no—
211C2×C42—21no—
212C2×C42C22×D311yescharacteristic
213#106—62no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—61no—
217C4⋊C8#6712yes—
218#103—62no—
219#103—62no—
220#102—62no—
221#102—62no—
222C22⋊C8—21no—
223C22⋊C8—61no—
224C4×C8—21no—
225C4×C8—32no—
226#95—32no—
227#95—62no—
228#95—61no—
229C23×C3#12011yescharacteristic
230C22×D3—22no—
231C22×D3—42no—
232C22×D3—42no—
233C22×D3—41no—
234C22×D3—41no—
235C22×D3—41no—
236C3×Q8—22no—
237C3×Q8—22no—
238C3×Q8—41no—
239C3×Q8—41no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
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×C3×C4C2×SD1611yescharacteristic
255C2×C3×C4#13511yescharacteristic
256C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
257C2×C3×C4#13711yescharacteristic
258#67—81no—
259#67—81no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—42no—
266D12—22no—
267D12—22no—
268D12—42no—
269D12—41no—
270D12—41no—
271D12—41no—
272D12—41no—
273D12—81no—
274C4×D3—82no—
275C4×D3—82no—
276C4×D3—81no—
277C4×D3—81no—
278Dic6—41no—
279Dic6—41no—
280C3×C8—42no—
281C3×C8—81no—
282C3⋊C8—42no—
283C24—61no—
284#41—121no—
285#41—121no—
286#41—121no—
287#41—121no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—42no—
291C2×Q8—61no—
292C2×D4—32no—
293C2×D4—61no—
294C2×D4—61no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—61no—
300C2×D4—61no—
301C2×D4—121no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—62no—
305C22×C4—61no—
306C22×C4—121no—
307C22×C4—121no—
308C22×C4—121no—
309C22×C4—242no—
310C22×C4C2×D1211yescharacteristic
311C22×C4D3×D411yescharacteristic
312C22×C4D3×D411yescharacteristic
313SD16—124no—
314SD16—124no—
315D8—124no—
316M16—41no—
317M16—41no—
318C2×C8—22no—
319C2×C8—41no—
320C2×C8—62no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—121no—
330C4⋊C4—121no—
331C4⋊C4—242no—
332C22⋊C4—124no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C22⋊C4—242no—
338C42—41no—
339C42—41no—
340C42#23511yescharacteristic
341C42D3×D411yescharacteristic
342C42D3×D411yescharacteristic
343C42C2×#6711yescharacteristic
344C22×C3—21no—
345C22×C3—21no—
346C22×C3#39211yescharacteristic
347C22×C3#44811yescharacteristic
348C22×C3#44911yescharacteristic
349C2×D3—44no—
350C2×D3—42no—
351C2×D3—82no—
352C2×D3—81no—
353C2×D3—81no—
354C2×D3—81no—
355C2×D3—81no—
356C2×D3—81no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—42no—
362C3×C4—81no—
363C3×C4#45011yescharacteristic
364C3×C4#45211yescharacteristic
365C3×C4#46511yescharacteristic
366C3×C4#46811yescharacteristic
367Dic3—82no—
368Dic3—81no—
369Dic3—81no—
370C23—62no—
371C23—122no—
372C23—122no—
373C23—121no—
374C23—121no—
375C23—121no—
376C23#88211yescharacteristic
377Q8—22no—
378Q8—22no—
379Q8—41no—
380Q8—41no—
381Q8—121no—
382Q8—121no—
383D4—62no—
384D4—62no—
385D4—122no—
386D4—121no—
387D4—121no—
388D4—121no—
389D4—121no—
390D4—241no—
391D4—241no—
392D4—241no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—122no—
402C2×C4—122no—
403C2×C4—122no—
404C2×C4—121no—
405C2×C4—121no—
406C2×C4—122no—
407C2×C4—242no—
408C2×C4—242no—
409C2×C4—241no—
410C2×C4—241no—
411C2×C4#88211yescharacteristic
412C2×C4#88411yescharacteristic
413C2×C4#88411yescharacteristic
414C2×C4D3×SD1611yescharacteristic
415C2×C4#91411yescharacteristic
416C2×C4#91811yescharacteristic
417C2×C4#91911yescharacteristic
418C2×C4#93011yescharacteristic
419C2×C4#93711yescharacteristic
420C2×C4#94011yescharacteristic
421C8—42no—
422C8—81no—
423C8—122no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3#200011yescharacteristic
427C2×C3#200811yescharacteristic
428C2×C3#201011yescharacteristic
429D3—82no—
430D3—161no—
431C22—21no—
432C22—21no—
433C22—124no—
434C22—122no—
435C22—242no—
436C22—241no—
437C22—241no—
438C22—241no—
439C22—241no—
440C22—241no—
441C22#480611yescharacteristic
442C22#481011yescharacteristic
443C22#496011yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—42no—
449C4—81no—
450C4—242no—
451C4—241no—
452C4—241no—
453C4#480711yescharacteristic
454C4#481211yescharacteristic
455C4#516911yescharacteristic
456C4#518711yescharacteristic
457C3#999811yescharacteristic, complemented, Sylow
458C2—21no—
459C2—21no—
460C2—242no—
461C2—481no—
462C2#6999311yescharacteristic
463C2#7027211yescharacteristic
464C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1487C2
socleC22×C3#392
prime radical#79055C2
hypercenter#1487D3
hypocenterC3#9998
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
1. #82501C2semidirect
2. #82442C2semidirect
4. C3×#1487C2semidirect
5. #70790C2semidirect
6. #70756C2semidirect
7. #67713C2semidirect
38. #1487D3semidirect
141. C2×D12SD16semidirect
457. C3#9998semidirect

Maximal subgroups

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

subgroupindex
#825012
#824422
#790552
C3×#14872
#707902
#707562
#677132
#99983