Finite Group Databasebeta

← #10697376

Subgroups — #10697376

Order 768.

440 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 77 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×#3216C211yesmaximal, characteristic
2#83372C211yesmaximal, characteristic, complemented
3#83274C211yesmaximal, characteristic
4#71837C211yesmaximal, characteristic, complemented
5#71813C211yesmaximal, characteristic
6#70837C211yesmaximal, characteristic, complemented
7C3⋊#1308C211yesmaximal, characteristic, complemented
8#10210—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#497—21no—
13C3×#489—21no—
14C3×#462—21no—
15C3×#460—21no—
16C3×#452—22no—
17C3×#432C2211yescharacteristic
18#5094—21no—
19#5091—21no—
20#5081—21no—
21#5071—21no—
22#5069—21no—
23#5064—21no—
24#5035—22no—
25#5008—21no—
26#5007—21no—
27#4996—22no—
28#4937C2211yescharacteristic
29#4926C2211yescharacteristic
30#4926C2211yescharacteristic
31#4538—22no—
32#4537—22no—
33#4467—22no—
34#3237—31no—
35#3235—31no—
36#3216D311yescharacteristic
37#1515—31no—
38#1515—31no—
39#1514—31no—
40#1308—31no—
41C2×C3×#41—21no—
42C2×C3×Q16—22no—
43C2×C3×SD16—22no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#127D411yescharacteristic
48C3×#125—21no—
49C3×#124—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×#117—21no—
59C2×C3×C42C2311yescharacteristic
60#926—21no—
61#924—21no—
62#924—21no—
63#870—21no—
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×#103—22no—
70C3×#102—22no—
71C3×C22⋊C8D412yes—
72C3×#97—21no—
73#832—22no—
74#832—22no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78#830D412yes—
79#830D412yes—
80#810—22no—
81#810—22no—
82#808—22no—
83#808—22no—
84#807—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89#803—21no—
90#533—31no—
91#533C2×D311yescharacteristic
92#532C2×D311yescharacteristic
93#501—61no—
94#501—61no—
95#497—21no—
96#491—61no—
97#490—61no—
98#489—21no—
99#485—62no—
100#482—62no—
101#477—61no—
102#476—61no—
103#475—61no—
104#474—61no—
105#462—21no—
106#460—21no—
107#452—22no—
108#432—31no—
109#432—31no—
110#432C2×D311yescharacteristic
111#383—31no—
112#328—62no—
113#328—62no—
114#327—62no—
115C3×#41—82no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—41no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×Dic3—41no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C3×Q16—82no—
131C3×SD16—82no—
132C2×C3×C8—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#217—41no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×#41—21no—
160C2×Q16—22no—
161C2×SD16—22no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164C4⋊Q8#6711yescharacteristic
165C4⋊Q8#6711yescharacteristic
166C4⋊Q8#6711yescharacteristic
167#127#6711yescharacteristic
168#125—21no—
169#125—61no—
170#124—21no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8—61no—
174C22⋊Q8—61no—
175#121—21no—
176#121—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—61no—
180C4×Q8—61no—
181C4×D4—21no—
182C4×D4—21no—
183#117—21no—
184#117—61no—
185C2×C42C22×D311yescharacteristic
186#107—62no—
187#107—62no—
188#106—62no—
189#106—62no—
190C4⋊C8—21no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#103—22no—
194#103—62no—
195#103—62no—
196#102—22no—
197#102—62no—
198#102—62no—
199C22⋊C8—32no—
200C22⋊C8—32no—
201C22⋊C8#6712yes—
202#97—21no—
203#97—61no—
204#97—61no—
205#95—61no—
206#95—61no—
207#95—61no—
208C23×C3—41no—
209C23×C3#12711yescharacteristic
210C3×Q8—41no—
211C3×Q8—41no—
212C3×Q8—42no—
213C3×Q8—81no—
214C3×D4—42no—
215C3×D4—42no—
216C3×D4—81no—
217C3×D4—82no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—82no—
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—81no—
234C2×Dic3—81no—
235C2×Dic3—81no—
236Dic6—42no—
237Dic6—81no—
238C3×C8—42no—
239C3⋊C8—42no—
240C3⋊C8—42no—
241#41—82no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C22×C4—41no—
252C22×C4—121no—
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4C2×#6711yescharacteristic
256Q16—82no—
257SD16—82no—
258C2×C8—22no—
259C2×C8—62no—
260C2×C8—62no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—121no—
279C42—41no—
280C42—121no—
281C42D3×D411yescharacteristic
282C42#23711yescharacteristic
283C42C2×#6711yescharacteristic
284C42C2×#6711yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3—81no—
291C22×C3#39211yescharacteristic
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—81no—
299C3×C4—81no—
300C3×C4—81no—
301Dic3—81no—
302Dic3—81no—
303Dic3—161no—
304C23—41no—
305C23#94011yescharacteristic
306Q8—41no—
307Q8—41no—
308Q8—42no—
309Q8—81no—
310Q8—122no—
311Q8—241no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4—82no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—82no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—241no—
326C2×C4—241no—
327C2×C4—241no—
328C2×C4#93711yescharacteristic
329C2×C4#93711yescharacteristic
330C2×C4#93811yescharacteristic
331C2×C4#93911yescharacteristic
332C2×C4#93911yescharacteristic
333C2×C4#95311yescharacteristic
334C8—42no—
335C8—122no—
336C8—122no—
337C2×C3—41no—
338C2×C3—161no—
339C2×C3#200811yescharacteristic
340C2×C3#200911yescharacteristic
341C2×C3#200911yescharacteristic
342C22—21no—
343C22—21no—
344C22—21no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#522311yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—241no—
359C4—241no—
360C4—481no—
361C3#1021011yescharacteristic, complemented, Sylow
362C2—41no—
363C2—161no—
364C2#7185811yescharacteristic
365C2#7211111yescharacteristic
366C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3216C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3216D3
hypocenterC3#10210
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3216

Complements

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

NH ≅ G/Ntype
2. #83372C2semidirect
4. #71837C2semidirect
6. #70837C2semidirect
7. C3⋊#1308C2semidirect
361. C3#10210semidirect

Maximal subgroups

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

subgroupindex
C3×#32162
#833722
#832742
#718372
#718132
#708372
C3⋊#13082
#102103