Finite Group Databasebeta

← #10697250

Subgroups — #10697250

Order 768.

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

normal subgroups only

This group has 464 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×#3078C211yesmaximal, characteristic
2#83324C211yesmaximal, characteristic
3C2×#5020C211yesmaximal, characteristic
4#71824C211yesmaximal, characteristic
5#71624C211yesmaximal, characteristic
6#70507C211yesmaximal, characteristic
7#67714C211yesmaximal, characteristic
8#10191—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#479—22no—
13C3×#477—21no—
14C3×#475—21no—
15C3×#428—21no—
16C3×#423C2211yescharacteristic
17C3×#419—21no—
18C2×#835—21no—
19#5079—21no—
20#5078—21no—
21#5044—22no—
22#5021—22no—
23#5020—22no—
24#5020—22no—
25C2×#808—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29#4535—22no—
30#4528—22no—
31#4481—22no—
32#3230—31no—
33#3078D311yescharacteristic
34C2×#475—31no—
35#1548—31no—
36#1481—31no—
37#1461—31no—
38#1301—31no—
39C22×C3×Q8D411yescharacteristic
40C22×Dic6—21no—
41C2×C3×M16—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C42—21no—
57C2×C3×C42C2311yescharacteristic
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C22×C3⋊C8—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869D412yes—
65C3×#107—22no—
66C3×#106—22no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D412yes—
69C3×#103—22no—
70C3×#103—22no—
71#835—24no—
72#835—24no—
73#830—21no—
74#830D412yes—
75#808—24no—
76#807—24no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84C2×C4⋊Q8—31no—
85C2×C4×Q8C2×D311yescharacteristic
86#500—61no—
87#487—61no—
88#484—62no—
89#479—22no—
90#477—21no—
91#477—62no—
92#475—21no—
93#475—62no—
94#475—62no—
95#431—31no—
96#431—31no—
97#428—21no—
98C2×#106—61no—
99#423C2×D311yescharacteristic
100C2×C4⋊C8—31no—
101#419—21no—
102C2×#103—61no—
103#333—62no—
104Q8⋊C8—62no—
105#324—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8Q1612yes—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—41no—
120C2×Dic6—22no—
121C2×Dic6—41no—
122C2×Dic6—82no—
123C3×M16—41no—
124C3×M16—41no—
125C2×C3×C8—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C42—41no—
138C3×C42—41no—
139C3×C42—41no—
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C4⋊Dic3—22no—
145C4⋊Dic3—22no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C2×C3⋊C8—41no—
152C2×C3⋊C8—41no—
153C22×Q8—61no—
154C22×Q8#6711yescharacteristic
155C2×M16—21no—
156C22×C8—61no—
157C4⋊Q8—32no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8—61no—
161C4⋊Q8#6711yescharacteristic
162C4⋊Q8#6711yescharacteristic
163#125—21no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172#117—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4#6711yescharacteristic
177C2×C42—21no—
178C2×C42C22×D311yescharacteristic
179#107—22no—
180#107—64no—
181#106—22no—
182#106—64no—
183C4⋊C8—21no—
184C4⋊C8—32no—
185C4⋊C8—61no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8#6712yes—
189#103—22no—
190#103—22no—
191#103—64no—
192#103—64no—
193C22⋊C8—32no—
194C22⋊C8—61no—
195C4×C8—61no—
196C4×C8—61no—
197C23×C3#12111yescharacteristic
198C3×Q8—22no—
199C3×Q8—22no—
200C3×Q8—42no—
201C3×Q8—41no—
202C3×Q8—41no—
203C3×Q8—42no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—81no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12711yescharacteristic
219C2×C3×C4C2×Q1611yescharacteristic
220C2×C3×C4#13511yescharacteristic
221C2×C3×C4#13711yescharacteristic
222C2×C3×C4#13711yescharacteristic
223C2×Dic3—42no—
224C2×Dic3—82no—
225Dic6—82no—
226Dic6—82no—
227C3×C8—42no—
228C3⋊C8—42no—
229C3⋊C8—42no—
230C3⋊C8—81no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—62no—
239C2×Q8—121no—
240C2×Q8—242no—
241C2×Q8#21612yes—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—121no—
245C22×C4D3×D411yescharacteristic
246C22×C4#23711yescharacteristic
247C22×C4C2×#6711yescharacteristic
248M16—41no—
249M16—41no—
250C2×C8—22no—
251C2×C8—62no—
252C2×C8—62no—
253C2×C8—121no—
254C2×C8—121no—
255C2×C8—121no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—62no—
267C4⋊C4—62no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C22⋊C4—41no—
271C42—41no—
272C42—41no—
273C42—41no—
274C42D3×D411yescharacteristic
275C42D3×D411yescharacteristic
276C42C2×#6711yescharacteristic
277C42C2×#6711yescharacteristic
278C22×C3—21no—
279C22×C3—21no—
280C22×C3#39211yescharacteristic
281C22×C3#46711yescharacteristic
282C22×C3#47011yescharacteristic
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—42no—
287C3×C4—81no—
288C3×C4—81no—
289C3×C4—81no—
290C3×C4#45111yescharacteristic
291C3×C4#45111yescharacteristic
292C3×C4#45211yescharacteristic
293C3×C4#45211yescharacteristic
294C3×C4#47112yes—
295Dic3—82no—
296C23#93911yescharacteristic
297Q8—22no—
298Q8—22no—
299Q8—42no—
300Q8—41no—
301Q8—41no—
302Q8—42no—
303Q8—242no—
304Q8—242no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—122no—
315C2×C4—242no—
316C2×C4#93711yescharacteristic
317C2×C4#93711yescharacteristic
318C2×C4#93811yescharacteristic
319C2×C4#93911yescharacteristic
320C2×C4#94011yescharacteristic
321C2×C4#94211yescharacteristic
322C2×C4C2×#21611yescharacteristic
323C2×C4#95011yescharacteristic
324C2×C4#95111yescharacteristic
325C2×C4#95311yescharacteristic
326C8—42no—
327C8—122no—
328C8—122no—
329C8—241no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3#200111yescharacteristic
333C2×C3#200911yescharacteristic
334C2×C3#201911yescharacteristic
335C22—21no—
336C22—21no—
337C22#505111yescharacteristic
338C22#505211yescharacteristic
339C22#522311yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—242no—
348C4#504711yescharacteristic
349C4#505011yescharacteristic
350C4#520312yes—
351C4#522811yescharacteristic
352C4#523911yescharacteristic
353C3#1019111yescharacteristic, complemented, Sylow
354C2—21no—
355C2—21no—
356C2#7170011yescharacteristic
357C2#7210711yescharacteristic
358C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3078C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3078D3
hypocenterC3#10191
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
353. C3#10191semidirect

Maximal subgroups

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

subgroupindex
C3×#30782
#833242
C2×#50202
#718242
#716242
#705072
#677142
#101913