Finite Group Databasebeta

← #10693030

Subgroups — #10693030

Order 768.

474 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 113 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#79263C211yesmaximal, characteristic
2#74191C211yesmaximal, characteristic
3#70770C211yesmaximal, characteristic
4#70528C211yesmaximal, characteristic
5#70500C211yesmaximal, characteristic
6C3×#1286C211yesmaximal, characteristic
7#67631C211yesmaximal, characteristic
8#9484—31nomaximal, Sylow
9C2×C4×Dic6C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×#382C2211yescharacteristic
14C2×#824—21no—
15#4984—21no—
16#4973—21no—
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19#4830—22no—
20#4718—22no—
21C8×Dic6—22no—
22#4659C412yes—
23#4656C412yes—
24#4653—22no—
25#4554—21no—
26#4484—22no—
27#4456—22no—
28#2982—31no—
29#2581—31no—
30#1801—31no—
31#1487—31no—
32#1487—31no—
33#1286—31no—
34#1286D311yescharacteristic
35C22×Dic6C2×C411yescharacteristic
36C22×Dic6D411yescharacteristic
37C2×C3×M16—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C42C2311yescharacteristic
40C2×C4⋊Dic3—22no—
41C2×C4⋊Dic3C2×C411yescharacteristic
42C2×C4⋊Dic3D411yescharacteristic
43C2×Dic3⋊C4—21no—
44C2×Dic3⋊C4C2×C412yes—
45C2×C4×Dic3—21no—
46C2×C4×Dic3D412yes—
47C2×#208—21no—
48C4×Dic6—21no—
49C4×Dic6—21no—
50C4×Dic6—21no—
51C4×Dic6—21no—
52C4×Dic6—21no—
53C4×Dic6—21no—
54C4×Dic6—21no—
55C4×Dic6—21no—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×C4×C8D412yes—
61C3×#95—21no—
62C3×#95C2×C412yes—
63C3⋊#95—22no—
64C3⋊#95—22no—
65C3⋊#95—22no—
66#830—21no—
67#824—42no—
68Dic3⋊C8—24no—
69C8×Dic3—24no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4×C3⋊C8D412yes—
74C2×C4×Q8—31no—
75C2×C4×Q8—31no—
76C8×Q8—62no—
77C8×Q8—62no—
78#431—31no—
79#431—62no—
80#431C2×D311yescharacteristic
81#423—31no—
82#423C2×D311yescharacteristic
83C2×#101—61no—
84#400—32no—
85#395—32no—
86#391—61no—
87#389—61no—
88#387—62no—
89#382C2×D311yescharacteristic
90#343—61no—
91Q8⋊C8—62no—
92Q8⋊C8—62no—
93C22×C3×C4—22no—
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3C22⋊C412yes—
100C22×Dic3C22⋊C412yes—
101C2×Dic6—21no—
102C2×Dic6—21no—
103C2×Dic6—21no—
104C2×Dic6—21no—
105C2×Dic6—21no—
106C2×Dic6—21no—
107C2×Dic6—42no—
108C2×Dic6SD1612yes—
109C2×Dic6Q1612yes—
110C3×M16—41no—
111C3×M16—41no—
112C2×C3×C8—22no—
113C2×C3×C8—41no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C42C2×D411yescharacteristic
117C3×C42C2×D411yescharacteristic
118C3×C42#4111yescharacteristic
119C3×C42#4111yescharacteristic
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3—44no—
125Dic3⋊C4—22no—
126Dic3⋊C4—22no—
127Dic3⋊C4—41no—
128Dic3⋊C4—41no—
129C4×Dic3—22no—
130C4×Dic3—22no—
131C4×Dic3—41no—
132C4×Dic3—41no—
133#208—41no—
134#208—41no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—41no—
137C22×Q8—31no—
138C22×Q8—31no—
139C2×M16—21no—
140C2×M16—61no—
141C4×Q8—61no—
142C4×Q8—61no—
143C4×Q8—61no—
144C4×Q8—61no—
145C4×Q8—61no—
146C4×Q8—61no—
147C4×Q8—61no—
148C4×Q8—61no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—32no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—62no—
154C2×C4⋊C4—61no—
155C2×C42—32no—
156C2×C42—61no—
157C2×C42C22×D311yescharacteristic
158C4⋊C8—21no—
159C4⋊C8—21no—
160C4⋊C8—21no—
161C4⋊C8—64no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164C4⋊C8—61no—
165#101—122no—
166C22⋊C8—21no—
167C22⋊C8—61no—
168C4×C8—32no—
169C4×C8—64no—
170C4×C8D1212yes—
171#95—21no—
172#95—62no—
173#95—62no—
174#95—62no—
175#95C4×D312yes—
176C23×C3C2×C22⋊C411yescharacteristic
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4C4×D411yescharacteristic
183C2×C3×C4C4×D411yescharacteristic
184C2×C3×C4#12011yescharacteristic
185C2×C3×C4#12111yescharacteristic
186C2×C3×C4#12111yescharacteristic
187C2×C3×C4#12311yescharacteristic
188C2×C3×C4C2×SD1611yescharacteristic
189C2×C3×C4C2×Q1611yescharacteristic
190C2×C3×C4#13511yescharacteristic
191C2×C3×C4#13511yescharacteristic
192C2×Dic3—22no—
193C2×Dic3—22no—
194C2×Dic3—22no—
195C2×Dic3—22no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—41no—
201C2×Dic3#10314yes—
202Dic6—22no—
203Dic6—22no—
204Dic6—22no—
205Dic6—22no—
206Dic6—22no—
207Dic6—22no—
208Dic6—41no—
209Dic6—41no—
210C3×C8—42no—
211C3×C8—81no—
212C3⋊C8—42no—
213C3⋊C8—81no—
214C2×Q8—32no—
215C2×Q8—32no—
216C2×Q8—61no—
217C2×Q8—61no—
218C2×Q8—61no—
219C2×Q8—61no—
220C2×Q8—61no—
221C2×Q8—61no—
222C2×Q8—122no—
223C22×C4—22no—
224C22×C4—32no—
225C22×C4—32no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4C2×C4×D311yescharacteristic
229C22×C4D3×D411yescharacteristic
230C22×C4D3×D411yescharacteristic
231M16—41no—
232M16—41no—
233M16—121no—
234M16—121no—
235C2×C8—22no—
236C2×C8—41no—
237C2×C8—62no—
238C2×C8—121no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—62no—
242C4⋊C4—61no—
243C4⋊C4—62no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—124no—
248C4⋊C4—121no—
249C4⋊C4—121no—
250C42—62no—
251C42—62no—
252C42—121no—
253C42—121no—
254C42C2×D1211yescharacteristic
255C42#23511yescharacteristic
256C42D3×D411yescharacteristic
257C42#23911yescharacteristic
258C22×C3—21no—
259C22×C3—21no—
260C22×C3#38611yescharacteristic
261C22×C3C2×#10311yescharacteristic
262C22×C3#41611yescharacteristic
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—82no—
268C3×C4#46511yescharacteristic
269C3×C4#46511yescharacteristic
270C3×C4#46711yescharacteristic
271C3×C4#46711yescharacteristic
272Dic3—24no—
273Dic3—42no—
274Dic3—42no—
275Dic3—81no—
276C23D3×C22⋊C411yescharacteristic
277Q8—62no—
278Q8—62no—
279Q8—62no—
280Q8—62no—
281Q8—62no—
282Q8—62no—
283Q8—121no—
284Q8—121no—
285C2×C4—34no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—62no—
292C2×C4—62no—
293C2×C4—62no—
294C2×C4—62no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—121no—
300C2×C4C4×D1211yescharacteristic
301C2×C4#88211yescharacteristic
302C2×C4#88411yescharacteristic
303C2×C4#89311yescharacteristic
304C2×C4#89411yescharacteristic
305C2×C4#89511yescharacteristic
306C2×C4#91211yescharacteristic
307C2×C4D3×SD1611yescharacteristic
308C2×C4#91611yescharacteristic
309C2×C4D3×Q1611yescharacteristic
310C8—42no—
311C8—81no—
312C8—122no—
313C8—241no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3#189111yescharacteristic
317C2×C3#189211yescharacteristic
318C2×C3#191211yescharacteristic
319C22—21no—
320C22—21no—
321C22#466511yescharacteristic
322C22#477211yescharacteristic
323C22D3×#10311yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—64no—
329C4—82no—
330C4—122no—
331C4—122no—
332C4—241no—
333C4#477811yescharacteristic
334C4#481311yescharacteristic
335C4#486411yescharacteristic
336C4#488711yescharacteristic
337C3#948411yescharacteristic, complemented, Sylow
338C2—21no—
339C2—21no—
340C2#6870811yescharacteristic
341C2#6888211yescharacteristic
342C2#6903611yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1286C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1286D3
hypocenterC3#9484
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
337. C3#9484semidirect

Maximal subgroups

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

subgroupindex
#792632
#741912
#707702
#705282
#705002
C3×#12862
#676312
#94843