Finite Group Databasebeta

← #10732003

Subgroups — #10732003

Order 768.

1778 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic3C211yesmaximal, characteristic
2C2×C3×#395C211yesmaximal, characteristic
3C2×#5230C211yesmaximal, characteristic
4C2×#4976C213yesmaximal
5#71565C213yesmaximal
6#69410C213yesmaximal
7Dic3⋊#95C213yesmaximal
8#15158—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×C3×C4⋊C4C2213yes—
11C2×Q8×Dic3—212no—
12C22×C4⋊Dic3C2213yes—
13C22×Dic3⋊C4C2213yes—
14C22×Dic3⋊C4C2211yescharacteristic
15C22×Dic3⋊C4C2213yes—
16C22×C4×Dic3C2213yes—
17C3×#395—24no—
18C2×C3×#95C2213yes—
19#5230—24no—
20C2×C3⋊#95C2213yes—
21C2×C3⋊#95C2213yes—
22C2×C3⋊#95C2213yes—
23C2×C3⋊#95C2216yes—
24#4976—212no—
25C22×C4×Q8—31no—
26C2×#395—31no—
27C2×#395D311yescharacteristic
28C2×#389—33no—
29#1441—33no—
30#1441—33no—
31C4⋊#95—33no—
32C22×C3×Q8—26no—
33C22×C3×Q8—26no—
34C22×C3×Q8D414yes—
35C23×C3×C4C2311yescharacteristic
36C23×C3×C4C2313yes—
37C23×C3×C4C2313yes—
38C23×Dic3C2311yescharacteristic
39C23×Dic3C2313yes—
40C23×Dic3C2313yes—
41C23×Dic3C2311yescharacteristic
42C2×C3×C4⋊C4—26no—
43C2×C3×C4⋊C4—26no—
44C2×C3×C4⋊C4—26no—
45Q8×Dic3—232no—
46C2×C4⋊Dic3—26no—
47C2×C4⋊Dic3—212no—
48C2×Dic3⋊C4—26no—
49C2×Dic3⋊C4—26no—
50C2×Dic3⋊C4—26no—
51C2×Dic3⋊C4—26no—
52C2×Dic3⋊C4—26no—
53C2×Dic3⋊C4—26no—
54C2×Dic3⋊C4Q8112yescomplemented
55C2×C4×Dic3—212no—
56C2×C4×Dic3D4112yes—
57C3×#95—26no—
58C3⋊#95—26no—
59C3⋊#95—26no—
60C3⋊#95—26no—
61C3⋊#95—46no—
62C23×Q8C2×D311yescharacteristic
63C2×C4×Q8—612no—
64C22×C4⋊C4—33no—
65C22×C4⋊C4—33no—
66C22×C4⋊C4—33no—
67C22×C4⋊C4—31no—
68C22×C4⋊C4C2×D313yes—
69C22×C42—33no—
70#395—24no—
71#395—64no—
72#389—612no—
73C2×#95—33no—
74C2×#95—33no—
75C2×#95—33no—
76C2×#95—36no—
77C2×#95C2×D313yes—
78C24×C3C2411yescharacteristic
79C2×C3×Q8—28no—
80C2×C3×Q8—224no—
81C2×C3×Q8—224no—
82C22×C3×C4—26no—
83C22×C3×C4—26no—
84C22×C3×C4—23no—
85C22×C3×C4—23no—
86C22×C3×C4—23no—
87C22×C3×C4—23no—
88C22×C3×C4—23no—
89C22×C3×C4—23no—
90C22×C3×C4—23no—
91C22×C3×C4—21no—
92C22×C3×C4—23no—
93C22×C3×C4C2×D416yes—
94C22×C3×C4C2×D416yes—
95C22×C3×C4C2×Q816yes—
96C22×C3×C4#4116yes—
97C22×Dic3—23no—
98C22×Dic3—23no—
99C22×Dic3—26no—
100C22×Dic3—23no—
101C22×Dic3—23no—
102C22×Dic3—23no—
103C22×Dic3—23no—
104C22×Dic3—26no—
105C22×Dic3—23no—
106C22×Dic3—23no—
107C22×Dic3—23no—
108C22×Dic3—21no—
109C22×Dic3—23no—
110C22×Dic3C2×D416yes—
111C22×Dic3C2×D416yes—
112C22×Dic3C2×Q816yes—
113C22×Dic3C2×Q816yes—
114C22×Dic3#4112yes—
115C3×C4⋊C4—412no—
116C4⋊Dic3—224no—
117Dic3⋊C4—224no—
118Dic3⋊C4—412no—
119Dic3⋊C4—44no—
120C4×Dic3—224no—
121C22×Q8—26no—
122C22×Q8—26no—
123C22×Q8#6714yes—
124C23×C4—31no—
125C23×C4—33no—
126C23×C4—33no—
127C23×C4—31no—
128C23×C4C22×D311yescharacteristic
129C23×C4C22×D313yes—
130C23×C4C22×D313yes—
131C4×Q8—632no—
132C2×C4⋊C4—26no—
133C2×C4⋊C4—26no—
134C2×C4⋊C4—26no—
135C2×C4⋊C4—312no—
136C2×C4⋊C4—612no—
137C2×C4⋊C4—66no—
138C2×C4⋊C4—66no—
139C2×C4⋊C4—66no—
140C2×C4⋊C4—66no—
141C2×C4⋊C4—66no—
142C2×C4⋊C4—66no—
143C2×C4⋊C4—66no—
144C2×C42—312no—
145C2×C42—612no—
146#95—26no—
147#95—66no—
148#95—66no—
149#95—66no—
150#95—126no—
151C23×C3C22×D413yes—
152C23×C3C22×D411yescharacteristic
153C23×C3C22×Q813yes—
154C23×C3C2×#4113yes—
155C23×C3C2×#4111yescharacteristic
156C23×C3H(5,𝔽2)11yescharacteristic
157C23×C32−1+413yes—
158C3×Q8—232no—
159C2×C3×C4—26no—
160C2×C3×C4—212no—
161C2×C3×C4—26no—
162C2×C3×C4—212no—
163C2×C3×C4—26no—
164C2×C3×C4—26no—
165C2×C3×C4—26no—
166C2×C3×C4—43no—
167C2×C3×C4—43no—
168C2×C3×C4—43no—
169C2×C3×C4—43no—
170C2×C3×C4—41no—
171C2×C3×C4—43no—
172C2×C3×C4—43no—
173C2×C3×C4#121112yes—
174C2×Dic3—26no—
175C2×Dic3—26no—
176C2×Dic3—26no—
177C2×Dic3—26no—
178C2×Dic3—26no—
179C2×Dic3—26no—
180C2×Dic3—43no—
181C2×Dic3—46no—
182C2×Dic3—46no—
183C2×Dic3—43no—
184C2×Dic3—43no—
185C2×Dic3—43no—
186C2×Dic3—41no—
187C2×Dic3—43no—
188C2×Dic3—43no—
189C2×Dic3#12014yes—
190C2×Dic3C22⋊Q8112yes—
191C2×Dic3C4⋊Q8112yes—
192C24C23×D311yescharacteristic
193C2×Q8—28no—
194C2×Q8—224no—
195C2×Q8—224no—
196C22×C4—26no—
197C22×C4—26no—
198C22×C4—23no—
199C22×C4—23no—
200C22×C4—23no—
201C22×C4—23no—
202C22×C4—23no—
203C22×C4—23no—
204C22×C4—23no—
205C22×C4—23no—
206C22×C4—21no—
207C22×C4—36no—
208C22×C4—36no—
209C22×C4—32no—
210C22×C4—36no—
211C22×C4—36no—
212C22×C4—63no—
213C22×C4—66no—
214C22×C4—63no—
215C22×C4—66no—
216C22×C4—63no—
217C22×C4—63no—
218C22×C4—63no—
219C22×C4—63no—
220C22×C4—63no—
221C22×C4—63no—
222C22×C4—63no—
223C22×C4—63no—
224C22×C4—61no—
225C22×C4D3×D416yes—
226C22×C4#23716yes—
227C22×C4D3×Q816yes—
228C22×C4C2×#6716yes—
229C4⋊C4—412no—
230C4⋊C4—624no—
231C4⋊C4—624no—
232C4⋊C4—124no—
233C4⋊C4—1212no—
234C42—624no—
235C22×C3C2×#12011yescharacteristic
236C22×C3C2×#12113yes—
237C22×C3C2×C22⋊Q813yes—
238C22×C3#52513yes—
239C22×C3C2×C4⋊Q813yes—
240C22×C3#53613yes—
241C22×C3#53713yes—
242C22×C3D4×Q813yes—
243C22×C3#55011yescharacteristic
244C22×C3#55413yes—
245C22×C3#55613yes—
246C22×C3#55716yes—
247C3×C4—212no—
248C3×C4—46no—
249C3×C4—81no—
250Dic3—46no—
251Dic3—46no—
252Dic3—81no—
253Dic3#39518yescomplemented
254C23C2×D3×D413yes—
255C23C2×#23713yes—
256C23C2×D3×Q813yes—
257C23#100713yes—
258C23D3×#4111yescharacteristic
259C23#100911yescharacteristic
260C23C22×#6711yescharacteristic
261Q8—232no—
262C2×C4—212no—
263C2×C4—26no—
264C2×C4—212no—
265C2×C4—26no—
266C2×C4—26no—
267C2×C4—26no—
268C2×C4—26no—
269C2×C4—312no—
270C2×C4—34no—
271C2×C4—312no—
272C2×C4—43no—
273C2×C4—43no—
274C2×C4—43no—
275C2×C4—43no—
276C2×C4—43no—
277C2×C4—43no—
278C2×C4—41no—
279C2×C4—66no—
280C2×C4—66no—
281C2×C4—66no—
282C2×C4—66no—
283C2×C4—66no—
284C2×C4—66no—
285C2×C4—123no—
286C2×C4—126no—
287C2×C4—123no—
288C2×C4—123no—
289C2×C4—123no—
290C2×C4—126no—
291C2×C4—123no—
292C2×C4—123no—
293C2×C4—121no—
294C2×C4#939112yes—
295C2×C3C2×#39511yescharacteristic
296C2×C3#253813yes—
297C2×C3#254613yes—
298C2×C3#258113yes—
299C2×C3#258413yes—
300C2×C3#261611yescharacteristic
301C2×C3#284711yescharacteristic
302C22#558613yes—
303C22D3×#12011yescharacteristic
304C22D3×C22⋊Q813yes—
305C22#562913yes—
306C22#563513yes—
307C22#564113yes—
308C22#572116yes—
309C22D3×C4⋊Q813yes—
310C22#572413yes—
311C22C2×#93913yes—
312C22Q8×#6713yes—
313C22#583411yescharacteristic
314C4—212no—
315C4—38no—
316C4—46no—
317C4—81no—
318C4—126no—
319C4—126no—
320C4—241no—
321C3#1515811yescharacteristic, complemented, Sylow
322C2#7719113yes—
323C2#7719213yes—
324C2#7720111yescharacteristic
325C2D3×#39511yescharacteristic
326C2#8260111yescharacteristic
327C2#8277813yes—
328C2#8323813yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC2×C3×#395C2
hypercenterC2×#395D3
hypocenterC3#15158
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
54. C2×Dic3⋊C4Q8semidirect
253. Dic3#395semidirect
321. C3#15158semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic32
C2×C3×#3952
C2×#52302
C2×#49762
#715652
#694102
Dic3⋊#952
#151583