Finite Group Databasebeta

← #10705996

Subgroups — #10705996

Order 768.

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

normal subgroups only

This group has 420 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#82346C211yesmaximal, characteristic
2C3×#1800C211yesmaximal, characteristic
3#71602C211yesmaximal, characteristic
4#67496C212yesmaximal, complemented
5#67495C212yesmaximal, complemented
6#11830—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#926C411yescharacteristic
9C3×#423—21no—
10C2×C3×#100—21no—
11#5222—21no—
12C4×#217—21no—
13C4×#217C411yescharacteristic
14#5207—21no—
15#4986C2211yescharacteristic
16#4981C2211yescharacteristic
17#4953C411yescharacteristic
18#4952—21no—
19#4949C411yescharacteristic
20C4×Dic3⋊C4—21no—
21C3×#343C2212yes—
22#4528C2212yes—
23#4525—21no—
24#2395—31no—
25#1800D311yescharacteristic
26#1465—31no—
27#1353—32no—
28#1346—32no—
29C22×C3×D4D411yescharacteristic
30C23×C3×C4C2×C411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C2×C3×M16—22no—
33C3×C4×D4—41no—
34C3×C4×D4—41no—
35C2×C3×C4⋊C4C2311yescharacteristic
36C2×C3×C22⋊C4C2×C411yescharacteristic
37C2×C3×C22⋊C4D411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×#217—21no—
40C2×#217—21no—
41C2×#217—21no—
42C2×#217—21no—
43#926—41no—
44#926—41no—
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×C4×Dic3—21no—
50C2×C4×Dic3—21no—
51C2×#208—41no—
52C3×C4⋊C8—41no—
53C3×C4⋊C8—41no—
54C3×#100—42no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95C2×C411yescharacteristic
60C3⋊#95C2×C411yescharacteristic
61#830—22no—
62#830—41no—
63C2×C4×D4C2×D311yescharacteristic
64C2×#117—31no—
65#423—21no—
66C2×#100—21no—
67#407—61no—
68#400—31no—
69#399—31no—
70#398—31no—
71#397—31no—
72#397—61no—
73#394—61no—
74C4×C4⋊C4—61no—
75C4×C22⋊C4—31no—
76C4×C22⋊C4—61no—
77#343C2×D312yes—
78#324—32no—
79#323—61no—
80C24×C3C22⋊C411yescharacteristic
81C24×C3C22⋊C411yescharacteristic
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—41no—
85C2×C3×D4—41no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C3×M16—42no—
102C3×M16—42no—
103C2×C3×C8—42no—
104C3×C4⋊C4—41no—
105C3×C22⋊C4—41no—
106C3×C22⋊C4—41no—
107C3×C42—21no—
108C3×C42—21no—
109#217—42no—
110#217—42no—
111#217—42no—
112#217—42no—
113C4⋊Dic3—41no—
114C4⋊Dic3—41no—
115Dic3⋊C4—42no—
116Dic3⋊C4—41no—
117Dic3⋊C4—41no—
118C4×Dic3—42no—
119C4×Dic3—41no—
120C4×Dic3—41no—
121#208—82no—
122C2×C3⋊C8—42no—
123C22×D4#6711yescharacteristic
124C23×C4C4×D311yescharacteristic
125C23×C4D1211yescharacteristic
126C2×M16—22no—
127C2×M16—121no—
128C4×D4—41no—
129C4×D4—41no—
130#117—121no—
131#117—121no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4C22×D311yescharacteristic
137C2×C22⋊C4—61no—
138C2×C22⋊C4—61no—
139C2×C22⋊C4—61no—
140C2×C22⋊C4—61no—
141C2×C22⋊C4C4×D311yescharacteristic
142C2×C22⋊C4D1211yescharacteristic
143C2×C42—61no—
144C2×C42—61no—
145C2×C42#6711yescharacteristic
146C4⋊C8—41no—
147C4⋊C8—41no—
148#100—42no—
149C22⋊C8—62no—
150C22⋊C8—121no—
151#95—31no—
152#95—31no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
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×D4—42no—
169C3×D4—42no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—22no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—81no—
183C2×C3×C4#9911yescharacteristic
184C2×C3×C4#9911yescharacteristic
185C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
186C2×C3×C4#13711yescharacteristic
187C2×Dic3—22no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C3×C8—82no—
200C3⋊C8—82no—
201C24(C2×D3)⋊C411yescharacteristic
202C24(C2×D3)⋊C411yescharacteristic
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—41no—
206C2×D4—41no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4(C2×D3)⋊C411yescharacteristic
218C22×C4(C2×D3)⋊C411yescharacteristic
219C22×C4C2×C4×D311yescharacteristic
220C22×C4C2×D1211yescharacteristic
221C22×C4C2×#6711yescharacteristic
222M16—42no—
223M16—42no—
224M16—242no—
225C2×C8—42no—
226C2×C8—122no—
227C4⋊C4—41no—
228C4⋊C4—121no—
229C4⋊C4—121no—
230C4⋊C4—122no—
231C4⋊C4—121no—
232C4⋊C4—121no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—122no—
236C22⋊C4—122no—
237C22⋊C4—122no—
238C22⋊C4—122no—
239C42—21no—
240C42—21no—
241C42—122no—
242C42—121no—
243C42—121no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—42no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—81no—
254C22×C3—81no—
255C22×C3—81no—
256C22×C3C2×#9911yescharacteristic
257C22×C3#41711yescharacteristic
258C22×C3#42111yescharacteristic
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4#35212yes—
266Dic3—42no—
267Dic3—81no—
268Dic3—81no—
269Dic3—81no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×(C2×D3)⋊C411yescharacteristic
281D4—42no—
282D4—42no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—62no—
296C2×C4—81no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4—121no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4#80611yescharacteristic
309C2×C4#80611yescharacteristic
310C2×C4#93211yescharacteristic
311C2×C4#94211yescharacteristic
312C8—82no—
313C8—242no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—81no—
317C2×C3—81no—
318C2×C3#151311yescharacteristic
319C2×C3C2×#35211yescharacteristic
320C2×C3#213211yescharacteristic
321C22—21no—
322C22—21no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22C2×#80611yescharacteristic
334C22#496611yescharacteristic
335C22#500511yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—122no—
343C4—241no—
344C4—241no—
345C4—241no—
346C4#447612yes—
347C3#1183011yescharacteristic, complemented, Sylow
348C2—21no—
349C2—21no—
350C2—81no—
351C2—81no—
352C2#6950811yescharacteristic
353C2C2×#447611yescharacteristic
354C2#7014811yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1800C2
socleC22×C3C2×#99
prime radicalC22×C3×D4D4
cosocleC2×C3×C4⋊C4C23
hypercenter#1800D3
hypocenterC3#11830
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
4. #67496C2semidirect
5. #67495C2semidirect
347. C3#11830semidirect

Maximal subgroups

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

subgroupindex
#823462
C3×#18002
#716022
#674962
#674952
#118303