Finite Group Databasebeta

← #10692760

Subgroups — #10692760

Order 768.

414 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3181C211yesmaximal, characteristic
2#83128C211yesmaximal, characteristic
3#71838C211yesmaximal, characteristic, complemented
4#71822C211yesmaximal, characteristic, complemented
5#71756C211yesmaximal, characteristic
6#67662C211yesmaximal, characteristic, complemented
7#67659C211yesmaximal, characteristic, complemented
8#9431—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#529C2211yescharacteristic
11C3×#501C411yescharacteristic, complemented
12C3×#497C411yescharacteristic, complemented
13C3×#491C411yescharacteristic, complemented
14C3×#489C411yescharacteristic, complemented
15C3×#467—21no—
16C3×#465—21no—
17C3×#441—21no—
18C3×#440—21no—
19C2×C3×#97C2211yescharacteristic
20#5024—21no—
21#5011—21no—
22#5011—21no—
23D4×C3⋊C8—21no—
24#5000—21no—
25#4997—21no—
26#4936C2211yescharacteristic
27#4926C2211yescharacteristic
28#4925C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4543—21no—
31#4543—21no—
32#4538—21no—
33#4538—21no—
34#4538—21no—
35#4537—21no—
36#3181D311yescharacteristic
37#2968—31no—
38#1531—31no—
39#1518—31no—
40#1512—31no—
41#1302—31no—
42#1295—31no—
43C2×C3×Q16—21no—
44C2×C3×SD16—21no—
45C22×C3×C8—21no—
46C3×C4⋊Q8C2×C411yescharacteristic
47C3×C4⋊Q8C2×C411yescharacteristic
48C3×#126—21no—
49C3×#125—21no—
50C3×#125C2×C411yescharacteristic
51C3×#124C2×C411yescharacteristic
52C3×#123—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8D411yescharacteristic
56C3×#121D411yescharacteristic
57C3×C4×Q8—21no—
58C3×C4×Q8—21no—
59C3×C4×Q8D411yescharacteristic
60C3×C4×D4—21no—
61C3×C4×D4D411yescharacteristic
62C3×#117—21no—
63C3×#117—21no—
64C2×C3×C42C2311yescharacteristic
65C2×C4⋊Dic3—21no—
66C2×#208—21no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C3×#107—21no—
70C3×#106—21no—
71C3×#103—21no—
72C3×#103—21no—
73C3×#103—21no—
74C3×#102—21no—
75C3×#97—21no—
76C3×#97C2×C411yescharacteristic
77C3×#97C2×C411yescharacteristic
78C3⋊#95—21no—
79C3⋊#95—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830D411yescharacteristic
84#830D411yescharacteristic
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89#803—21no—
90#803—21no—
91#803—21no—
92#803D411yescharacteristic
93#803D411yescharacteristic
94C4×C3⋊C8—21no—
95C4×C3⋊C8—21no—
96#533C2×D311yescharacteristic
97#529C2×D311yescharacteristic
98#501Dic311yescharacteristic, complemented
99#497Dic311yescharacteristic, complemented
100#491Dic311yescharacteristic, complemented
101#489Dic311yescharacteristic, complemented
102#467—21no—
103#465—21no—
104#446—61no—
105#441—21no—
106#440—21no—
107#435—61no—
108#435—61no—
109C8×D4—61no—
110#433—61no—
111#432—31no—
112#431—31no—
113#405—61no—
114C2×#97—31no—
115C2×#97C2×D311yescharacteristic
116#382—31no—
117#330—61no—
118#330—61no—
119#328—61no—
120#328—61no—
121#328—61no—
122#327—61no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8C22⋊C411yescharacteristic
126C2×C3×D4—21no—
127C2×C3×D4C22⋊C411yescharacteristic
128C22×C3×C4—21no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×Dic3—41no—
133C3×Q16—42no—
134C3×SD16—42no—
135C2×C3×C8—21no—
136C2×C3×C8—21no—
137C2×C3×C8—42no—
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⋊C4—21no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4C22⋊C411yescharacteristic
149C3×C4⋊C4C22⋊C411yescharacteristic
150C3×C22⋊C4—21no—
151C3×C22⋊C4—21no—
152C3×C22⋊C4—21no—
153C3×C22⋊C4—21no—
154C3×C22⋊C4—21no—
155C3×C22⋊C4—41no—
156C3×C42—21no—
157C3×C42—41no—
158C3×C42C22×C411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42#4111yescharacteristic
162C4⋊Dic3—81no—
163#208—42no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—42no—
169C2×C3⋊C8—42no—
170C2×C3⋊C8—41no—
171C2×Q16—21no—
172C2×SD16—21no—
173C2×M16—61no—
174C22×C8—21no—
175C22×C8—61no—
176C22×C8—61no—
177C4⋊Q8C2×Dic311yescharacteristic
178C4⋊Q8C2×Dic311yescharacteristic
179#126—21no—
180#125—21no—
181#125C2×Dic311yescharacteristic
182#124C2×Dic311yescharacteristic
183#123—21no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8#6711yescharacteristic
187#121#6711yescharacteristic
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8#6711yescharacteristic
191C4×D4—21no—
192C4×D4#6711yescharacteristic
193#117—21no—
194#117—21no—
195C2×C4⋊C4—61no—
196C2×C42C22×D311yescharacteristic
197#107—21no—
198#106—21no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203#103—21no—
204#103—21no—
205#103—21no—
206#102—21no—
207C22⋊C8—31no—
208C22⋊C8—31no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212#97—21no—
213#97—31no—
214#97—31no—
215#97—61no—
216#97—61no—
217#97—61no—
218#97C2×Dic311yescharacteristic
219#97C2×Dic311yescharacteristic
220C4×C8—61no—
221C4×C8—61no—
222#95—61no—
223#95—61no—
224C23×C3—21no—
225C23×C3#12111yescharacteristic
226C3×Q8—41no—
227C3×Q8—41no—
228C3×Q8—41no—
229C3×Q8—81no—
230C3×D4—41no—
231C3×D4—41no—
232C3×D4—41no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4C2×C22⋊C411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12311yescharacteristic
250C2×Dic3—81no—
251C2×Dic3—81no—
252C2×Dic3—81no—
253C3×C8—22no—
254C3×C8—41no—
255C3⋊C8—22no—
256C3⋊C8—41no—
257C3⋊C8—41no—
258C3⋊C8—41no—
259C3⋊C8—81no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8#21711yescharacteristic
263C2×D4—21no—
264C2×D4#21711yescharacteristic
265C22×C4—21no—
266C22×C4—121no—
267C22×C4D3×D411yescharacteristic
268C22×C4#23711yescharacteristic
269C22×C4C2×#6711yescharacteristic
270Q16—42no—
271SD16—42no—
272M16—122no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—42no—
276C2×C8—61no—
277C2×C8—61no—
278C2×C8—61no—
279C2×C8—61no—
280C2×C8—122no—
281C2×C8—122no—
282C2×C8—121no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—241no—
294C4⋊C4#21711yescharacteristic
295C4⋊C4#21711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—41no—
302C42—21no—
303C42—41no—
304C42D3×D411yescharacteristic
305C42#23711yescharacteristic
306C42C22×Dic311yescharacteristic
307C42C2×#6711yescharacteristic
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3#38611yescharacteristic
314C22×C3#47111yescharacteristic
315C22×C3#47311yescharacteristic
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—41no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—81no—
326C3×C4—81no—
327Dic3—161no—
328C23—21no—
329C23#93911yescharacteristic
330Q8—41no—
331Q8—41no—
332Q8—41no—
333Q8—81no—
334D4—41no—
335D4—41no—
336D4—41no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—241no—
349C2×C4—241no—
350C2×C4—241no—
351C2×C4D4×Dic311yescharacteristic
352C2×C4D4×Dic311yescharacteristic
353C2×C4#93511yescharacteristic
354C2×C4#93711yescharacteristic
355C2×C4#93911yescharacteristic
356C2×C4C2×#21711yescharacteristic
357C8—22no—
358C8—41no—
359C8—62no—
360C8—121no—
361C8—121no—
362C8—121no—
363C8—241no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—81no—
367C2×C3#188411yescharacteristic
368C2×C3#190611yescharacteristic
369C2×C3#190811yescharacteristic
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22#520311yescharacteristic
376C22#520411yescharacteristic
377C22#522011yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—81no—
388C4—81no—
389C4—481no—
390C3#943111yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—81no—
394C2#7181711yescharacteristic
395C2#7210311yescharacteristic
396C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3181C2
socleC22×C3#386
prime radicalC3×#121D4
cosocleC2×C3×C42C23
hypercenter#3181D3
hypocenterC3#9431
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3181

Complements

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

NH ≅ G/Ntype
3. #71838C2semidirect
4. #71822C2semidirect
6. #67662C2semidirect
7. #67659C2semidirect
11. C3×#501C4semidirect
12. C3×#497C4semidirect
13. C3×#491C4semidirect
14. C3×#489C4semidirect
98. #501Dic3semidirect
99. #497Dic3semidirect
100. #491Dic3semidirect
101. #489Dic3semidirect
390. C3#9431semidirect

Maximal subgroups

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

subgroupindex
C3×#31812
#831282
#718382
#718222
#717562
#676622
#676592
#94313