Finite Group Databasebeta

← #10697652

Subgroups — #10697652

Order 768.

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

normal subgroups only

This group has 504 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×#2567C211yesmaximal, characteristic
2#83224C211yesmaximal, characteristic
3#82431C211yesmaximal, characteristic
4#71775C211yesmaximal, characteristic
5#71621C211yesmaximal, characteristic
6#69471C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic
8#10253—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#395—22no—
13C3×#391—21no—
14C3×#389—22no—
15C3×#384—21no—
16C3×C4×C4⋊C4C2211yescharacteristic
17#5225—21no—
18#5093—21no—
19#5091—22no—
20#5078—21no—
21#5022—22no—
22#5021—21no—
23#5020—21no—
24#4970—22no—
25#4964—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4547—22no—
30#4535—22no—
31#4452—22no—
32#3230—31no—
33#3078—31no—
34#2567D311yescharacteristic
35#1549—31no—
36#1503—31no—
37#1479—31no—
38#1303—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D412yes—
55C2×C3×C42—21no—
56C2×C3×C42—21no—
57C2×C3×C42C2311yescharacteristic
58#926—21no—
59#924—21no—
60#924—21no—
61C2×#208—21no—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×#95—24no—
68C3×#95D412yes—
69#835—22no—
70#835—22no—
71#830—21no—
72#830—21no—
73#810—24no—
74#808—22no—
75#807—22no—
76#807—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D412yes—
80C4⋊(C3⋊C8)D412yes—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×C4×Q8C2×D311yescharacteristic
86#500—61no—
87#487—61no—
88#484—62no—
89#479—62no—
90#477—62no—
91#477—61no—
92#475—61no—
93#431—31no—
94#431—31no—
95#428—61no—
96#423—31no—
97#419—61no—
98#395—22no—
99#391—21no—
100#389—22no—
101#384—21no—
102C4×C4⋊C4C2×D311yescharacteristic
103#332—62no—
104#328—62no—
105Q8⋊C8—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—82no—
114C2×C3×Q8Q1612yes—
115C22×C3×C4—21no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C2×Dic6—21no—
123C2×Dic6—21no—
124C3×C4⋊C4—24no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—84no—
134C3×C42—41no—
135C3×C42—41no—
136C3×C42—41no—
137C3×C42—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142#217—41no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—41no—
150#208—41no—
151#208—41no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C22×Q8—21no—
156C22×Q8#6711yescharacteristic
157C2×M16—61no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164#125—61no—
165C22⋊Q8—61no—
166C22⋊Q8—61no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—61no—
172C4×Q8—61no—
173#117—61no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4#6711yescharacteristic
179C2×C4⋊C4#6712yes—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42C22×D311yescharacteristic
183#107—62no—
184#107—64no—
185#106—62no—
186C4⋊C8—32no—
187C4⋊C8—32no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190#103—64no—
191#103—62no—
192#103—62no—
193C22⋊C8—61no—
194C22⋊C8—61no—
195C4×C8—61no—
196C4×C8—61no—
197#95—24no—
198#95#6712yes—
199C23×C3#12011yescharacteristic
200C3×Q8—22no—
201C3×Q8—22no—
202C3×Q8—41no—
203C3×Q8—41no—
204C3×Q8—82no—
205C3×Q8—82no—
206C2×C3×C4—22no—
207C2×C3×C4—24no—
208C2×C3×C4—22no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—41no—
212C2×C3×C4—42no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—44no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12711yescharacteristic
224C2×C3×C4C2×Q1611yescharacteristic
225C2×C3×C4#13511yescharacteristic
226C2×C3×C4#13711yescharacteristic
227C2×C3×C4#13711yescharacteristic
228C2×Dic3—41no—
229C2×Dic3—41no—
230Dic6—42no—
231Dic6—81no—
232C3⋊C8—42no—
233C3⋊C8—42no—
234C3⋊C8—81no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×Q8—82no—
245C2×Q8#21612yes—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4C2×#6711yescharacteristic
251C22×C4C2×#6711yescharacteristic
252C22×C4C2×#6711yescharacteristic
253M16—121no—
254M16—121no—
255C2×C8—62no—
256C2×C8—62no—
257C2×C8—121no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—22no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—84no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C22⋊C4—121no—
275C42—41no—
276C42—41no—
277C42—41no—
278C42—41no—
279C42—121no—
280C42D3×D411yescharacteristic
281C42D3×D411yescharacteristic
282C42D3×D411yescharacteristic
283C42#23711yescharacteristic
284C22×C3—21no—
285C22×C3—21no—
286C22×C3#39211yescharacteristic
287C22×C3#45111yescharacteristic
288C22×C3#45211yescharacteristic
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—44no—
293C3×C4—42no—
294C3×C4—81no—
295C3×C4—82no—
296C3×C4#45111yescharacteristic
297C3×C4#45211yescharacteristic
298C3×C4#45612yes—
299C3×C4#46711yescharacteristic
300C3×C4#47011yescharacteristic
301Dic3—81no—
302Dic3—81no—
303C23#95311yescharacteristic
304Q8—22no—
305Q8—22no—
306Q8—41no—
307Q8—41no—
308Q8—82no—
309Q8—82no—
310Q8—122no—
311Q8—241no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4#93711yescharacteristic
327C2×C4#93711yescharacteristic
328C2×C4#93811yescharacteristic
329C2×C4#93911yescharacteristic
330C2×C4#93911yescharacteristic
331C2×C4#94011yescharacteristic
332C2×C4#94211yescharacteristic
333C2×C4C2×#21611yescharacteristic
334C2×C4#95011yescharacteristic
335C2×C4#95111yescharacteristic
336C8—122no—
337C8—122no—
338C8—241no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3#200111yescharacteristic
342C2×C3#200511yescharacteristic
343C2×C3#200811yescharacteristic
344C22—21no—
345C22—21no—
346C22#522311yescharacteristic
347C22#522811yescharacteristic
348C22#523911yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—44no—
353C4—42no—
354C4—82no—
355C4—81no—
356C4—241no—
357C4—241no—
358C4#504711yescharacteristic
359C4#505011yescharacteristic
360C4#505111yescharacteristic
361C4#505211yescharacteristic
362C4#509512yes—
363C3#1025311yescharacteristic, complemented, Sylow
364C2—21no—
365C2—21no—
366C2#7170011yescharacteristic
367C2#7179511yescharacteristic
368C2#7185811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2567C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#2567D3
hypocenterC3#10253
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2567

Complements

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

NH ≅ G/Ntype
363. C3#10253semidirect

Maximal subgroups

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

subgroupindex
C3×#25672
#832242
#824312
#717752
#716212
#694712
#676202
#102533