Finite Group Databasebeta

← #10697334

Subgroups — #10697334

Order 768.

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

normal subgroups only

This group has 490 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×#3249C211yesmaximal, characteristic
2#83341C211yesmaximal, characteristic
3#83324C211yesmaximal, characteristic
4#71841C211yesmaximal, characteristic
5#71835C211yesmaximal, characteristic
6#70456C211yesmaximal, characteristic
7#67706C211yesmaximal, characteristic
8#10203—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#533C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#477—21no—
13C3×#477—21no—
14C3×#467—22no—
15C3×#462—21no—
16C3×#462—21no—
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×#816—22no—
19#5080—21no—
20#5079—21no—
21#5078—21no—
22#5078—21no—
23#5064—22no—
24#5044—22no—
25#5021—22no—
26#4995—22no—
27C4×C4⋊Dic3C2211yescharacteristic
28#4925C2211yescharacteristic
29C4×#208C2211yescharacteristic
30#4538—22no—
31#4532—22no—
32#3249D311yescharacteristic
33#3230—31no—
34#3165—31no—
35#1986—31no—
36#1981—31no—
37#1503—31no—
38#1283—31no—
39C22×Dic6—21no—
40C2×C3×Q16—22no—
41C22×C3×C8D412yes—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×#125—21no—
48C3×C22⋊Q8—21no—
49C3×C22⋊Q8—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×#117—21no—
57C3×#117—21no—
58C2×C3×C42C2311yescharacteristic
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4×Dic3—21no—
63C2×#208D412yes—
64#869—21no—
65#869—21no—
66#869—21no—
67#869D412yes—
68C3×#107—22no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×#103—22no—
72C3×#103—22no—
73#835—24no—
74C3⋊#95—21no—
75#830D412yes—
76#816—44no—
77#808—24no—
78#807—24no—
79#807—24no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84#533C2×D311yescharacteristic
85#533C2×D311yescharacteristic
86C2×C4⋊Q8—31no—
87#501—62no—
88#500—61no—
89#500—61no—
90#498—61no—
91#487—61no—
92#484—62no—
93#477—21no—
94#477—21no—
95#477—62no—
96#467—22no—
97#462—21no—
98#462—21no—
99#431—31no—
100#428—62no—
101C2×C4⋊C8C2×D311yescharacteristic
102C2×#103—62no—
103C4×M16—31no—
104C4×C4⋊C4—31no—
105#339—62no—
106#328—62no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×Dic3—41no—
115C22×Dic3—41no—
116C2×Dic6—22no—
117C2×Dic6—42no—
118C2×Dic6—42no—
119C2×Dic6—41no—
120C3×Q16—82no—
121C2×C3×C8—22no—
122C2×C3×C8—42no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—41no—
138C3×C42—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143C4⋊Dic3—22no—
144C4⋊Dic3—22no—
145C4⋊Dic3—22no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4×Dic3—42no—
150#208—22no—
151#208Q1614yescomplemented
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C22×Q8—61no—
155C2×Q16—22no—
156C2×M16—32no—
157C22×C8#6712yes—
158C4⋊Q8—32no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8#6711yescharacteristic
163C4⋊Q8#6711yescharacteristic
164C4⋊Q8#6711yescharacteristic
165C4⋊Q8#6711yescharacteristic
166#125—21no—
167#125—21no—
168C22⋊Q8—21no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176#117—21no—
177#117—21no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C42—61no—
182C2×C42C22×D311yescharacteristic
183#107—22no—
184#107—64no—
185#107—64no—
186#106—64no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—61no—
190#103—22no—
191#103—22no—
192#103—64no—
193#103—124no—
194C22⋊C8—32no—
195#97—61no—
196C4×C8—61no—
197C4×C8—61no—
198#95—61no—
199C23×C3#12711yescharacteristic
200C3×Q8—42no—
201C3×Q8—41no—
202C3×Q8—41no—
203C3×Q8—41no—
204C3×Q8—41no—
205C3×Q8—81no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4C2×Q1612yes—
220C2×C3×C4#13712yes—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—82no—
224C2×Dic3—81no—
225C2×Dic3—81no—
226Dic6—44no—
227Dic6—81no—
228Dic6—81no—
229C3×C8—42no—
230C3⋊C8—24no—
231C3⋊C8—42no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—62no—
237C2×Q8—122no—
238C2×Q8—122no—
239C2×Q8—121no—
240C22×C4—121no—
241C22×C4—121no—
242C22×C4D3×D411yescharacteristic
243C22×C4D3×D411yescharacteristic
244C22×C4C2×#6711yescharacteristic
245Q16—82no—
246M16—34no—
247M16—62no—
248C2×C8—22no—
249C2×C8—42no—
250C2×C8—62no—
251C2×C8—62no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C4⋊C4—62no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C42—41no—
273C42—41no—
274C42—122no—
275C42D3×D411yescharacteristic
276C42#23711yescharacteristic
277C42C2×#6711yescharacteristic
278C42C2×#6711yescharacteristic
279C22×C3—21no—
280C22×C3—21no—
281C22×C3#39211yescharacteristic
282C22×C3#49411yescharacteristic
283C22×C3#49711yescharacteristic
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—81no—
288C3×C4—81no—
289C3×C4—81no—
290C3×C4—81no—
291C3×C4#45112yes—
292C3×C4#45311yescharacteristic
293C3×C4#45411yescharacteristic
294C3×C4#46212yes—
295Dic3—82no—
296Dic3—82no—
297C23#94011yescharacteristic
298Q8—42no—
299Q8—41no—
300Q8—41no—
301Q8—41no—
302Q8—41no—
303Q8—81no—
304Q8—124no—
305Q8—241no—
306Q8—241no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4—242no—
317C2×C4—241no—
318C2×C4—241no—
319C2×C4D3×Q1612yes—
320C2×C4#91812yes—
321C2×C4#93711yescharacteristic
322C2×C4#93711yescharacteristic
323C2×C4#93811yescharacteristic
324C2×C4#93911yescharacteristic
325C2×C4#93911yescharacteristic
326C2×C4#95311yescharacteristic
327C8—42no—
328C8—64no—
329C8—122no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3#200611yescharacteristic
333C2×C3#200911yescharacteristic
334C2×C3#201611yescharacteristic
335C22—21no—
336C22—21no—
337C22#518311yescharacteristic
338C22#519011yescharacteristic
339C22#522311yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—242no—
348C4—242no—
349C4#518212yes—
350C4#518612yes—
351C4#524111yescharacteristic
352C4#524511yescharacteristic
353C3#1020311yescharacteristic, complemented, Sylow
354C2—21no—
355C2—21no—
356C2#7187911yescharacteristic
357C2#7207311yescharacteristic
358C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3249C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#3249D3
hypocenterC3#10203
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
151. #208Q16semidirect
353. C3#10203semidirect

Maximal subgroups

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

subgroupindex
C3×#32492
#833412
#833242
#718412
#718352
#704562
#677062
#102033