Finite Group Databasebeta

← #10706018

Subgroups — #10706018

Order 768.

400 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2638C211yesmaximal, characteristic
2#71617C211yesmaximal, characteristic
3#71602C211yesmaximal, characteristic
4#67650C212yesmaximal, complemented
5#67648C212yesmaximal, complemented
6#11834—31nomaximal, Sylow
7C2×C3×C22⋊Q8C411yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C3×#398C2211yescharacteristic
10C3×#397—21no—
11C3×#397C411yescharacteristic
12C3×#395C2211yescharacteristic
13C3×#394—21no—
14C3×#393C411yescharacteristic
15C3×#387—21no—
16C3×#385—21no—
17C3×#381C411yescharacteristic
18C2×#833—21no—
19#5207—21no—
20#4924—21no—
21#4530C2212yes—
22#4528C2212yes—
23#4525—21no—
24#2638D311yescharacteristic
25#1800—31no—
26#1465—31no—
27#1354—32no—
28#1346—32no—
29C22×C3×Q8C2×C411yescharacteristic
30C22×C3×D4D411yescharacteristic
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C3×C22⋊Q8—41no—
34C3×C22⋊Q8—41no—
35C3×C4×D4—41no—
36C3×C4×D4—41no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4C2311yescharacteristic
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4D411yescharacteristic
48C2×C3×C42D411yescharacteristic
49C2×#208—22no—
50C2×#208—41no—
51C3×#95—21no—
52C3×#95—21no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95C2×C411yescharacteristic
57C3×#95C2×C411yescharacteristic
58#833—42no—
59#830—22no—
60#830—41no—
61C4⋊(C3⋊C8)—41no—
62C4⋊(C3⋊C8)—41no—
63C2×C22⋊Q8Dic311yescharacteristic
64C2×C4×D4C2×D311yescharacteristic
65#423—61no—
66C2×#100—61no—
67#407—61no—
68#398C2×D311yescharacteristic
69#397—21no—
70#397Dic311yescharacteristic
71#395C2×D311yescharacteristic
72#394—21no—
73#393Dic311yescharacteristic
74#387—21no—
75#385—21no—
76#381Dic311yescharacteristic
77#343—32no—
78#324—32no—
79#323—61no—
80C24×C3C22⋊C411yescharacteristic
81C24×C3C22⋊C411yescharacteristic
82C2×C3×Q8—41no—
83C2×C3×Q8—81no—
84C2×C3×D4—21no—
85C2×C3×D4—21no—
86C2×C3×D4—41no—
87C2×C3×D4—41no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—81no—
97C22×C3×C4—81no—
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—81no—
110C3×C22⋊C4—41no—
111C3×C22⋊C4—41no—
112C3×C22⋊C4—42no—
113C3×C22⋊C4—42no—
114C3×C22⋊C4—81no—
115C3×C22⋊C4—81no—
116C3×C42—21no—
117C3×C42—21no—
118#208—42no—
119#208—42no—
120#208—82no—
121C2×C3⋊C8—42no—
122C2×C3⋊C8—42no—
123C22×Q8C2×Dic311yescharacteristic
124C22×D4#6711yescharacteristic
125C23×C4C2×Dic311yescharacteristic
126C23×C4#6711yescharacteristic
127C2×M16—62no—
128C2×M16—121no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C4×D4—41no—
132C4×D4—41no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4C2×Dic311yescharacteristic
137C2×C4⋊C4C22×D311yescharacteristic
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4C2×Dic311yescharacteristic
143C2×C22⋊C4#6711yescharacteristic
144C2×C42#6711yescharacteristic
145C4⋊C8—121no—
146C4⋊C8—121no—
147#100—122no—
148C22⋊C8—62no—
149C22⋊C8—121no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95C2×Dic311yescharacteristic
156#95C2×Dic311yescharacteristic
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—41no—
164C23×C3—41no—
165C23×C3—41no—
166C23×C3—41no—
167C23×C3C2×C22⋊C411yescharacteristic
168C3×Q8—82no—
169C3×D4—42no—
170C3×D4—42no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—41no—
177C2×C3×C4—42no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—81no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
196C2×C3×C4#9911yescharacteristic
197C2×C3×C4#9911yescharacteristic
198C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
199C2×C3×C4#13711yescharacteristic
200C3⋊C8—82no—
201C3⋊C8—82no—
202C24#21711yescharacteristic
203C24#21711yescharacteristic
204C2×Q8—41no—
205C2×Q8—81no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—41no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4#21711yescharacteristic
221C22×C4#21711yescharacteristic
222C22×C4C22×Dic311yescharacteristic
223C22×C4C2×#6711yescharacteristic
224C22×C4C2×#6711yescharacteristic
225M16—122no—
226M16—122no—
227M16—242no—
228C2×C8—122no—
229C2×C8—122no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—81no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C42—21no—
244C42—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—42no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—81no—
255C22×C3—81no—
256C22×C3—81no—
257C22×C3C2×#9911yescharacteristic
258C22×C3#41711yescharacteristic
259C22×C3#42111yescharacteristic
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—81no—
268C3×C4—81no—
269C3×C4—81no—
270C3×C4—81no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23C2×#21711yescharacteristic
282Q8—82no—
283D4—42no—
284D4—42no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4#83411yescharacteristic
311C2×C4#83411yescharacteristic
312C2×C4#94911yescharacteristic
313C2×C4#95111yescharacteristic
314C8—242no—
315C8—242no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—81no—
319C2×C3—81no—
320C2×C3#151311yescharacteristic
321C2×C3#212111yescharacteristic
322C2×C3#213511yescharacteristic
323C22—21no—
324C22—21no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22C2×#83411yescharacteristic
336C22#523211yescharacteristic
337C22#523611yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C3#1183411yescharacteristic, complemented, Sylow
350C2—21no—
351C2—21no—
352C2—81no—
353C2—81no—
354C2#7170911yescharacteristic
355C2#7184711yescharacteristic
356C2#7187311yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2638C2
socleC22×C3C2×#99
prime radicalC22×C3×D4D4
cosocleC2×C3×C4⋊C4C23
hypercenter#2638D3
hypocenterC3#11834
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2638

Complements

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

NH ≅ G/Ntype
4. #67650C2semidirect
5. #67648C2semidirect
349. C3#11834semidirect

Maximal subgroups

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

subgroupindex
C3×#26382
#716172
#716022
#676502
#676482
#118343