Finite Group Databasebeta

← #10690984

Subgroups — #10690984

Order 768.

452 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×#3116C211yesmaximal, characteristic
2#82518C211yesmaximal, characteristic
3#71836C211yesmaximal, characteristic, complemented
4#71821C211yesmaximal, characteristic, complemented
5#71643C211yesmaximal, characteristic
6#67753C211yesmaximal, characteristic, complemented
7#67748C211yesmaximal, characteristic, complemented
8#9159—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×C4×#41C2211yescharacteristic
11C3×#485—21no—
12C3×#477C411yescharacteristic, complemented
13C3×#476C411yescharacteristic, complemented
14C3×#462C411yescharacteristic, complemented
15C3×#461C411yescharacteristic, complemented
16C3×#452—21no—
17C3×#441—21no—
18C3×#440—21no—
19C3×#432C2211yescharacteristic
20C2×#837—21no—
21#5024—21no—
22#5011—22no—
23D4×C3⋊C8—21no—
24#4932C2211yescharacteristic
25#4926—21no—
26#4926C2211yescharacteristic
27C4×#208C2211yescharacteristic
28C4×#208C2211yescharacteristic
29C2×#803—21no—
30#4543—21no—
31#4543—21no—
32#4538—22no—
33#4532—21no—
34#3116D311yescharacteristic
35#2970—31no—
36#1759—31no—
37#1526—31no—
38#1506—31no—
39#1300—31no—
40#1291—31no—
41C2×C3×#41D411yescharacteristic
42C2×C3×Q16—21no—
43C2×C3×SD16—21no—
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8C2×C411yescharacteristic
52C3×C4×Q8D411yescharacteristic
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4C2×C411yescharacteristic
56C3×C4×D4D411yescharacteristic
57C3×#117—21no—
58C3×#117—21no—
59C3×#117D411yescharacteristic
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×C4×Dic3—21no—
63C2×#208—21no—
64C2×#208—21no—
65C22×C3⋊C8—22no—
66C3×#107—21no—
67C3×#106—21no—
68C3×C4⋊C8C2×C411yescharacteristic
69C3×C4⋊C8C2×C411yescharacteristic
70C3×#103—21no—
71C3×#103—21no—
72C3×#103—21no—
73C3×#102—21no—
74C3×C22⋊C8—21no—
75C3×#97—21no—
76#837—24no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79#830—22no—
80#830D412yes—
81C4⋊(C3⋊C8)—22no—
82C4⋊(C3⋊C8)—21no—
83#803—22no—
84#803—21no—
85#803—21no—
86#803D411yescharacteristic
87#803D411yescharacteristic
88C4×C3⋊C8—21no—
89C4×C3⋊C8—21no—
90#533C2×D311yescharacteristic
91C4×#41C2×D311yescharacteristic
92#485—21no—
93#477Dic311yescharacteristic, complemented
94#476Dic311yescharacteristic, complemented
95#462Dic311yescharacteristic, complemented
96#461Dic311yescharacteristic, complemented
97#452—21no—
98#446—61no—
99#441—21no—
100#440—21no—
101#435—62no—
102C8×D4—61no—
103#432—31no—
104#432—61no—
105#432C2×D311yescharacteristic
106C2×#104—61no—
107C4×M16—31no—
108C4×M16—31no—
109C2×#97—61no—
110#376—31no—
111#339—61no—
112#330—61no—
113#330—61no—
114#328—62no—
115C3×#41—22no—
116C3×#41—41no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8C22⋊C411yescharacteristic
120C2×C3×D4—21no—
121C2×C3×D4C22⋊C411yescharacteristic
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C22×Dic3—41no—
129C3×Q16—42no—
130C3×SD16—42no—
131C2×C3×C8—21no—
132C2×C3×C8—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4C22⋊C411yescharacteristic
142C3×C4⋊C4C22⋊C411yescharacteristic
143C3×C22⋊C4—21no—
144C3×C22⋊C4—21no—
145C3×C22⋊C4—21no—
146C3×C22⋊C4—41no—
147C3×C42—21no—
148C3×C42—21no—
149C3×C42—21no—
150C3×C42—41no—
151C3×C42C22×C411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42#4111yescharacteristic
155C4×Dic3—22no—
156C4×Dic3—41no—
157#208—22no—
158#208—42no—
159#208—41no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—44no—
164C2×C3⋊C8—41no—
165C2×#41#6711yescharacteristic
166C2×Q16—21no—
167C2×SD16—21no—
168C2×M16—61no—
169C2×M16—61no—
170C22×C8—62no—
171C4⋊Q8C2×Dic311yescharacteristic
172C4⋊Q8C2×Dic311yescharacteristic
173#125—21no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8C2×Dic311yescharacteristic
179C4×Q8#6711yescharacteristic
180C4×D4—21no—
181C4×D4—21no—
182C4×D4C2×Dic311yescharacteristic
183C4×D4#6711yescharacteristic
184#117—21no—
185#117—21no—
186#117#6711yescharacteristic
187C2×C42—21no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190#107—21no—
191#106—21no—
192C4⋊C8—62no—
193C4⋊C8—61no—
194C4⋊C8C2×Dic311yescharacteristic
195C4⋊C8C2×Dic311yescharacteristic
196#104—64no—
197#103—21no—
198#103—21no—
199#103—21no—
200#102—21no—
201C22⋊C8—21no—
202C22⋊C8—32no—
203C22⋊C8—62no—
204#97—21no—
205#97—31no—
206#97—31no—
207#97—62no—
208#97—61no—
209#97—61no—
210C4×C8—61no—
211C4×C8—61no—
212#95—61no—
213#95—61no—
214C23×C3—21no—
215C23×C3#12111yescharacteristic
216C3×Q8—21no—
217C3×Q8—21no—
218C3×Q8—41no—
219C3×Q8—41no—
220C3×Q8—81no—
221C3×D4—21no—
222C3×D4—42no—
223C3×D4#10412yes—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—81no—
236C2×C3×C4C2×C22⋊C411yescharacteristic
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4C4×D411yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12311yescharacteristic
242C2×Dic3—42no—
243C2×Dic3—81no—
244C2×Dic3—81no—
245C3×C8—41no—
246C3×C8—41no—
247C3⋊C8—22no—
248C3⋊C8—42no—
249C3⋊C8—41no—
250C3⋊C8—42no—
251#41—22no—
252#41—41no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8#21711yescharacteristic
256C2×D4—21no—
257C2×D4#21711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—121no—
262C22×C4D3×D411yescharacteristic
263C22×C4#23711yescharacteristic
264C22×C4C2×#6711yescharacteristic
265Q16—42no—
266SD16—42no—
267M16—62no—
268M16—122no—
269M16—121no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—61no—
273C2×C8—61no—
274C2×C8—62no—
275C2×C8—124no—
276C2×C8—121no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4#21711yescharacteristic
286C4⋊C4#21711yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—41no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—41no—
295C42—62no—
296C42—121no—
297C42D3×D411yescharacteristic
298C42#23711yescharacteristic
299C42C22×Dic311yescharacteristic
300C42C2×#6711yescharacteristic
301C22×C3—21no—
302C22×C3—22no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3#38611yescharacteristic
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—41no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4—81no—
317C3×C4—81no—
318C3×C4C2×#10411yescharacteristic
319C3×C4#42111yescharacteristic
320Dic3—82no—
321C23—21no—
322C23#93911yescharacteristic
323Q8—21no—
324Q8—21no—
325Q8—41no—
326Q8—41no—
327Q8—81no—
328D4—21no—
329D4—42no—
330D4#83712yes—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—122no—
344C2×C4—241no—
345C2×C4—241no—
346C2×C4D4×Dic311yescharacteristic
347C2×C4D4×Dic311yescharacteristic
348C2×C4#93511yescharacteristic
349C2×C4#93711yescharacteristic
350C2×C4#93911yescharacteristic
351C2×C4C2×#21711yescharacteristic
352C8—41no—
353C8—41no—
354C8—62no—
355C8—122no—
356C8—121no—
357C8—122no—
358C2×C3—41no—
359C2×C3—42no—
360C2×C3#187111yescharacteristic
361C2×C3#190211yescharacteristic
362C2×C3#190611yescharacteristic
363C22—21no—
364C22—22no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22#522011yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—81no—
379C4—81no—
380C4—242no—
381C4C2×#83711yescharacteristic
382C4#523611yescharacteristic
383C3#915911yescharacteristic, complemented, Sylow
384C2—41no—
385C2—42no—
386C2#7171311yescharacteristic
387C2#7210211yescharacteristic
388C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3116C2
socleC22×C3#386
prime radicalC2×C3×#41D4
cosocleC2×C3×C42C23
hypercenter#3116D3
hypocenterC3#9159
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3116

Complements

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

NH ≅ G/Ntype
3. #71836C2semidirect
4. #71821C2semidirect
6. #67753C2semidirect
7. #67748C2semidirect
12. C3×#477C4semidirect
13. C3×#476C4semidirect
14. C3×#462C4semidirect
15. C3×#461C4semidirect
93. #477Dic3semidirect
94. #476Dic3semidirect
95. #462Dic3semidirect
96. #461Dic3semidirect
383. C3#9159semidirect

Maximal subgroups

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

subgroupindex
C3×#31162
#825182
#718362
#718212
#716432
#677532
#677482
#91593