Finite Group Databasebeta

← #10691019

Subgroups — #10691019

Order 768.

458 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3112C211yesmaximal, characteristic
2#82437C211yesmaximal, characteristic, complemented
3#71811C211yesmaximal, characteristic
4#71798C211yesmaximal, characteristic
5#71631C211yesmaximal, characteristic, complemented
6#67751C211yesmaximal, characteristic, complemented
7#67750C211yesmaximal, characteristic, complemented
8#9164—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C3×#480—21no—
12C3×#477C411yescharacteristic, complemented
13C3×#477C411yescharacteristic, complemented
14C3×#460C411yescharacteristic, complemented
15C3×#460C411yescharacteristic, complemented
16C3×#448—21no—
17C3×#440—21no—
18C3×#440—21no—
19C3×#432C2211yescharacteristic
20C2×#835—21no—
21#5024—21no—
22#5024—22no—
23Q8×C3⋊C8—21no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4926—21no—
26#4926C2211yescharacteristic
27#4925—21no—
28#4925C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4540—22no—
31#4537—21no—
32#4537—21no—
33#4529—21no—
34#3112D311yescharacteristic
35#2984—31no—
36#1763—31no—
37#1530—31no—
38#1502—31no—
39#1302—31no—
40#1285—31no—
41C2×C3×#41—21no—
42C22×C3×Q8D411yescharacteristic
43C2×C3×SD16—21no—
44C2×C3×SD16—21no—
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×#127C2×C411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8C2×C411yescharacteristic
53C3×C4×Q8C2×C411yescharacteristic
54C3×C4×Q8D411yescharacteristic
55C3×C4×Q8D411yescharacteristic
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×C4⋊Dic3—21no—
63C2×C4×Dic3—21no—
64C22×C3⋊C8—21no—
65C3×#107—21no—
66C3×#107—21no—
67C3×C4⋊C8C2×C411yescharacteristic
68C3×C4⋊C8C2×C411yescharacteristic
69C3×#103—21no—
70C3×#103—21no—
71C3×#102—21no—
72C3×#102—21no—
73C3×C22⋊C8—21no—
74C3×#97—21no—
75#835—24no—
76C3⋊#95—21no—
77#830—21no—
78#830D411yescharacteristic
79#830D411yescharacteristic
80C4⋊(C3⋊C8)—22no—
81C4⋊(C3⋊C8)—22no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84#803—22no—
85#803—21no—
86#803—21no—
87#803D412yes—
88C4×C3⋊C8—22no—
89C4×C3⋊C8—21no—
90#532C2×D311yescharacteristic
91C2×C4×Q8C2×D311yescharacteristic
92#480—21no—
93#477Dic311yescharacteristic, complemented
94#477Dic311yescharacteristic, complemented
95#460Dic311yescharacteristic, complemented
96#460Dic311yescharacteristic, complemented
97#448—21no—
98#446—62no—
99#446—61no—
100C8×Q8—61no—
101#440—21no—
102#440—21no—
103#432—31no—
104#432—61no—
105#432C2×D311yescharacteristic
106#431—31no—
107#431—61no—
108C2×#103—61no—
109C2×#97—31no—
110C4×C4⋊C4—31no—
111#340—61no—
112#329—62no—
113#327—61no—
114#327—61no—
115C3×#41—81no—
116C3×#41—81no—
117C2×C3×Q8—21no—
118C2×C3×Q8—22no—
119C2×C3×Q8—21no—
120C2×C3×Q8—41no—
121C2×C3×Q8C22⋊C411yescharacteristic
122C2×C3×Q8C22⋊C411yescharacteristic
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—41no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—41no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×Dic3—41no—
134C3×SD16—42no—
135C3×SD16—42no—
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4C22⋊C411yescharacteristic
146C3×C4⋊C4C22⋊C411yescharacteristic
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C42—21no—
150C3×C42—21no—
151C3×C42—21no—
152C3×C42—21no—
153C3×C42C22×C411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4111yescharacteristic
157C4⋊Dic3—22no—
158C4⋊Dic3—41no—
159C4×Dic3—81no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×#41—21no—
167C22×Q8#6711yescharacteristic
168C2×SD16—21no—
169C2×SD16—21no—
170C22×C8—61no—
171C4⋊Q8C2×Dic311yescharacteristic
172#127C2×Dic311yescharacteristic
173#124—21no—
174#121—21no—
175#121—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8C2×Dic311yescharacteristic
179C4×Q8C2×Dic311yescharacteristic
180C4×Q8#6711yescharacteristic
181C4×Q8#6711yescharacteristic
182C4×D4—21no—
183C4×D4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4#6711yescharacteristic
187C2×C42—21no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190#107—21no—
191#107—21no—
192C4⋊C8—62no—
193C4⋊C8—62no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8C2×Dic311yescharacteristic
197C4⋊C8C2×Dic311yescharacteristic
198#103—21no—
199#103—21no—
200#103—64no—
201#102—21no—
202#102—21no—
203C22⋊C8—21no—
204C22⋊C8—31no—
205C22⋊C8—31no—
206C22⋊C8—61no—
207#97—21no—
208#97—32no—
209#97—62no—
210#97—61no—
211#97—61no—
212C4×C8—62no—
213C4×C8—61no—
214#95—61no—
215C23×C3—41no—
216C23×C3#12111yescharacteristic
217C3×Q8—21no—
218C3×Q8—21no—
219C3×Q8—21no—
220C3×Q8—42no—
221C3×Q8—81no—
222C3×Q8#10412yes—
223C3×D4—41no—
224C3×D4—41no—
225C3×D4—81no—
226C3×D4—81no—
227C3×D4—81no—
228C3×D4—81no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—82no—
243C2×C3×C4C2×C22⋊C411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12311yescharacteristic
249C2×Dic3—42no—
250C2×Dic3—81no—
251C2×Dic3—81no—
252C3×C8—41no—
253C3×C8—41no—
254C3⋊C8—41no—
255C3⋊C8—41no—
256C3⋊C8—42no—
257C3⋊C8—42no—
258#41—81no—
259#41—81no—
260C2×Q8—21no—
261C2×Q8—22no—
262C2×Q8—21no—
263C2×Q8—41no—
264C2×Q8#21711yescharacteristic
265C2×Q8#21711yescharacteristic
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—41no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—121no—
274C22×C4D3×D411yescharacteristic
275C22×C4#23711yescharacteristic
276C22×C4C2×#6711yescharacteristic
277SD16—42no—
278SD16—42no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—62no—
282C2×C8—61no—
283C2×C8—62no—
284C2×C8—61no—
285C2×C8—121no—
286C2×C8—121no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—62no—
295C4⋊C4—121no—
296C4⋊C4#21711yescharacteristic
297C4⋊C4#21711yescharacteristic
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—241no—
305C42D3×D411yescharacteristic
306C42#23711yescharacteristic
307C42C22×Dic311yescharacteristic
308C42C2×#6711yescharacteristic
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3#38611yescharacteristic
315C22×C3#47111yescharacteristic
316C22×C3#47211yescharacteristic
317C3×C4—21no—
318C3×C4—22no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—41no—
326C3×C4—81no—
327C3×C4C2×#10411yescharacteristic
328C3×C4#42111yescharacteristic
329Dic3—82no—
330C23—41no—
331C23#93911yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—21no—
335Q8—42no—
336Q8—81no—
337Q8#83712yes—
338D4—41no—
339D4—41no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—81no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—82no—
358C2×C4—122no—
359C2×C4—241no—
360C2×C4—241no—
361C2×C4D4×Dic311yescharacteristic
362C2×C4D4×Dic311yescharacteristic
363C2×C4#93511yescharacteristic
364C2×C4#93711yescharacteristic
365C2×C4#93911yescharacteristic
366C2×C4C2×#21711yescharacteristic
367C8—41no—
368C8—41no—
369C8—122no—
370C8—121no—
371C8—122no—
372C8—121no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—161no—
376C2×C3#187111yescharacteristic
377C2×C3#190511yescharacteristic
378C2×C3#190711yescharacteristic
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#519911yescharacteristic
385C22#520011yescharacteristic
386C22#522011yescharacteristic
387C4—21no—
388C4—22no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—81no—
397C4—242no—
398C4C2×#83711yescharacteristic
399C4#523611yescharacteristic
400C3#916411yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—161no—
404C2#7171311yescharacteristic
405C2#7209111yescharacteristic
406C2#7210111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3112C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3112D3
hypocenterC3#9164
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3112

Complements

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

NH ≅ G/Ntype
2. #82437C2semidirect
5. #71631C2semidirect
6. #67751C2semidirect
7. #67750C2semidirect
12. C3×#477C4semidirect
13. C3×#477C4semidirect
14. C3×#460C4semidirect
15. C3×#460C4semidirect
93. #477Dic3semidirect
94. #477Dic3semidirect
95. #460Dic3semidirect
96. #460Dic3semidirect
400. C3#9164semidirect

Maximal subgroups

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

subgroupindex
C3×#31122
#824372
#718112
#717982
#716312
#677512
#677502
#91643