Finite Group Databasebeta

← #10704761

Subgroups — #10704761

Order 768.

440 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#82180C211yesmaximal, characteristic
2C2×C3×#324C211yesmaximal, characteristic, complemented
3#71552C211yesmaximal, characteristic, complemented
4#67454C212yesmaximal, complemented
5#67452C212yesmaximal, complemented
6#11553—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#930C411yescharacteristic
9C2×C3×C22⋊C8—21no—
10#5213—21no—
11#5213—21no—
12C4×#217C411yescharacteristic
13#5207—21no—
14#4992—21no—
15#4992—21no—
16#4988C411yescharacteristic
17#4988C2211yescharacteristic
18#4977C411yescharacteristic
19#4977C2211yescharacteristic
20C3×#324—21no—
21C3×#324C2212yes—
22#4528—21no—
23#4528C2212yes—
24#2504—31no—
25#1463—31no—
26C2×#324D311yescharacteristic, complemented
27#1324—32no—
28#1318—32no—
29C23×C3×C4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C22×#67—21no—
33C22×D12C2×C411yescharacteristic
34C22×C3×C8—41no—
35C2×C3×C4⋊C4—41no—
36C2×C3×C4⋊C4—41no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C4⋊C4C2311yescharacteristic
41C2×#217—21no—
42#930—41no—
43#930—41no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×C4⋊Dic3C2×C411yescharacteristic
50C2×C4×Dic3C2×C411yescharacteristic
51C2×C4×Dic3C2×C411yescharacteristic
52C2×#208—41no—
53C3×C22⋊C8—22no—
54C3×C22⋊C8—41no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60#830—22no—
61#830—41no—
62C2×#121—31no—
63C22×C4⋊C4C2×D311yescharacteristic
64#407—61no—
65C2×C22⋊C8—21no—
66#397—61no—
67#397—61no—
68#390—31no—
69#390—31no—
70#386—61no—
71#386—61no—
72#382—31no—
73#382—31no—
74C4×C22⋊C4—31no—
75#324—21no—
76#324—32no—
77#324—61no—
78#324C2×D312yes—
79C24×C3C22⋊C411yescharacteristic
80C23×D3—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—22no—
88C22×C3×C4—41no—
89C22×C3×C4—81no—
90C22×C3×C4—81no—
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C2×#67—42no—
98C2×#67—42no—
99C2×#67—41no—
100C2×#67—41no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C2×D12—22no—
105C2×D12—41no—
106C2×D12—41no—
107C2×C3×C8—42no—
108C2×C3×C8—82no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113#217—42no—
114(C2×D3)⋊C4—42no—
115(C2×D3)⋊C4—81no—
116(C2×D3)⋊C4—81no—
117(C2×D3)⋊C4—81no—
118(C2×D3)⋊C4—81no—
119C4⋊Dic3—41no—
120C4×Dic3—22no—
121C4×Dic3—41no—
122#208—82no—
123C2×C3⋊C8—42no—
124C22×D4—31no—
125C22×D4—61no—
126C23×C4C4×D311yescharacteristic
127C23×C4D1211yescharacteristic
128C23×C4#6711yescharacteristic
129C2×M16—121no—
130C22×C8—41no—
131#121—121no—
132#121—121no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4C4×D311yescharacteristic
137C2×C4⋊C4D1211yescharacteristic
138C2×C4⋊C4#6711yescharacteristic
139C2×C4⋊C4C22×D311yescharacteristic
140C2×C22⋊C4—61no—
141C2×C22⋊C4—61no—
142C2×C22⋊C4—61no—
143C2×C22⋊C4—61no—
144C2×C22⋊C4—61no—
145C2×C22⋊C4—61no—
146C2×C42—31no—
147C2×C42—31no—
148C22⋊C8—22no—
149C22⋊C8—41no—
150C22⋊C8—62no—
151C22⋊C8—121no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157C23×C3—22no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—41no—
161C23×C3#9911yescharacteristic
162C23×C3#9911yescharacteristic
163C23×C3C2×C22⋊C411yescharacteristic
164C22×D3—41no—
165C22×D3—41no—
166C22×D3—41no—
167C22×D3—41no—
168C22×D3—41no—
169C22×D3—41no—
170C22×D3—41no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—81no—
189C2×C3×C4—81no—
190C2×C3×C4#13112yes—
191#67—82no—
192#67—82no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202D12—82no—
203C3×C8—82no—
204C3⋊C8—82no—
205C24—61no—
206C24(C2×D3)⋊C411yescharacteristic
207C2×D4—62no—
208C2×D4—121no—
209C2×D4—121no—
210C2×D4—122no—
211C2×D4—122no—
212C2×D4—121no—
213C2×D4—121no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—41no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4(C2×D3)⋊C411yescharacteristic
228C22×C4(C2×D3)⋊C411yescharacteristic
229C22×C4(C2×D3)⋊C411yescharacteristic
230C22×C4C2×C4×D311yescharacteristic
231C22×C4C2×D1211yescharacteristic
232C22×C4C2×#6711yescharacteristic
233M16—242no—
234C2×C8—42no—
235C2×C8—82no—
236C2×C8—122no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—121no—
242C22⋊C4—122no—
243C22⋊C4—122no—
244C22⋊C4—241no—
245C22⋊C4—241no—
246C22⋊C4—241no—
247C22⋊C4—241no—
248C42—62no—
249C42—121no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—42no—
258C22×C3—42no—
259C22×C3#35512yes—
260C22×C3#40811yescharacteristic
261C22×C3C2×#9911yescharacteristic
262C22×C3#41311yescharacteristic
263C2×D3—81no—
264C2×D3—81no—
265C2×D3—81no—
266C2×D3—81no—
267C2×D3—81no—
268C2×D3—81no—
269C2×D3—81no—
270C3×C4—22no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—41no—
276Dic3—81no—
277Dic3—81no—
278Dic3—81no—
279C23—21no—
280C23—22no—
281C23—21no—
282C23—41no—
283C23—121no—
284C23—121no—
285C23—121no—
286C23—121no—
287C23—121no—
288C23—121no—
289C23—121no—
290C23#80611yescharacteristic
291C23#80611yescharacteristic
292C23C2×(C2×D3)⋊C411yescharacteristic
293D4—242no—
294D4—242no—
295D4—242no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4#90112yes—
325C8—82no—
326C8—242no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—22no—
330C2×C3—41no—
331C2×C3#146811yescharacteristic
332C2×C3#212711yescharacteristic
333C2×C3C2×#35511yescharacteristic
334D3—161no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—42no—
343C22—42no—
344C22—241no—
345C22—241no—
346C22—241no—
347C22—241no—
348C22—241no—
349C22—241no—
350C22—241no—
351C22#447712yes—
352C22C2×#80611yescharacteristic
353C22#511011yescharacteristic
354C22#513211yescharacteristic
355C4—22no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—241no—
362C4—241no—
363C4—241no—
364C3#1155311yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—22no—
368C2—41no—
369C2—481no—
370C2#6992011yescharacteristic
371C2#7001911yescharacteristic
372C2C2×#447711yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC2×C3×#324C2
socleC22×C3C2×#99
prime radicalC2×#930C4
cosocleC2×C3×C4⋊C4C23
hypercenterC2×#324D3
hypocenterC3#11553
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. C2×C3×#324C2semidirect
3. #71552C2semidirect
4. #67454C2semidirect
5. #67452C2semidirect
26. C2×#324D3semidirect
364. C3#11553semidirect

Maximal subgroups

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

subgroupindex
#821802
C2×C3×#3242
#715522
#674542
#674522
#115533