Finite Group Databasebeta

← #10691727

Subgroups — #10691727

Order 768.

428 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3229C211yesmaximal, characteristic
2#83330C211yesmaximal, characteristic
3#71830C211yesmaximal, characteristic
4#71789C211yesmaximal, characteristic
5C2×#4547C211yesmaximal, characteristic
6#67745C211yesmaximal, characteristic
7#67744C211yesmaximal, characteristic
8#9272—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C3×#498—21no—
12C3×#487—21no—
13C3×#483C414yes—
14C3×#475—22no—
15C3×#431C2211yescharacteristic
16C2×#836—21no—
17C2×#836—21no—
18#5024—22no—
19#5024—21no—
20Q8×C3⋊C8—21no—
21#4999—21no—
22#4997—21no—
23#4924C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4547—21no—
28#4547—21no—
29#4547—21no—
30#4546—21no—
31#3229D311yescharacteristic
32#2962—31no—
33#1966—31no—
34#1550—31no—
35C2×#332—31no—
36#1301—31no—
37#1283—31no—
38C22×C3×Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8C2×C412yes—
48C3×C4×Q8—21no—
49C3×C4×Q8D412yes—
50C3×#117—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4C2×C412yes—
53C2×C3×C42C2311yescharacteristic
54C2×#208—21no—
55C2×#208D411yescharacteristic
56C2×#208D411yescharacteristic
57C22×C3⋊C8—21no—
58C22×C3⋊C8—21no—
59C3×#106—24no—
60C3×C4⋊C8—21no—
61C3×#103—24no—
62C3×C22⋊C8C2×C412yes—
63C3×C4×C8—21no—
64#836—42no—
65#836—42no—
66C4⋊(C3⋊C8)—22no—
67C4⋊(C3⋊C8)—22no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)D411yescharacteristic
74C4⋊(C3⋊C8)D411yescharacteristic
75#803—22no—
76#803—21no—
77C4×C3⋊C8—22no—
78C4×C3⋊C8—21no—
79#533C2×D311yescharacteristic
80C2×C4⋊Q8C2×D311yescharacteristic
81#498—21no—
82#487—21no—
83#483Dic314yes—
84#475—22no—
85#446—61no—
86#446—62no—
87C8×Q8—61no—
88#431C2×D311yescharacteristic
89#424—61no—
90#423—31no—
91C2×C4⋊C8—31no—
92C2×C4⋊C8—31no—
93C2×#101—61no—
94C2×#101—61no—
95#405—61no—
96C4×M16—31no—
97#333—61no—
98#332—61no—
99#332—61no—
100#332—61no—
101C2×C3×Q8—21no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8C22⋊C412yes—
109C22×C3×C4—22no—
110C22×C3×C4C22×C411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C2×C3×C8—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—21no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4C22⋊C412yes—
124C3×C22⋊C4—22no—
125C3×C42—22no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130#208—21no—
131#208—21no—
132#208—21no—
133#208—41no—
134#208—41no—
135#208SD1612yes—
136C2×C3⋊C8—21no—
137C2×C3⋊C8—21no—
138C2×C3⋊C8—21no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8—41no—
142C2×C3⋊C8—41no—
143C2×C3⋊C8—41no—
144C2×C3⋊C8—81no—
145C22×Q8—21no—
146C2×M16—31no—
147C2×M16—31no—
148C2×M16—61no—
149C22×C8—61no—
150C22×C8—61no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8#6711yescharacteristic
156C4⋊Q8#6711yescharacteristic
157#125—21no—
158C22⋊Q8—21no—
159C22⋊Q8C2×Dic312yes—
160C4×Q8—21no—
161C4×Q8#6712yes—
162#117—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4C2×Dic312yes—
165C2×C42C22×D311yescharacteristic
166#106—24no—
167C4⋊C8—21no—
168C4⋊C8—31no—
169C4⋊C8—31no—
170C4⋊C8—62no—
171C4⋊C8—62no—
172C4⋊C8—61no—
173C4⋊C8—61no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176C4⋊C8—61no—
177#103—24no—
178#101—122no—
179#101—122no—
180C22⋊C8C2×Dic312yes—
181#97—62no—
182#97—61no—
183C4×C8—21no—
184C4×C8—62no—
185C4×C8—61no—
186C23×C3C4×D411yescharacteristic
187C3×Q8—42no—
188C3×Q8—41no—
189C3×Q8—41no—
190C3×Q8—81no—
191C3×Q8—81no—
192C3×Q8—81no—
193C3×Q8—81no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4C2×C22⋊C411yescharacteristic
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12311yescharacteristic
208C2×C3×C4C2×SD1611yescharacteristic
209C2×C3×C4#13511yescharacteristic
210C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
211C2×C3×C4#13711yescharacteristic
212C3×C8—42no—
213C3⋊C8—22no—
214C3⋊C8—41no—
215C3⋊C8—41no—
216C3⋊C8—41no—
217C3⋊C8—81no—
218C3⋊C8—81no—
219C2×Q8—21no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8#21712yes—
227C22×C4—22no—
228C22×C4D3×D411yescharacteristic
229C22×C4#23711yescharacteristic
230C22×C4C22×Dic311yescharacteristic
231M16—32no—
232M16—61no—
233M16—61no—
234M16—61no—
235M16—121no—
236M16—121no—
237C2×C8—22no—
238C2×C8—61no—
239C2×C8—61no—
240C2×C8—61no—
241C2×C8—61no—
242C2×C8—121no—
243C2×C8—121no—
244C2×C8—121no—
245C2×C8—121no—
246C2×C8—241no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4#21712yes—
257C22⋊C4—22no—
258C42—22no—
259C42D3×D411yescharacteristic
260C42#23711yescharacteristic
261C42C2×#6711yescharacteristic
262C42C2×#6711yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3#38611yescharacteristic
266C22×C3C4×SD1611yescharacteristic
267C22×C3#44011yescharacteristic
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—42no—
272C3×C4—42no—
273C3×C4—82no—
274C3×C4#44811yescharacteristic
275C3×C4#44911yescharacteristic
276C3×C4#46011yescharacteristic
277C3×C4#46411yescharacteristic
278C3×C4#47111yescharacteristic
279C3×C4#47311yescharacteristic
280C23D4×Dic311yescharacteristic
281Q8—42no—
282Q8—41no—
283Q8—41no—
284Q8—81no—
285Q8—81no—
286Q8—81no—
287Q8—81no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4D3×SD1611yescharacteristic
297C2×C4#91411yescharacteristic
298C2×C4#91511yescharacteristic
299C2×C4#91611yescharacteristic
300C2×C4D4×Dic311yescharacteristic
301C2×C4#93511yescharacteristic
302C2×C4#93711yescharacteristic
303C2×C4#93911yescharacteristic
304C2×C4#93911yescharacteristic
305C2×C4C2×#21711yescharacteristic
306C8—42no—
307C8—62no—
308C8—121no—
309C8—121no—
310C8—121no—
311C8—241no—
312C8—241no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3#188011yescharacteristic
316C2×C3#190811yescharacteristic
317C2×C3#191511yescharacteristic
318C22—21no—
319C22—21no—
320C22Dic3×SD1611yescharacteristic
321C22#516411yescharacteristic
322C22#522011yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—42no—
328C4—82no—
329C4#516311yescharacteristic
330C4#516611yescharacteristic
331C4#517011yescharacteristic
332C4#517211yescharacteristic
333C4#520311yescharacteristic
334C4#520411yescharacteristic
335C3#927211yescharacteristic, complemented, Sylow
336C2—21no—
337C2—21no—
338C2#7186611yescharacteristic
339C2#7204811yescharacteristic
340C2#7210311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3229C2
socleC22×C3#386
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenter#3229D3
hypocenterC3#9272
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
335. C3#9272semidirect

Maximal subgroups

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

subgroupindex
C3×#32292
#833302
#718302
#717892
C2×#45472
#677452
#677442
#92723