Finite Group Databasebeta

← #10691622

Subgroups — #10691622

Order 768.

524 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#484C211yesmaximal, characteristic
2#82160C211yesmaximal, characteristic
3#71701C211yesmaximal, characteristic
4#71690C211yesmaximal, characteristic
5#71554C211yesmaximal, characteristic
6#67677C211yesmaximal, characteristic
7#67676C211yesmaximal, characteristic
8#9257—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C3×#484—21no—
12C3×#484—21no—
13C3×#484—22no—
14C2×C3×#107C412yescomplemented
15C2×C3×#103C412yescomplemented
16C2×C3×C22⋊C8C2211yescharacteristic
17#5212—21no—
18#5212C2211yescharacteristic
19#5212C2211yescharacteristic
20C4×#217—21no—
21C4×#217C2211yescharacteristic
22C2×#830C2211yescharacteristic
23#4984—21no—
24#4980—22no—
25Dic3×C4⋊C4—21no—
26#4538—22no—
27#4529—22no—
28#4526—21no—
29#4526—21no—
30C2×#484D311yescharacteristic
31#2345—31no—
32#1996—31no—
33#1824—31no—
34#1520—31no—
35#1289—31no—
36#1282—31no—
37C22×C3×Q8C2×C411yescharacteristic
38C23×C3×C4—21no—
39C23×C3×C4C2311yescharacteristic
40C22×C3×C8C2×C412yes—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C4⋊C4C2×C412yes—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C4⋊C4D412yes—
54C2×C3×C22⋊C4—21no—
55C2×#217—21no—
56C2×#217—21no—
57C2×#217D411yescharacteristic
58C2×#217D411yescharacteristic
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—22no—
62C2×C4×Dic3—21no—
63C2×C4×Dic3—22no—
64C22×C3⋊C8—21no—
65C3×#107—22no—
66C3×#107—22no—
67C3×#103—22no—
68C3×#103—22no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3⋊#95—22no—
72C3⋊#95—22no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95D412yes—
76#830—21no—
77#830—21no—
78C2×C22⋊Q8C2×D311yescharacteristic
79C22×C4⋊C4C2×D311yescharacteristic
80#484—22no—
81#484—21no—
82#484—21no—
83C2×#107Dic312yescomplemented
84C2×#103Dic312yescomplemented
85C2×C22⋊C8—31no—
86C2×C22⋊C8C2×D311yescharacteristic
87#389—61no—
88#387—62no—
89#380—31no—
90#380—31no—
91#380—61no—
92C4×C4⋊C4—61no—
93C4×C22⋊C4—31no—
94C4×C22⋊C4—61no—
95#342—61no—
96#342—61no—
97#340—62no—
98#328—62no—
99C24×C3C2×D411yescharacteristic
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—42no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—42no—
110C22×C3×C4—82no—
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4#4111yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—22no—
120C22×Dic3—41no—
121C2×C3×C8—22no—
122C2×C3×C8—42no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4C22⋊C412yes—
132C3×C4⋊C4C22⋊C412yes—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135#217—21no—
136#217—21no—
137#217—21no—
138#217—41no—
139#217—41no—
140#217—41no—
141#217—41no—
142#217Q1612yes—
143C4⋊Dic3—42no—
144C4⋊Dic3—42no—
145C4⋊Dic3—82no—
146C4×Dic3—42no—
147C4×Dic3—82no—
148C2×C3⋊C8—42no—
149C2×C3⋊C8—41no—
150C22×Q8C2×Dic311yescharacteristic
151C23×C4—21no—
152C23×C4C22×D311yescharacteristic
153C22×C8—61no—
154C22×C8C2×Dic312yes—
155C22⋊Q8—21no—
156C22⋊Q8—21no—
157C22⋊Q8—21no—
158C22⋊Q8—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—62no—
166C2×C4⋊C4C2×Dic311yescharacteristic
167C2×C4⋊C4C2×Dic312yes—
168C2×C4⋊C4#6712yes—
169C2×C4⋊C4#6711yescharacteristic
170C2×C4⋊C4#6711yescharacteristic
171C2×C22⋊C4—21no—
172C2×C22⋊C4—31no—
173C2×C22⋊C4—31no—
174C2×C22⋊C4—61no—
175C2×C22⋊C4—61no—
176C2×C42—61no—
177C2×C42—62no—
178#107—22no—
179#107—22no—
180#103—22no—
181#103—22no—
182C22⋊C8—21no—
183C22⋊C8—21no—
184C22⋊C8—61no—
185C22⋊C8—61no—
186#95—32no—
187#95—62no—
188#95—62no—
189#95—61no—
190#95—61no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3C4×D411yescharacteristic
194C23×C3#12011yescharacteristic
195C23×C3#12111yescharacteristic
196C23×C3C2×Q1611yescharacteristic
197C23×C3#13511yescharacteristic
198C23×C3#13711yescharacteristic
199C23×C3#13711yescharacteristic
200C3×Q8—42no—
201C3×Q8—81no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—82no—
215C2×C3×C4—82no—
216C2×C3×C4—81no—
217C2×C3×C4—81no—
218C2×C3×C4C2×C22⋊C411yescharacteristic
219C2×C3×C4C4×D411yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12311yescharacteristic
222C2×Dic3—22no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—42no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233C2×Dic3—81no—
234C3×C8—42no—
235C3⋊C8—81no—
236C24D3×D411yescharacteristic
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—42no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—42no—
247C22×C4—61no—
248C22×C4—62no—
249C22×C4—61no—
250C22×C4—82no—
251C22×C4—121no—
252C22×C4D3×D411yescharacteristic
253C22×C4#23711yescharacteristic
254C22×C4#23711yescharacteristic
255C22×C4C22×Dic311yescharacteristic
256C22×C4C2×#6711yescharacteristic
257C22×C4C2×#6711yescharacteristic
258C2×C8—22no—
259C2×C8—42no—
260C2×C8—122no—
261C2×C8—121no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—122no—
271C4⋊C4—122no—
272C4⋊C4—242no—
273C4⋊C4#21712yes—
274C4⋊C4#21712yes—
275C22⋊C4—32no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—61no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C22⋊C4—121no—
285C42—122no—
286C42—242no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3#38611yescharacteristic
298C22×C3C4×Q1611yescharacteristic
299C22×C3#44111yescharacteristic
300C22×C3#45111yescharacteristic
301C22×C3#45211yescharacteristic
302C22×C3#46211yescharacteristic
303C22×C3#46411yescharacteristic
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—42no—
308C3×C4—42no—
309C3×C4—81no—
310C3×C4—81no—
311C3×C4#41012yes—
312Dic3—42no—
313Dic3—81no—
314Dic3—82no—
315Dic3—161no—
316C23—41no—
317C23—41no—
318C23D3×Q1611yescharacteristic
319C23#91811yescharacteristic
320C23#91811yescharacteristic
321C23#91911yescharacteristic
322C23D4×Dic311yescharacteristic
323C23#93711yescharacteristic
324C23#93911yescharacteristic
325Q8—42no—
326Q8—81no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—62no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—122no—
352C2×C4—241no—
353C2×C4—241no—
354C2×C4—241no—
355C2×C4D4×Dic311yescharacteristic
356C2×C4#93511yescharacteristic
357C2×C4#93911yescharacteristic
358C2×C4C2×#21711yescharacteristic
359C8—42no—
360C8—241no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3#187811yescharacteristic
368C2×C3#189911yescharacteristic
369C2×C3#190411yescharacteristic
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22Dic3×Q1611yescharacteristic
381C22#518211yescharacteristic
382C22#518411yescharacteristic
383C22#518511yescharacteristic
384C22#518611yescharacteristic
385C22#518711yescharacteristic
386C22#522011yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—42no—
392C4—81no—
393C4—81no—
394C4—122no—
395C4—241no—
396C4—242no—
397C4—481no—
398C4#524012yes—
399C3#925711yescharacteristic, complemented, Sylow
400C2—21no—
401C2—21no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2#7170511yescharacteristic
407C2#7206511yescharacteristic
408C2#7208911yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#484C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#484D3
hypocenterC3#9257
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
14. C2×C3×#107C4semidirect
15. C2×C3×#103C4semidirect
83. C2×#107Dic3semidirect
84. C2×#103Dic3semidirect
399. C3#9257semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4842
#821602
#717012
#716902
#715542
#676772
#676762
#92573