Finite Group Databasebeta

← #10706346

Subgroups — #10706346

Order 768.

440 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2564C211yesmaximal, characteristic
2#71617C211yesmaximal, characteristic
3#71600C211yesmaximal, characteristic
4#67652C212yesmaximal, complemented
5#67648C212yesmaximal, complemented
6#11906—31nomaximal, Sylow
7C2×C3×C22⋊Q8C411yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C3×#400C2211yescharacteristic
10C3×#399C411yescharacteristic
11C3×#395C2211yescharacteristic
12C3×#393C411yescharacteristic
13C3×#387—21no—
14C3×#386—21no—
15C3×#385—21no—
16C3×#380—21no—
17C2×C3×#95C411yescharacteristic
18C2×#833—21no—
19C2×#830—21no—
20#4924—21no—
21#4530C2212yes—
22#4528C2212yes—
23#4525—21no—
24#2564D311yescharacteristic
25#1800—31no—
26#1459—31no—
27#1354—32no—
28#1353—32no—
29C22×C3×Q8C2×C411yescharacteristic
30C22×C3×D4D411yescharacteristic
31C23×C3×C4—21no—
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C3×C22⋊Q8—41no—
35C3×C22⋊Q8—41no—
36C3×C4×D4—41no—
37C3×C4×D4—41no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4C2311yescharacteristic
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—
50C22×C3⋊C8—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—
67C2×C22⋊C8—61no—
68#400C2×D311yescharacteristic
69#399Dic311yescharacteristic
70#395C2×D311yescharacteristic
71#393Dic311yescharacteristic
72#387—21no—
73#386—21no—
74#385—21no—
75#380—21no—
76C2×#95Dic311yescharacteristic
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—41no—
85C2×C3×D4—41no—
86C2×C3×D4SD1612yes—
87C2×C3×D4Q1612yes—
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—42no—
97C22×C3×C4—42no—
98C22×C3×C4—81no—
99C22×C3×C4—81no—
100C22×C3×C4—81no—
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—42no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—81no—
112C3×C22⋊C4—41no—
113C3×C22⋊C4—41no—
114C3×C22⋊C4—42no—
115C3×C22⋊C4—42no—
116C3×C22⋊C4—81no—
117C3×C42—21no—
118C3×C42—21no—
119#208—42no—
120#208—42no—
121C2×C3⋊C8—42no—
122C2×C3⋊C8—42no—
123C2×C3⋊C8—82no—
124C22×Q8C2×Dic311yescharacteristic
125C22×D4#6711yescharacteristic
126C23×C4—21no—
127C23×C4C2×Dic311yescharacteristic
128C23×C4#6711yescharacteristic
129C2×M16—62no—
130C22×C8—121no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C4×D4—41no—
134C4×D4—41no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4C2×Dic311yescharacteristic
139C2×C4⋊C4C22×D311yescharacteristic
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4C2×Dic311yescharacteristic
144C2×C22⋊C4#6711yescharacteristic
145C2×C42#6711yescharacteristic
146C4⋊C8—121no—
147C4⋊C8—121no—
148#100—122no—
149C22⋊C8—62no—
150C22⋊C8—121no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—21no—
156#95C2×Dic311yescharacteristic
157#95C2×Dic311yescharacteristic
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—41no—
163C23×C3—41no—
164C23×C3—41no—
165C23×C3—41no—
166C23×C3#10312yes—
167C23×C3#10312yes—
168C23×C3C2×C22⋊C411yescharacteristic
169C3×Q8—82no—
170C3×D4—42no—
171C3×D4—42no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—22no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—41no—
179C2×C3×C4—42no—
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—41no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
196C2×C3×C4—81no—
197C2×C3×C4—84no—
198C2×C3×C4#9911yescharacteristic
199C2×C3×C4#9911yescharacteristic
200C2×C3×C4C2×SD1611yescharacteristic
201C2×C3×C4C2×Q1611yescharacteristic
202C3⋊C8—82no—
203C3⋊C8—82no—
204C24#21711yescharacteristic
205C24#21711yescharacteristic
206C2×Q8—41no—
207C2×Q8—81no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4#21512yes—
211C2×D4#21612yes—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4#21711yescharacteristic
226C22×C4#21711yescharacteristic
227C22×C4C22×Dic311yescharacteristic
228C22×C4C2×#6711yescharacteristic
229C22×C4C2×#6711yescharacteristic
230M16—122no—
231M16—122no—
232C2×C8—122no—
233C2×C8—122no—
234C2×C8—242no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—81no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—81no—
246C42—21no—
247C42—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3—42no—
253C22×C3—42no—
254C22×C3—42no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—81no—
260C22×C3C2×#9911yescharacteristic
261C22×C3C2×#10311yescharacteristic
262C22×C3#42111yescharacteristic
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—42no—
268C3×C4—41no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—81no—
272C3×C4—81no—
273C3×C4—81no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23#83512yes—
283C23#83512yes—
284C23C2×#21711yescharacteristic
285Q8—82no—
286D4—42no—
287D4—42no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
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—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—84no—
314C2×C4#83411yescharacteristic
315C2×C4#83411yescharacteristic
316C2×C4C2×#21511yescharacteristic
317C2×C4C2×#21611yescharacteristic
318C8—242no—
319C8—242no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—42no—
323C2×C3—42no—
324C2×C3#152011yescharacteristic
325C2×C3#212111yescharacteristic
326C2×C3#213211yescharacteristic
327C22—21no—
328C22—21no—
329C22—22no—
330C22—22no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—81no—
339C22C2×#83411yescharacteristic
340C22C2×#83511yescharacteristic
341C22#523611yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C3#1190611yescharacteristic, complemented, Sylow
354C2—21no—
355C2—21no—
356C2—42no—
357C2—42no—
358C2#7169011yescharacteristic
359C2#7182011yescharacteristic
360C2#7184711yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2564C2
socleC22×C3C2×#99
prime radicalC22×C3×D4D4
cosocleC2×C3×C4⋊C4C23
hypercenter#2564D3
hypocenterC3#11906
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2564

Complements

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

NH ≅ G/Ntype
4. #67652C2semidirect
5. #67648C2semidirect
353. C3#11906semidirect

Maximal subgroups

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

subgroupindex
C3×#25642
#716172
#716002
#676522
#676482
#119063