Finite Group Databasebeta

← #10690166

Subgroups — #10690166

Order 768.

560 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2973C211yesmaximal, characteristic
2#82346C211yesmaximal, characteristic
3C4×#834C211yesmaximal, characteristic
4#69975C212yesmaximal, complemented
5#67683C212yesmaximal, complemented
6#9011—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#926C2211yescharacteristic
9C3×#435—24no—
10C3×#432—21no—
11C3×#407C2212yes—
12C2×C3×C22⋊C8—21no—
13#5222—21no—
14C2×#834—21no—
15C4×#217—21no—
16C4×#217C2212yes—
17C4×#217C2211yescharacteristic
18#4986C411yescharacteristic, complemented
19#4981C411yescharacteristic, complemented
20#4953C411yescharacteristic, complemented
21#4952—21no—
22#4949C411yescharacteristic, complemented
23C4×Dic3⋊C4—21no—
24#4555—24no—
25#4526—21no—
26#2973D311yescharacteristic
27#2395—31no—
28#1826—32no—
29C4×#99—31no—
30#1297—32no—
31C22×C3×D4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C2×C3×M16D414yes—
35C22×C3×C8—22no—
36C3×C4×D4—22no—
37C3×C4×D4—22no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C22⋊C4C2×C411yescharacteristic
40C2×C3×C22⋊C4D411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×#217—22no—
43C2×#217—21no—
44C2×#217—21no—
45C2×#217—21no—
46C2×#217—21no—
47#926—22no—
48#926—22no—
49C2×C4⋊Dic3C2×C411yescharacteristic
50C2×C4⋊Dic3C2×C411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×C4×Dic3—21no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3—22no—
56C3×C4⋊C8—22no—
57C3×C22⋊C8—22no—
58C3×C22⋊C8—22no—
59C3×C22⋊C8—22no—
60C3×#97—22no—
61#834—24no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—22no—
67C3⋊#95C2×C411yescharacteristic
68C3⋊#95C2×C411yescharacteristic
69C2×C4×D4C2×D311yescharacteristic
70C2×#117—31no—
71#435—24no—
72#432—21no—
73C2×#99—61no—
74#407C2×D312yes—
75C2×C22⋊C8—21no—
76#400—31no—
77#399—31no—
78#398—31no—
79#397—31no—
80#397—61no—
81#394—61no—
82C4×C4⋊C4—61no—
83C4×C22⋊C4—31no—
84C4×C22⋊C4—32no—
85C4×C22⋊C4—61no—
86#342—61no—
87#339—64no—
88C24×C3C22⋊C411yescharacteristic
89C24×C3C2×D411yescharacteristic
90C2×C3×D4—22no—
91C2×C3×D4—22no—
92C2×C3×D4—41no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—42no—
96C22×C3×C4—42no—
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C2×D412yes—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4112yes—
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—42no—
109C3×M16—44no—
110C2×C3×C8—24no—
111C2×C3×C8—82no—
112C3×C4⋊C4—22no—
113C3×C22⋊C4—22no—
114C3×C22⋊C4—22no—
115C3×C42—21no—
116C3×C42—21no—
117#217—24no—
118#217—42no—
119#217—42no—
120#217—42no—
121#217—42no—
122#217—42no—
123C4⋊Dic3—22no—
124C4⋊Dic3—22no—
125Dic3⋊C4—42no—
126Dic3⋊C4—42no—
127C4×Dic3—42no—
128C4×Dic3—41no—
129C4×Dic3—41no—
130C4×Dic3—44no—
131C22×D4D1211yescharacteristic
132C23×C4#6711yescharacteristic
133C23×C4C22×D311yescharacteristic
134C2×M16#6714yes—
135C22×C8—22no—
136C4×D4—22no—
137C4×D4—22no—
138#117—62no—
139#117—62no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4C4×D311yescharacteristic
145C2×C22⋊C4—62no—
146C2×C22⋊C4—61no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4C4×D311yescharacteristic
151C2×C22⋊C4D1211yescharacteristic
152C2×C42—61no—
153C2×C42—61no—
154C2×C42—62no—
155C2×C42#6711yescharacteristic
156C4⋊C8—22no—
157#99—64no—
158C22⋊C8—22no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161#97—22no—
162#95—31no—
163#95—31no—
164#95—61no—
165#95—61no—
166#95—61no—
167#95—61no—
168#95—62no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3#10412yes—
178C23×C3C2×C22⋊C411yescharacteristic
179C23×C3#12112yes—
180C3×D4—44no—
181C2×C3×C4—24no—
182C2×C3×C4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—41no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—81no—
194C2×C3×C4C4×D412yes—
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12311yescharacteristic
197C2×Dic3—22no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—44no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—82no—
211C2×Dic3—82no—
212C3×C8—44no—
213C24(C2×D3)⋊C411yescharacteristic
214C24C2×D1211yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—61no—
226C22×C4—122no—
227C22×C4(C2×D3)⋊C411yescharacteristic
228C22×C4(C2×D3)⋊C411yescharacteristic
229C22×C4(C2×D3)⋊C411yescharacteristic
230C22×C4C2×C4×D311yescharacteristic
231C22×C4#23512yes—
232C22×C4C2×#6712yes—
233C22×C4C2×#6711yescharacteristic
234M16—44no—
235C2×C8—24no—
236C2×C8—82no—
237C4⋊C4—22no—
238C4⋊C4—62no—
239C4⋊C4—62no—
240C4⋊C4—122no—
241C4⋊C4—122no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—64no—
245C22⋊C4—122no—
246C22⋊C4—122no—
247C22⋊C4—122no—
248C22⋊C4—122no—
249C22⋊C4—122no—
250C42—21no—
251C42—21no—
252C42—122no—
253C42—124no—
254C42—121no—
255C42—121no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—22no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—42no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—81no—
266C22×C3—81no—
267C22×C3—81no—
268C22×C3#38611yescharacteristic
269C22×C3C2×#10411yescharacteristic
270C22×C3#42111yescharacteristic
271C3×C4—21no—
272C3×C4—22no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4#45812yes—
277Dic3—42no—
278Dic3—81no—
279Dic3—84no—
280Dic3—81no—
281Dic3—81no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23#82512yes—
291C23C2×(C2×D3)⋊C411yescharacteristic
292C23#93012yes—
293D4—44no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—62no—
307C2×C4—81no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—124no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—121no—
320C2×C4—242no—
321C2×C4—242no—
322C2×C4C4×#6712yes—
323C2×C4#92911yescharacteristic
324C2×C4#95311yescharacteristic
325C8—44no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—42no—
329C2×C3—42no—
330C2×C3#187111yescharacteristic
331C2×C3#188411yescharacteristic
332C2×C3#189011yescharacteristic
333C22—21no—
334C22—21no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×#82511yescharacteristic
346C22#513911yescharacteristic
347C22#521311yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—122no—
354C4—241no—
355C4—244no—
356C4—241no—
357C4—241no—
358C4#524312yes—
359C3#901111yescharacteristic, complemented, Sylow
360C2—21no—
361C2—21no—
362C2—42no—
363C2—42no—
364C2#7165511yescharacteristic
365C2#7198411yescharacteristic
366C2#7202111yescharacteristic
subgroupG/H
centerC2×C4#953
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2973C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC23×C3×C4C23
hypercenter#2973D3
hypocenterC3#9011
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. #69975C2semidirect
5. #67683C2semidirect
18. #4986C4semidirect
19. #4981C4semidirect
20. #4953C4semidirect
22. #4949C4semidirect
359. C3#9011semidirect

Maximal subgroups

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

subgroupindex
C3×#29732
#823462
C4×#8342
#699752
#676832
#90113