Finite Group Databasebeta

← #10691888

Subgroups — #10691888

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3208C211yesmaximal, characteristic
2#83172C211yesmaximal, characteristic
3#71840C211yesmaximal, characteristic, complemented
4#71838C211yesmaximal, characteristic, complemented
5#71759C211yesmaximal, characteristic
6#67738C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#9295—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#127C2211yescharacteristic
11C2×D4×Dic3—21no—
12C3×#492C412yescomplemented
13C3×#486C412yescomplemented
14C3×#460—22no—
15C3×#450—22no—
16C3×#431C2211yescharacteristic
17#5221—21no—
18#5220—22no—
19#5220—22no—
20#4936C2211yescharacteristic
21#4932C2211yescharacteristic
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4544—22no—
25#4538—22no—
26#4532—21no—
27#4532—21no—
28#3208D311yescharacteristic
29#2328—31no—
30#1995—31no—
31#1545—31no—
32#1518—31no—
33#1286—31no—
34#1274—31no—
35C22×C3×D4—21no—
36C22×C3×D4—21no—
37C22×C3×D4D412yes—
38C23×Dic3—22no—
39C2×C3×SD16—24no—
40C3×C4⋊Q8C2×C411yescharacteristic, complemented
41C3×C4⋊Q8C2×C411yescharacteristic, complemented
42C3×#127—21no—
43C3×#127—21no—
44C3×#127C2×C412yes—
45C3×#127D412yes—
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×#117—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#217—22no—
54C2×#217—22no—
55C2×#217—22no—
56D4×Dic3—42no—
57D4×Dic3—42no—
58C2×C4⋊Dic3—21no—
59C2×C4×Dic3D411yescharacteristic
60C2×C4×Dic3D411yescharacteristic
61C2×#208—21no—
62C3×C4⋊C8—21no—
63C3×#102—24no—
64C3×C22⋊C8—21no—
65C3×C4×C8C2×C412yescomplemented
66C3⋊#95—21no—
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95D412yes—
70#830—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75#533C2×D311yescharacteristic
76C2×#127C2×D311yescharacteristic
77C2×C4×D4—61no—
78#492Dic312yescomplemented
79#486Dic312yescomplemented
80#460—22no—
81#450—22no—
82#431—31no—
83#431C2×D311yescharacteristic
84#423—31no—
85#390—61no—
86#386—62no—
87#386—62no—
88#382—31no—
89#376—31no—
90#339—61no—
91#339—61no—
92#331—62no—
93#328—62no—
94C24×C3—22no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×D4—22no—
98C2×C3×D4—22no—
99C2×C3×D4—22no—
100C2×C3×D4—22no—
101C2×C3×D4—22no—
102C2×C3×D4—41no—
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C2×C3×D4—41no—
106C2×C3×D4—82no—
107C2×C3×D4—82no—
108C2×C3×D4C22⋊C414yes—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—22no—
115C22×Dic3—44no—
116C22×Dic3—82no—
117C22×Dic3—82no—
118C3×SD16—84no—
119C2×C3×C8—22no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C42—41no—
128C3×C42C22×C411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131C3×C42#4111yescharacteristic
132#217—44no—
133#217—82no—
134#217—82no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137C4×Dic3—21no—
138C4×Dic3—21no—
139C4×Dic3—21no—
140C4×Dic3SD1612yescomplemented
141#208—41no—
142#208—41no—
143C2×C3⋊C8—41no—
144C2×C3⋊C8—41no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4#6712yes—
148C23×C4—62no—
149C2×SD16—24no—
150C2×M16—61no—
151C4⋊Q8C2×Dic311yescharacteristic, complemented
152C4⋊Q8C2×Dic311yescharacteristic, complemented
153#127—21no—
154#127—21no—
155#127C2×Dic312yes—
156#127#6712yes—
157#125—21no—
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160C4×Q8—21no—
161C4×Q8—21no—
162C4×D4—122no—
163C4×D4—122no—
164#117—21no—
165C2×C4⋊C4—61no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—62no—
168C2×C22⋊C4—62no—
169C2×C42—31no—
170C2×C42—31no—
171C2×C42C22×D311yescharacteristic
172C4⋊C8—21no—
173C4⋊C8—61no—
174C4⋊C8—61no—
175C4⋊C8—61no—
176#102—24no—
177C22⋊C8—21no—
178C22⋊C8—61no—
179C4×C8—61no—
180C4×C8C2×Dic312yescomplemented
181#95—32no—
182#95—61no—
183#95—61no—
184#95—61no—
185C23×C3—24no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—42no—
189C23×C3—42no—
190C23×C3—42no—
191C23×C3—42no—
192C23×C3#12011yescharacteristic
193C3×Q8—42no—
194C3×Q8—81no—
195C3×D4—44no—
196C3×D4—42no—
197C3×D4—42no—
198C3×D4—42no—
199C3×D4—42no—
200C3×D4—82no—
201C3×D4—82no—
202C3×D4—82no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4C2×C22⋊C411yescharacteristic
208C2×C3×C4C4×D411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12311yescharacteristic
213C2×C3×C4C2×SD1611yescharacteristic
214C2×C3×C4#13511yescharacteristic
215C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
216C2×C3×C4#13711yescharacteristic
217C2×Dic3—22no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—41no—
226C2×Dic3—88no—
227C3×C8—42no—
228C3⋊C8—81no—
229C3⋊C8—81no—
230C24—22no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—82no—
243C2×D4—82no—
244C2×D4#21714yes—
245C22×C4—61no—
246C22×C4—62no—
247C22×C4—61no—
248C22×C4—124no—
249C22×C4—242no—
250C22×C4—242no—
251C22×C4D3×D411yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4C2×#6711yescharacteristic
254SD16—84no—
255M16—121no—
256M16—121no—
257C2×C8—22no—
258C2×C8—121no—
259C2×C8—121no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—121no—
267C4⋊C4—121no—
268C22⋊C4—41no—
269C22⋊C4—124no—
270C22⋊C4—242no—
271C22⋊C4—242no—
272C42—32no—
273C42—41no—
274C42—61no—
275C42—61no—
276C42—61no—
277C42#23711yescharacteristic
278C42#23711yescharacteristic
279C42C22×Dic311yescharacteristic
280C42C2×#6711yescharacteristic
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—44no—
284C22×C3—44no—
285C22×C3—44no—
286C22×C3—82no—
287C22×C3—82no—
288C22×C3—82no—
289C22×C3—82no—
290C22×C3#38611yescharacteristic
291C22×C3#45011yescharacteristic
292C22×C3#45211yescharacteristic
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—81no—
298C3×C4—81no—
299C3×C4C4×SD1611yescharacteristic
300C3×C4#44011yescharacteristic
301C3×C4#46111yescharacteristic
302C3×C4#46311yescharacteristic
303Dic3—42no—
304Dic3—81no—
305Dic3—82no—
306C23—24no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23#93711yescharacteristic
314Q8—42no—
315Q8—81no—
316D4—44no—
317D4—42no—
318D4—42no—
319D4—42no—
320D4—42no—
321D4—82no—
322D4—82no—
323D4—82no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—62no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—121no—
337C2×C4—248no—
338C2×C4D3×SD1611yescharacteristic
339C2×C4#91411yescharacteristic
340C2×C4#91511yescharacteristic
341C2×C4#91611yescharacteristic
342C2×C4D4×Dic311yescharacteristic
343C2×C4D4×Dic311yescharacteristic
344C2×C4#93511yescharacteristic
345C2×C4#93911yescharacteristic
346C2×C4#93911yescharacteristic
347C2×C4C2×#21711yescharacteristic
348C8—42no—
349C8—241no—
350C8—241no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—84no—
354C2×C3#187711yescharacteristic
355C2×C3#190411yescharacteristic
356C2×C3#191111yescharacteristic
357C22—21no—
358C22—21no—
359C22—44no—
360C22—44no—
361C22—44no—
362C22—82no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22#516911yescharacteristic
367C22#517311yescharacteristic
368C22#522011yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—81no—
374C4—81no—
375C4—122no—
376C4—242no—
377C4—241no—
378C4Dic3×SD1611yescharacteristic
379C4#516211yescharacteristic
380C4#516411yescharacteristic
381C4#516511yescharacteristic
382C3#929511yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—84no—
386C2#7185311yescharacteristic
387C2#7204711yescharacteristic
388C2#7208911yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3208C2
socleC22×C3#386
prime radicalC2×C3×#127C22
cosocleC2×C3×C42C23
hypercenter#3208D3
hypocenterC3#9295
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
3. #71840C2semidirect
4. #71838C2semidirect
6. #67738C2semidirect
7. #67737C2semidirect
12. C3×#492C4semidirect
13. C3×#486C4semidirect
40. C3×C4⋊Q8C2×C4semidirect
41. C3×C4⋊Q8C2×C4semidirect
65. C3×C4×C8C2×C4semidirect
78. #492Dic3semidirect
79. #486Dic3semidirect
140. C4×Dic3SD16semidirect
151. C4⋊Q8C2×Dic3semidirect
152. C4⋊Q8C2×Dic3semidirect
180. C4×C8C2×Dic3semidirect
382. C3#9295semidirect

Maximal subgroups

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

subgroupindex
C3×#32082
#831722
#718402
#718382
#717592
#677382
#677372
#92953