Finite Group Databasebeta

← #10692856

Subgroups — #10692856

Order 768.

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

normal subgroups only

This group has 490 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×#2973C211yesmaximal, characteristic
2#82386C211yesmaximal, characteristic
3#71603C211yesmaximal, characteristic
4#70814C212yesmaximal, complemented
5#67749C212yesmaximal, complemented
6#9449—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#870C2211yescharacteristic
9C3×#435—24no—
10C3×#432C2212yes—
11C3×#407—21no—
12C2×C3×C22⋊C8—21no—
13#5221C411yescharacteristic
14#5221C411yescharacteristic
15#5210—21no—
16#5210—21no—
17#4977C411yescharacteristic
18#4975C411yescharacteristic
19#4948—21no—
20#4948—21no—
21C4×C4⋊Dic3C2211yescharacteristic
22#4926C2212yes—
23#4525—21no—
24#4525—21no—
25#4465—24no—
26#2973D311yescharacteristic
27#2566—31no—
28#1534—32no—
29#1466—31no—
30#1285—32no—
31C22×C3×D4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C2×C3×M16—22no—
35C22×C3×C8—22no—
36C3×C4×D4—22no—
37C3×C4×D4—22no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C22⋊C4D411yescharacteristic
40C2×C3×C22⋊C4D411yescharacteristic
41C2×C3×C42C2311yescharacteristic
42C2×#217—21no—
43C2×#217—21no—
44C2×#217—21no—
45C2×#217—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×C4⋊Dic3C2×C411yescharacteristic
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×C4×Dic3C2×C411yescharacteristic
54C2×C4×Dic3C2×C411yescharacteristic
55#870—22no—
56#870—22no—
57C3×C4⋊C8—22no—
58C3×C22⋊C8—22no—
59C3×C22⋊C8—22no—
60C3×C22⋊C8—22no—
61C3×#97D414yes—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65#830—22no—
66C4⋊(C3⋊C8)—22no—
67#803—22no—
68C2×#125—31no—
69C2×C4×D4C2×D311yescharacteristic
70#435—24no—
71#432—32no—
72#432C2×D312yes—
73#407—21no—
74C2×C22⋊C8—21no—
75#396—61no—
76#396—61no—
77#390—31no—
78#390—31no—
79#385—61no—
80#385—61no—
81#384—31no—
82#382—31no—
83C4×C4⋊C4—31no—
84#330—64no—
85#323—61no—
86#323—61no—
87C24×C3C22⋊C411yescharacteristic
88C24×C3C22⋊C411yescharacteristic
89C2×C3×D4—22no—
90C2×C3×D4—22no—
91C2×C3×D4—41no—
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
103C22×Dic3—21no—
104C22×Dic3—21no—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C3×M16—82no—
108C2×C3×C8—24no—
109C2×C3×C8—82no—
110C3×C4⋊C4—22no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C42C2×D412yes—
114C3×C42#4112yes—
115#217—42no—
116#217—42no—
117#217—81no—
118#217—81no—
119C4⋊Dic3—22no—
120C4⋊Dic3—22no—
121C4⋊Dic3—42no—
122Dic3⋊C4—41no—
123Dic3⋊C4—42no—
124Dic3⋊C4—41no—
125Dic3⋊C4—81no—
126Dic3⋊C4—81no—
127C4×Dic3—41no—
128C4×Dic3—41no—
129C2×C3⋊C8—42no—
130C22×D4C4×D311yescharacteristic
131C23×C4#6711yescharacteristic
132C23×C4#6711yescharacteristic
133C2×M16—22no—
134C22×C8—22no—
135#125—62no—
136#125—62no—
137C4×D4—22no—
138C4×D4—22no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4C4×D311yescharacteristic
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149C2×C22⋊C4—61no—
150C2×C22⋊C4—61no—
151C2×C22⋊C4D1211yescharacteristic
152C2×C22⋊C4D1211yescharacteristic
153C2×C42—31no—
154C2×C42—31no—
155C2×C42C22×D311yescharacteristic
156C4⋊C8—22no—
157C4⋊C8—62no—
158C22⋊C8—22no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—62no—
162#97—62no—
163#97#6714yes—
164#95—61no—
165#95—61no—
166#95—61no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3#10112yes—
177C23×C3C2×C22⋊C411yescharacteristic
178C3×D4—44no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—81no—
190C2×C3×C4—81no—
191C2×C3×C4C4×D412yes—
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12112yes—
194C2×C3×C4#12311yescharacteristic
195C2×Dic3—22no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—41no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—41no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C3×C8—44no—
208C3⋊C8—82no—
209C24(C2×D3)⋊C411yescharacteristic
210C24(C2×D3)⋊C411yescharacteristic
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—41no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4(C2×D3)⋊C411yescharacteristic
225C22×C4(C2×D3)⋊C411yescharacteristic
226C22×C4C2×C4×D311yescharacteristic
227C22×C4C2×D1211yescharacteristic
228C22×C4C2×#6711yescharacteristic
229M16—82no—
230C2×C8—24no—
231C2×C8—82no—
232C2×C8—122no—
233C4⋊C4—22no—
234C4⋊C4—62no—
235C4⋊C4—62no—
236C4⋊C4—122no—
237C4⋊C4—121no—
238C4⋊C4—122no—
239C4⋊C4—121no—
240C4⋊C4—241no—
241C4⋊C4—241no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—122no—
245C22⋊C4—122no—
246C22⋊C4—241no—
247C22⋊C4—241no—
248C42—121no—
249C42—121no—
250C42#23512yes—
251C42C2×#6712yes—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—22no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—81no—
263C22×C3—81no—
264C22×C3#38611yescharacteristic
265C22×C3C2×#10111yescharacteristic
266C22×C3#41311yescharacteristic
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—41no—
273Dic3—42no—
274Dic3—81no—
275Dic3—81no—
276Dic3—81no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#82412yes—
287C23C2×(C2×D3)⋊C411yescharacteristic
288D4—44no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—62no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4—121no—
311C2×C4—121no—
312C2×C4—121no—
313C2×C4C4×#6712yes—
314C2×C4#92911yescharacteristic
315C2×C4#93012yes—
316C2×C4#95311yescharacteristic
317C8—44no—
318C8—242no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—42no—
322C2×C3—81no—
323C2×C3#188411yescharacteristic
324C2×C3#190111yescharacteristic
325C2×C3#191111yescharacteristic
326C22—21no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—42no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—81no—
337C22—81no—
338C22#513211yescharacteristic
339C22C2×#82411yescharacteristic
340C22#521311yescharacteristic
341C4—22no—
342C4—22no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—122no—
348C4—241no—
349C4—241no—
350C4—241no—
351C3#944911yescharacteristic, complemented, Sylow
352C2—21no—
353C2—21no—
354C2—42no—
355C2—81no—
356C2#7165211yescharacteristic
357C2#7200311yescharacteristic
358C2#7202111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2973C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#2973D3
hypocenterC3#9449
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. #70814C2semidirect
5. #67749C2semidirect
351. C3#9449semidirect

Maximal subgroups

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

subgroupindex
C3×#29732
#823862
#716032
#708142
#677492
#94493