Finite Group Databasebeta

← #10691160

Subgroups — #10691160

Order 768.

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

normal subgroups only

This group has 582 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, complemented
2#82422C211yesmaximal, characteristic
3C4×#834C211yesmaximal, characteristic, complemented
4#69977C212yesmaximal, complemented
5#67682C212yesmaximal, complemented
6#9185—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C3×#435—24no—
10C3×#432—21no—
11C3×#407C2212yes—
12C2×C3×C22⋊C8—21no—
13#5223—21no—
14C2×#834—21no—
15#5213—21no—
16#5211C2211yescharacteristic
17C4×#217C2212yes—
18#4991C411yescharacteristic, complemented
19#4986C411yescharacteristic, complemented
20#4960C411yescharacteristic, complemented
21#4959—21no—
22#4949C411yescharacteristic, complemented
23#4940—21no—
24#4526—21no—
25#4470—24no—
26#2973D311yescharacteristic, complemented
27#2635—31no—
28C4×#99—31no—
29#1507—32no—
30#1284—32no—
31C22×C3×D4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C22×#67—21no—
35C22×#67—21no—
36C22×D12C2×C411yescharacteristic
37C2×C3×M16—22no—
38C22×C3×C8—22no—
39C3×C4×D4—22no—
40C3×C4×D4—22no—
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C22⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4D411yescharacteristic
44C2×C3×C42D411yescharacteristic
45C2×#217—22no—
46C2×#217—21no—
47C2×#217—21no—
48#930—22no—
49#930—22no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×C4⋊Dic3C2×C411yescharacteristic
55C2×Dic3⋊C4—21no—
56C2×C4×Dic3—22no—
57C3×C4⋊C8—22no—
58C3×C22⋊C8—22no—
59C3×C22⋊C8—22no—
60C3×C22⋊C8D414yescomplemented
61C3×#97—22no—
62#834—24no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—22no—
67C3⋊#95C2×C411yescharacteristic
68C3⋊#95C2×C411yescharacteristic
69C2×#121—31no—
70C2×C4×D4C2×D311yescharacteristic
71#435—24no—
72#432—21no—
73C2×#99—61no—
74#407C2×D312yes—
75C2×C22⋊C8—21no—
76#400—31no—
77#399—31no—
78#394—31no—
79#394—61no—
80#392—31no—
81#392—61no—
82#388—61no—
83#386—61no—
84#381—31no—
85C4×C22⋊C4—32no—
86#342—61no—
87#327—64no—
88C24×C3C22⋊C411yescharacteristic
89C24×C3C2×D411yescharacteristic
90C23×D3—21no—
91C2×C3×D4—22no—
92C2×C3×D4—22no—
93C2×C3×D4—41no—
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×C4C22×C411yescharacteristic
102C22×C3×C4C2×D412yes—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4112yes—
105C2×#67—42no—
106C2×#67—42no—
107C2×#67—81no—
108C2×#67—81no—
109C2×#67—81no—
110C2×#67—81no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—42no—
115C2×D12—22no—
116C2×D12—81no—
117C3×M16—82no—
118C2×C3×C8—24no—
119C2×C3×C8—82no—
120C3×C4⋊C4—22no—
121C3×C22⋊C4—22no—
122C3×C22⋊C4—22no—
123C3×C42—21no—
124C3×C42—21no—
125#217—24no—
126#217—42no—
127#217—81no—
128#217—81no—
129(C2×D3)⋊C4—42no—
130(C2×D3)⋊C4—41no—
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—81no—
133(C2×D3)⋊C4—81no—
134C4⋊Dic3—22no—
135Dic3⋊C4—81no—
136C4×Dic3—82no—
137C22×D4—31no—
138C22×D4—61no—
139C22×D4—61no—
140C22×D4D1211yescharacteristic
141C23×C4#6711yescharacteristic
142C23×C4C22×D311yescharacteristic
143C2×M16—22no—
144C22×C8—22no—
145#121—62no—
146#121—62no—
147C4×D4—22no—
148C4×D4—22no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—61no—
151C2×C4⋊C4C4×D311yescharacteristic
152C2×C22⋊C4—62no—
153C2×C22⋊C4—61no—
154C2×C22⋊C4—61no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4C4×D311yescharacteristic
160C2×C22⋊C4D1211yescharacteristic
161C2×C42—62no—
162C2×C42#6711yescharacteristic
163C4⋊C8—22no—
164#99—64no—
165C22⋊C8—22no—
166C22⋊C8—22no—
167C22⋊C8#6714yescomplemented
168#97—22no—
169#95—31no—
170#95—31no—
171#95—61no—
172#95—61no—
173#95—61no—
174#95—62no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—41no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3—41no—
183C23×C3#10412yes—
184C23×C3C2×C22⋊C411yescharacteristic
185C23×C3#12112yes—
186C22×D3—41no—
187C22×D3—41no—
188C22×D3—41no—
189C22×D3—41no—
190C22×D3—41no—
191C22×D3—41no—
192C22×D3—41no—
193C3×D4—44no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—81no—
207C2×C3×C4C4×D412yes—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12311yescharacteristic
210#67—84no—
211#67—84no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—44no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—82no—
223C2×Dic3—82no—
224D12—82no—
225C3×C8—44no—
226C24—61no—
227C24(C2×D3)⋊C411yescharacteristic
228C24C2×D1211yescharacteristic
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—41no—
232C2×D4—62no—
233C2×D4—122no—
234C2×D4—122no—
235C2×D4—241no—
236C2×D4—241no—
237C2×D4—241no—
238C2×D4—241no—
239C2×D4—241no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—122no—
248C22×C4(C2×D3)⋊C411yescharacteristic
249C22×C4(C2×D3)⋊C411yescharacteristic
250C22×C4(C2×D3)⋊C411yescharacteristic
251C22×C4C2×C4×D311yescharacteristic
252C22×C4#23512yes—
253C22×C4C2×#6712yes—
254C22×C4C2×#6711yescharacteristic
255M16—82no—
256C2×C8—24no—
257C2×C8—82no—
258C4⋊C4—22no—
259C4⋊C4—62no—
260C4⋊C4—241no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—64no—
264C22⋊C4—122no—
265C22⋊C4—122no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C22⋊C4—241no—
269C22⋊C4—241no—
270C22⋊C4—241no—
271C22⋊C4—241no—
272C42—21no—
273C42—21no—
274C42—242no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—22no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—22no—
281C22×C3—42no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3#38611yescharacteristic
288C22×C3C2×#10411yescharacteristic
289C22×C3#42111yescharacteristic
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C2×D3—81no—
294C2×D3—81no—
295C2×D3—81no—
296C2×D3—81no—
297C3×C4—21no—
298C3×C4—22no—
299C3×C4—21no—
300C3×C4—41no—
301C3×C4—41no—
302C3×C4—41no—
303Dic3—81no—
304Dic3—81no—
305Dic3—84no—
306Dic3—81no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—121no—
316C23—121no—
317C23—121no—
318C23—121no—
319C23—121no—
320C23—121no—
321C23—121no—
322C23#82512yes—
323C23C2×(C2×D3)⋊C411yescharacteristic
324C23#93012yes—
325D4—44no—
326D4—242no—
327D4—244no—
328D4—244no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—124no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—242no—
353C2×C4—242no—
354C2×C4C4×#6712yes—
355C2×C4#92911yescharacteristic
356C2×C4#95311yescharacteristic
357C8—44no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—42no—
361C2×C3—42no—
362C2×C3#187111yescharacteristic
363C2×C3#191011yescharacteristic
364C2×C3#191311yescharacteristic
365D3—161no—
366C22—21no—
367C22—21no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—241no—
379C22—241no—
380C22—241no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22—241no—
385C22C2×#82511yescharacteristic
386C22#513911yescharacteristic
387C22#521311yescharacteristic
388C4—21no—
389C4—21no—
390C4—22no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—241no—
395C4—241no—
396C4—244no—
397C4—241no—
398C3#918511yescharacteristic, complemented, Sylow
399C2—21no—
400C2—21no—
401C2—42no—
402C2—42no—
403C2—481no—
404C2#7165611yescharacteristic
405C2#7198411yescharacteristic
406C2#7202511yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2973C2
socleC22×C3#386
prime radical#82422C2
cosocleC23×C3×C4C23
hypercenter#2973D3
hypocenterC3#9185
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
1. C3×#2973C2semidirect
3. C4×#834C2semidirect
4. #69977C2semidirect
5. #67682C2semidirect
18. #4991C4semidirect
19. #4986C4semidirect
20. #4960C4semidirect
22. #4949C4semidirect
26. #2973D3semidirect
60. C3×C22⋊C8D4semidirect
167. C22⋊C8#67semidirect
398. C3#9185semidirect

Maximal subgroups

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

subgroupindex
C3×#29732
#824222
C4×#8342
#699772
#676822
#91853