Finite Group Databasebeta

← #10697617

Subgroups — #10697617

Order 768.

604 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 105 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×#2542C211yesmaximal, characteristic
2#83222C211yesmaximal, characteristic, complemented
3#82296C211yesmaximal, characteristic
4#71774C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic
6#69471C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#10248—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C22⋊Q8—22no—
11C2×C3×C4×D4C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#399—22no—
14C3×#390—21no—
15C3×#384—21no—
16C3×C4×C4⋊C4C2211yescharacteristic
17#5228—22no—
18#5215—21no—
19#5092—21no—
20#5090—22no—
21#5079—21no—
22#5009—22no—
23#5008—21no—
24#5007—21no—
25#4964—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4546—22no—
30D4⋊(C3⋊C8)—22no—
31#4452—22no—
32#3209—31no—
33#3077—31no—
34#2542D311yescharacteristic
35#1548—31no—
36#1503—31no—
37#1478—31no—
38#1303—31no—
39C22×C3×Q8D412yes—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C22⋊Q8—48no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C22⋊C4—24no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C42—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#216—24no—
61#926—21no—
62#924—21no—
63#924—21no—
64C2×#208—21no—
65#870—21no—
66#869D411yescharacteristic
67#869D411yescharacteristic
68C4×Dic6—21no—
69C4×Dic6—21no—
70C3×#95D412yes—
71#835—24no—
72#832—22no—
73#832—22no—
74#830—21no—
75#830—21no—
76#808—22no—
77#807—22no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81C4⋊(C3⋊C8)D412yes—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84#533—31no—
85C2×C4⋊Q8C2×D311yescharacteristic
86C2×C22⋊Q8—22no—
87C2×C4×D4C2×D311yescharacteristic
88#494—61no—
89#487—61no—
90#478—62no—
91#476—61no—
92#474—61no—
93#462—62no—
94#451—62no—
95#431—31no—
96#431—31no—
97#428—61no—
98#423—31no—
99#418—61no—
100#399—22no—
101#390—21no—
102#384—21no—
103C4×C4⋊C4C2×D311yescharacteristic
104#333—62no—
105#328—62no—
106D4⋊C8—62no—
107C24×C3—21no—
108C2×C3×Q8—22no—
109C2×C3×Q8—22no—
110C2×C3×Q8—44no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4Q1612yescomplemented
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—44no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C2×Dic6—21no—
127C2×Dic6—21no—
128C3×C4⋊C4—24no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—48no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C22⋊C4—48no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C42—41no—
140C3×C42—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42#4111yescharacteristic
145#217—41no—
146#216—84no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153C4×Dic3—41no—
154#208—41no—
155#208—41no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—41no—
159C22×Q8#6712yes—
160C22×D4#6711yescharacteristic
161C23×C4—21no—
162C2×Q16—64no—
163C2×M16—61no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—31no—
169C4⋊Q8—31no—
170#125—61no—
171C22⋊Q8—48no—
172C22⋊Q8—61no—
173C22⋊Q8—61no—
174C4×Q8—61no—
175C4×Q8—61no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180#117—61no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4#6711yescharacteristic
186C2×C22⋊C4—24no—
187C2×C22⋊C4—21no—
188C2×C42—21no—
189C2×C42C22×D311yescharacteristic
190#107—62no—
191#106—62no—
192C4⋊C8—32no—
193C4⋊C8—32no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196#103—64no—
197#102—62no—
198#102—62no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201C4×C8—61no—
202C4×C8—61no—
203#95#6712yes—
204C23×C3—22no—
205C23×C3—42no—
206C23×C3—42no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3#12011yescharacteristic
210C3×Q8—44no—
211C3×Q8—82no—
212C3×D4—22no—
213C3×D4—22no—
214C3×D4—41no—
215C3×D4—41no—
216C2×C3×C4—22no—
217C2×C3×C4—24no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—44no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—82no—
226C2×C3×C4—82no—
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×Q1611yescharacteristic
234C2×C3×C4#13511yescharacteristic
235C2×C3×C4#13711yescharacteristic
236C2×C3×C4#13711yescharacteristic
237C2×Dic3—41no—
238C2×Dic3—41no—
239Dic6—42no—
240Dic6—81no—
241C3⋊C8—42no—
242C3⋊C8—42no—
243C3⋊C8—81no—
244C24—21no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—44no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4#21612yescomplemented
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—44no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4C2×#6711yescharacteristic
263C22×C4C2×#6711yescharacteristic
264C22×C4C2×#6711yescharacteristic
265Q16—244no—
266M16—121no—
267M16—121no—
268C2×C8—62no—
269C2×C8—62no—
270C2×C8—121no—
271C4⋊C4—24no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—48no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C22⋊C4—48no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—121no—
289C42—41no—
290C42—41no—
291C42—121no—
292C42D3×D411yescharacteristic
293C42D3×D411yescharacteristic
294C42D3×D411yescharacteristic
295C42#23711yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—44no—
299C22×C3—42no—
300C22×C3—82no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3#39211yescharacteristic
304C22×C3#45111yescharacteristic
305C22×C3#45211yescharacteristic
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—44no—
310C3×C4—81no—
311C3×C4—82no—
312C3×C4#45111yescharacteristic
313C3×C4#45211yescharacteristic
314C3×C4#45312yes—
315C3×C4#46711yescharacteristic
316C3×C4#47011yescharacteristic
317Dic3—81no—
318Dic3—81no—
319C23—22no—
320C23—42no—
321C23—42no—
322C23—41no—
323C23—41no—
324C23#95311yescharacteristic
325Q8—44no—
326Q8—82no—
327Q8—122no—
328Q8—241no—
329D4—22no—
330D4—22no—
331D4—41no—
332D4—41no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4#93711yescharacteristic
347C2×C4#93711yescharacteristic
348C2×C4#93811yescharacteristic
349C2×C4#93911yescharacteristic
350C2×C4#93911yescharacteristic
351C2×C4#94011yescharacteristic
352C2×C4#94211yescharacteristic
353C2×C4C2×#21611yescharacteristic
354C2×C4#95011yescharacteristic
355C2×C4#95111yescharacteristic
356C8—122no—
357C8—122no—
358C8—241no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—82no—
362C2×C3#200111yescharacteristic
363C2×C3#200211yescharacteristic
364C2×C3#200911yescharacteristic
365C22—21no—
366C22—21no—
367C22—44no—
368C22—42no—
369C22—82no—
370C22—81no—
371C22—81no—
372C22#522311yescharacteristic
373C22#522811yescharacteristic
374C22#523911yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—44no—
379C4—81no—
380C4—82no—
381C4—241no—
382C4—241no—
383C4#504711yescharacteristic
384C4#505011yescharacteristic
385C4#505111yescharacteristic
386C4#505211yescharacteristic
387C4#507712yes—
388C3#1024811yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—82no—
392C2#7170011yescharacteristic
393C2#7176911yescharacteristic
394C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2542C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#2542D3
hypocenterC3#10248
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2542

Complements

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

NH ≅ G/Ntype
2. #83222C2semidirect
4. #71774C2semidirect
6. #69471C2semidirect
7. #67620C2semidirect
116. C2×C3×D4Q16semidirect
255. C2×D4#216semidirect
388. C3#10248semidirect

Maximal subgroups

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

subgroupindex
C3×#25422
#832222
#822962
#717742
#715972
#694712
#676202
#102483