Finite Group Databasebeta

← #10711080

Subgroups — #10711080

Order 768.

860 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 127 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×#3549C211yesmaximal, characteristic
2#71861C212yesmaximal, complemented
3#71854C212yesmaximal, complemented
4#71819C211yesmaximal, characteristic, complemented
5#71817C211yesmaximal, characteristic, complemented
6#12778—31nomaximal, Sylow
7C3×H(3,𝔽4)—21no—
8C3×H(3,𝔽4)C412yescomplemented
9C3×#555—22no—
10C3×#555C412yescomplemented
11C3×D42—22no—
12C3×D42—21no—
13C3×#535—21no—
14C3×#535—24no—
15C3×#535C2212yes—
16C3×#532—21no—
17C3×#532C2211yescharacteristic
18#5819C2211yescharacteristic, complemented
19#5240C2211yescharacteristic, complemented
20#5240C2212yescomplemented
21#5236—21no—
22#5215—22no—
23#4545—21no—
24#4545—24no—
25#4541—21no—
26#4531—22no—
27#4523—21no—
28#3549D311yescharacteristic
29#2128—32no—
30#2126—31no—
31#1910—32no—
32#1884—31no—
33C2×C3×#41—22no—
34C2×C3×#41D411yescharacteristic
35C2×C3×#41C2311yescharacteristic
36C22×C3×D4—24no—
37C22×C3×D4—21no—
38C22×C3×D4D412yes—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#127—21no—
42C3×#127—22no—
43C3×#127D411yescharacteristic
44C3×#127D414yes—
45C3×#124—24no—
46C3×#124—22no—
47C3×#124—22no—
48C3×#124—24no—
49C3×#124C2×C411yescharacteristic
50C3×#124C2×C414yes—
51C3×#124D411yescharacteristic
52C3×#121—22no—
53C3×#121—22no—
54C3×#121—24no—
55C3×#121—24no—
56C3×#121—22no—
57C3×#121—22no—
58C3×#121—22no—
59C3×#120—24no—
60C3×#120—22no—
61C3×#120—22no—
62C3×#120—24no—
63C3×#120—22no—
64C3×#120—22no—
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C3×#117—21no—
69C3×#117—24no—
70C3×#117D412yes—
71C2×C3×C42D411yescharacteristic
72#948—42no—
73#926—22no—
74#926—21no—
75C2×#208—22no—
76C2×#208—21no—
77#837—42no—
78#834—22no—
79#834—22no—
80#834—22no—
81#832—44no—
82#568—31no—
83H(3,𝔽4)—21no—
84H(3,𝔽4)Dic312yescomplemented
85#555—22no—
86#555Dic312yescomplemented
87D42—22no—
88D42—21no—
89#535—21no—
90#535—24no—
91#535C2×D312yes—
92#532—21no—
93#532C2×D311yescharacteristic
94#421—61no—
95#418—62no—
96#410—32no—
97#410—31no—
98#354—61no—
99#353—61no—
100#353—64no—
101#339—61no—
102H(3,ℤ/4)—62no—
103C24×C3—22no—
104C24×C3—22no—
105C3×#41—41no—
106C3×#41—41no—
107C3×#41—41no—
108C3×#41—41no—
109C3×#41—84no—
110C2×C3×Q8—22no—
111C2×C3×Q8C22⋊C411yescharacteristic
112C2×C3×Q8C22×C411yescharacteristic
113C2×C3×D4—22no—
114C2×C3×D4—22no—
115C2×C3×D4—22no—
116C2×C3×D4—22no—
117C2×C3×D4—24no—
118C2×C3×D4—24no—
119C2×C3×D4—21no—
120C2×C3×D4—24no—
121C2×C3×D4—22no—
122C2×C3×D4—42no—
123C2×C3×D4—42no—
124C2×C3×D4—88no—
125C2×C3×D4—81no—
126C2×C3×D4—81no—
127C2×C3×D4C22⋊C411yescharacteristic
128C2×C3×D4C22⋊C414yes—
129C2×C3×D4C2×D411yescharacteristic
130C2×C3×D4C2×D412yes—
131C22×C3×C4—24no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D412yes—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C22⋊C4—24no—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—24no—
142C3×C22⋊C4—22no—
143C3×C22⋊C4—22no—
144C3×C22⋊C4—24no—
145C3×C22⋊C4—22no—
146C3×C22⋊C4—24no—
147C3×C42—22no—
148C3×C42—21no—
149C3×C42C22⋊C411yescharacteristic
150C3×C42C22⋊C411yescharacteristic
151C3×C42C22⋊C414yes—
152#217—42no—
153#217—41no—
154C4⋊Dic3—42no—
155C4⋊Dic3—41no—
156C4×Dic3—41no—
157C4×Dic3—42no—
158#208—22no—
159#208—42no—
160#208—41no—
161C2×C3⋊C8—24no—
162C2×C3⋊C8—41no—
163C2×#41—22no—
164C2×#41#6711yescharacteristic
165C2×#41C22×D311yescharacteristic
166C22×D4—24no—
167C22×D4—21no—
168C22×D4#6712yes—
169#131—122no—
170C2×M16—62no—
171C2×M16—61no—
172C4⋊Q8—21no—
173C4⋊Q8C2×Dic311yescharacteristic
174#127—21no—
175#127—22no—
176#127#6711yescharacteristic
177#127#6714yes—
178#124—24no—
179#124—22no—
180#124—22no—
181#124—24no—
182#124C2×Dic311yescharacteristic
183#124C2×Dic314yes—
184#124#6711yescharacteristic
185#121—22no—
186#121—22no—
187#121—24no—
188#121—24no—
189#121—22no—
190#121—22no—
191#121—22no—
192#120—22no—
193#120—24no—
194#120—22no—
195#120—24no—
196#120—22no—
197#120—22no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201#117—21no—
202#117—24no—
203#117—62no—
204#117—61no—
205#117#6712yes—
206C2×C42#6711yescharacteristic
207#104—122no—
208#102—124no—
209#99—62no—
210#99—62no—
211#99—62no—
212C23×C3—24no—
213C23×C3—21no—
214C23×C3—82no—
215C23×C3—84no—
216C23×C3—84no—
217C23×C3—82no—
218C23×C3#12012yes—
219C23×C3#12011yescharacteristic
220C3×Q8—41no—
221C3×Q8—41no—
222C3×Q8—82no—
223C3×D4—42no—
224C3×D4—41no—
225C3×D4—44no—
226C3×D4—41no—
227C3×D4—44no—
228C3×D4—44no—
229C3×D4—42no—
230C3×D4—42no—
231C3×D4—81no—
232C3×D4—84no—
233C3×D4—84no—
234C3×D4—82no—
235C3×D4—84no—
236C3×D4—82no—
237C2×C3×C4—21no—
238C2×C3×C4—22no—
239C2×C3×C4—21no—
240C2×C3×C4—22no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—24no—
245C2×C3×C4—48no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4C2×C22⋊C411yescharacteristic
250C2×C3×C4C2×C22⋊C412yes—
251C2×C3×C4#12011yescharacteristic
252C2×Dic3—81no—
253C2×Dic3—82no—
254C3⋊C8—42no—
255C24—22no—
256C24—22no—
257#41—41no—
258#41—41no—
259#41—41no—
260#41—41no—
261#41—84no—
262C2×Q8—22no—
263C2×Q8#21711yescharacteristic
264C2×Q8C22×Dic311yescharacteristic
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—21no—
272C2×D4—24no—
273C2×D4—22no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—88no—
277C2×D4—81no—
278C2×D4—81no—
279C2×D4#21711yescharacteristic
280C2×D4#21714yes—
281C2×D4C2×#6711yescharacteristic
282C2×D4C2×#6712yes—
283C22×C4—24no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4C2×#6711yescharacteristic
287C22×C4C2×#6712yes—
288M16—62no—
289M16—122no—
290M16—121no—
291C2×C8—64no—
292C2×C8—121no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—122no—
297C4⋊C4—121no—
298C22⋊C4—24no—
299C22⋊C4—22no—
300C22⋊C4—24no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—24no—
304C22⋊C4—22no—
305C22⋊C4—24no—
306C22⋊C4—122no—
307C22⋊C4—121no—
308C42—22no—
309C42—21no—
310C42—121no—
311C42—122no—
312C42#21711yescharacteristic
313C42#21711yescharacteristic
314C42#21714yes—
315C22×C3—21no—
316C22×C3—22no—
317C22×C3—41no—
318C22×C3—42no—
319C22×C3—44no—
320C22×C3—44no—
321C22×C3—42no—
322C22×C3—41no—
323C22×C3—44no—
324C22×C3—164no—
325C22×C3—164no—
326C22×C3#37911yescharacteristic
327C3×C4—22no—
328C3×C4—22no—
329C3×C4—21no—
330C3×C4—44no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4B(4,𝔽2)11yescharacteristic
334C3×C4#45811yescharacteristic
335Dic3—161no—
336Dic3—162no—
337C23—24no—
338C23—21no—
339C23—82no—
340C23—84no—
341C23—84no—
342C23—82no—
343C23#95312yes—
344C23#95311yescharacteristic
345Q8—41no—
346Q8—41no—
347Q8—82no—
348D4—42no—
349D4—41no—
350D4—44no—
351D4—41no—
352D4—44no—
353D4—44no—
354D4—42no—
355D4—42no—
356D4—81no—
357D4—84no—
358D4—84no—
359D4—82no—
360D4—84no—
361D4—82no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—24no—
370C2×C4—48no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—241no—
375C2×C4—242no—
376C2×C4C2×#21711yescharacteristic
377C2×C4C2×#21712yes—
378C2×C4#95311yescharacteristic
379C8—122no—
380C2×C3—21no—
381C2×C3—41no—
382C2×C3—42no—
383C2×C3—84no—
384C2×C3—81no—
385C2×C3#179011yescharacteristic
386C22—21no—
387C22—22no—
388C22—41no—
389C22—42no—
390C22—44no—
391C22—44no—
392C22—42no—
393C22—41no—
394C22—44no—
395C22—164no—
396C22—164no—
397C22#524711yescharacteristic
398C4—22no—
399C4—22no—
400C4—21no—
401C4—44no—
402C4—41no—
403C4—41no—
404C4—482no—
405C4—481no—
406C4#524311yescharacteristic
407C4#524411yescharacteristic
408C3#1277811yescharacteristic, complemented, Sylow
409C2—21no—
410C2—42no—
411C2—41no—
412C2—84no—
413C2—81no—
414C2#7214211yescharacteristic
subgroupG/H
centerC2#72142
commutatorC2×C3×Q8C22×C4
FrattiniC2×#41C22×D3
Fitting, prime radicalC3×#3549C2
socleC2×C3#1790
cosocleC2×C3×#41C23
hypercenter#3549D3
hypocenterC3#12778
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #3549

Complements

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

NH ≅ G/Ntype
2. #71861C2semidirect
3. #71854C2semidirect
4. #71819C2semidirect
5. #71817C2semidirect
8. C3×H(3,𝔽4)C4semidirect
10. C3×#555C4semidirect
18. #5819C22semidirect
19. #5240C22semidirect
20. #5240C22semidirect
84. H(3,𝔽4)Dic3semidirect
86. #555Dic3semidirect
408. C3#12778semidirect

Maximal subgroups

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

subgroupindex
C3×#35492
#718612
#718542
#718192
#718172
#127783