Finite Group Databasebeta

← #10690536

Subgroups — #10690536

Order 768.

606 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#924C211yesmaximal, characteristic
2C2×#4925C211yesmaximal, characteristic
3C3×#1797C211yesmaximal, characteristic
4C2×#4484C212yesmaximal, complemented
5#67704C212yesmaximal, complemented
6#9075—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#924C411yescharacteristic
9C2×C4×Dic6C2211yescharacteristic
10C2×C3×C4⋊C8C2212yes—
11C2×C3×C22⋊C8—21no—
12#5212C411yescharacteristic
13#5210—21no—
14C4×#217—21no—
15C2×#830—21no—
16C4×C4⋊Dic3C2211yescharacteristic
17#4933—21no—
18C4×Dic3⋊C4—21no—
19#4929C411yescharacteristic
20#4928C411yescharacteristic
21#4925—24no—
22C2×C4⋊(C3⋊C8)—21no—
23C2×C4×C3⋊C8C2212yes—
24C3×#336—21no—
25#4484—24no—
26C2×#431—31no—
27C4×C22⋊Q8—31no—
28#1797D311yescharacteristic
29C2×Q8⋊C8—32no—
30#1279—32no—
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C22×Dic6C2×C411yescharacteristic
35C22×C3×C8—22no—
36C2×C3×C42—22no—
37C2×C3×C42—22no—
38C2×C3×C42C2×C411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×C3×C42C2311yescharacteristic
42C2×#217—21no—
43C2×#217—21no—
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3C2×C411yescharacteristic
47#924—22no—
48#924—22no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3—21no—
54C22×C3⋊C8—22no—
55C4×Dic6—22no—
56C4×Dic6—22no—
57C3×C4⋊C8—22no—
58C3×C4⋊C8—22no—
59C3×C22⋊C8—42no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65#830—24no—
66C4⋊(C3⋊C8)—24no—
67C4×C3⋊C8—22no—
68C4×C3⋊C8D414yes—
69C2×C22⋊Q8—31no—
70C2×C4×Q8—31no—
71C22×C42C2×D311yescharacteristic
72#431—64no—
73C2×C4⋊C8—61no—
74C2×C4⋊C8C2×D312yes—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—61no—
77C2×C4×C8—32no—
78#391—31no—
79#389—31no—
80#387—61no—
81#385—61no—
82#380—31no—
83C4×C4⋊C4—31no—
84C4×C4⋊C4—61no—
85C4×C22⋊C4—61no—
86#336—21no—
87Q8⋊C8—64no—
88C24×C3C22⋊C411yescharacteristic
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—22no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4SD1612yes—
103C22×C3×C4Q1612yes—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C2×Dic6—22no—
112C2×Dic6—22no—
113C2×Dic6—21no—
114C2×Dic6M1612yes—
115C2×C3×C8—42no—
116C2×C3×C8—42no—
117C2×C3×C8—42no—
118C3×C42—22no—
119C3×C42—21no—
120C3×C42—22no—
121C3×C42—21no—
122C3×C42C2×D412yes—
123C3×C42#4112yes—
124#217—24no—
125#217—42no—
126C4⋊Dic3—21no—
127C4⋊Dic3—22no—
128C4⋊Dic3—22no—
129C4⋊Dic3M1612yes—
130Dic3⋊C4—24no—
131Dic3⋊C4—42no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134C4×Dic3—42no—
135C4×Dic3—42no—
136C2×C3⋊C8—24no—
137C2×C3⋊C8—42no—
138C2×C3⋊C8—42no—
139C22×Q8—31no—
140C23×C4C4×D311yescharacteristic
141C23×C4D1211yescharacteristic
142C23×C4#6711yescharacteristic
143C22×C8—22no—
144C22×C8—62no—
145C22⋊Q8—62no—
146C22⋊Q8—62no—
147C4×Q8—62no—
148C4×Q8—62no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—61no—
153C2×C4⋊C4—61no—
154C2×C4⋊C4—61no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4—61no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—61no—
160C2×C42—61no—
161C2×C42C4×D311yescharacteristic
162C2×C42D1211yescharacteristic
163C2×C42#6711yescharacteristic
164C2×C42C22×D311yescharacteristic
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—64no—
168C22⋊C8—42no—
169C22⋊C8—64no—
170C4×C8—34no—
171C4×C8—62no—
172#95—61no—
173#95—61no—
174#95—61no—
175#95—61no—
176#95—61no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3#10312yes—
184C23×C3C2×C22⋊C411yescharacteristic
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—24no—
190C2×C3×C4—24no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4#10312yes—
202C2×C3×C4C4×D412yes—
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12112yes—
205C2×C3×C4#12311yescharacteristic
206C2×C3×C4C2×M1611yescharacteristic
207C2×C3×C4C2×M1611yescharacteristic
208C2×C3×C4#13112yes—
209C2×C3×C4C2×SD1611yescharacteristic
210C2×C3×C4C2×Q1611yescharacteristic
211C2×C3×C4#13512yes—
212C2×Dic3—22no—
213C2×Dic3—22no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224Dic6—24no—
225Dic6—42no—
226C3×C8—82no—
227C3⋊C8—44no—
228C24(C2×D3)⋊C411yescharacteristic
229C2×Q8—32no—
230C2×Q8—61no—
231C2×Q8—62no—
232C2×Q8—62no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4(C2×D3)⋊C411yescharacteristic
248C22×C4(C2×D3)⋊C411yescharacteristic
249C22×C4(C2×D3)⋊C411yescharacteristic
250C22×C4#21412yes—
251C22×C4#21612yes—
252C22×C4C2×C4×D311yescharacteristic
253C22×C4C2×D1211yescharacteristic
254C22×C4C2×#6711yescharacteristic
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—64no—
259C2×C8—122no—
260C2×C8—122no—
261C4⋊C4—32no—
262C4⋊C4—64no—
263C4⋊C4—61no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C4⋊C4—122no—
267C4⋊C4—121no—
268C4⋊C4—121no—
269C22⋊C4—64no—
270C22⋊C4—122no—
271C42—22no—
272C42—21no—
273C42—21no—
274C42—22no—
275C42—122no—
276C42—122no—
277C42D3×D412yes—
278C42#23712yes—
279C22×C3—22no—
280C22×C3—22no—
281C22×C3—22no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3#38611yescharacteristic
287C22×C3#40711yescharacteristic
288C22×C3#40811yescharacteristic
289C22×C3C2×#10311yescharacteristic
290C22×C3#41611yescharacteristic
291C22×C3#42111yescharacteristic
292C22×C3#42111yescharacteristic
293C3×C4—22no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—41no—
297C3×C4—41no—
298C3×C4#43512yes—
299C3×C4#43612yes—
300C3×C4#46512yes—
301C3×C4#46712yes—
302Dic3—42no—
303Dic3—42no—
304Dic3—81no—
305Dic3—81no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23#81012yes—
313C23C2×(C2×D3)⋊C411yescharacteristic
314Q8—64no—
315Q8—122no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—62no—
333C2×C4—62no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4#81012yes—
345C2×C4#67⋊C412yes—
346C2×C4#88211yescharacteristic
347C2×C4#88611yescharacteristic
348C2×C4#90112yes—
349C2×C4D3×M1611yescharacteristic
350C2×C4D3×M1611yescharacteristic
351C2×C4C2×#21411yescharacteristic
352C2×C4#93912yes—
353C2×C4C2×#21611yescharacteristic
354C2×C4#95012yes—
355C8—82no—
356C8—124no—
357C2×C3—42no—
358C2×C3#148111yescharacteristic
359C2×C3#148211yescharacteristic
360C2×C3#148711yescharacteristic
361C2×C3#148811yescharacteristic
362C2×C3#187011yescharacteristic
363C2×C3#189211yescharacteristic
364C2×C3#189711yescharacteristic
365C22—22no—
366C22—22no—
367C22—22no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22#495811yescharacteristic
373C22C2×#81011yescharacteristic
374C22#500211yescharacteristic
375C22#511011yescharacteristic
376C22#512611yescharacteristic
377C22#513911yescharacteristic
378C22#513911yescharacteristic
379C4—21no—
380C4—21no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—122no—
385C4—122no—
386C4—241no—
387C4—241no—
388C4#472812yes—
389C4#472912yes—
390C4#504112yes—
391C4#505112yes—
392C3#907511yescharacteristic, complemented, Sylow
393C2—42no—
394C2#6951411yescharacteristic
395C2#6995311yescharacteristic
396C2#6996611yescharacteristic
397C2#7042911yescharacteristic
398C2#7050011yescharacteristic
399C2#7075911yescharacteristic
400C2#7079811yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1797C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1797D3
hypocenterC3#9075
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
4. C2×#4484C2semidirect
5. #67704C2semidirect
392. C3#9075semidirect

Maximal subgroups

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

subgroupindex
C4×#9242
C2×#49252
C3×#17972
C2×#44842
#677042
#90753