Finite Group Databasebeta

← #10691314

Subgroups — #10691314

Order 768.

450 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3106C211yesmaximal, characteristic
2#82513C211yesmaximal, characteristic, complemented
3#71802C211yesmaximal, characteristic, complemented
4#71797C211yesmaximal, characteristic, complemented
5#71641C211yesmaximal, characteristic, complemented
6#67743C211yesmaximal, characteristic, complemented
7#67739C211yesmaximal, characteristic, complemented
8#9213—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C3×C4×#41C2211yescharacteristic
11C3×#482—21no—
12C3×#477C411yescharacteristic
13C3×#476C411yescharacteristic
14C3×#460C411yescharacteristic
15C3×#459C411yescharacteristic
16C3×#449—21no—
17C3×C4×SD16—21no—
18C3×C4×D8—21no—
19C3×#431C2211yescharacteristic
20#5234—21no—
21#5000—22no—
22#4999—21no—
23#4925—22no—
24#4924C2211yescharacteristic
25C4×#208C2211yescharacteristic
26C4×#208C2211yescharacteristic
27C2×C4×C3⋊C8C2211yescharacteristic
28#4540—22no—
29#4535—21no—
30D4⋊(C3⋊C8)—21no—
31D4⋊(C3⋊C8)—22no—
32#3106D311yescharacteristic
33#2922—31no—
34#1525—31no—
35#1492—31no—
36#1489—31no—
37#1299—31no—
38#1278—31no—
39C2×C3×#41—21no—
40C2×C3×#41D411yescharacteristic
41C2×C3×SD16—21no—
42C2×C3×D8—21no—
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×#127C2×C411yescharacteristic
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×Q8C2×C411yescharacteristic
49C3×C4×Q8D411yescharacteristic
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4C2×C411yescharacteristic
55C3×C4×D4D411yescharacteristic
56C3×#117—21no—
57C3×#117D411yescharacteristic
58C2×C3×C42—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#208—21no—
61C2×#208—21no—
62C2×#208—21no—
63C22×C3⋊C8—21no—
64C3×#107—21no—
65C3×#106—21no—
66C3×C4⋊C8C2×C411yescharacteristic
67C3×C4⋊C8C2×C411yescharacteristic
68C3×#103—21no—
69C3×#102—21no—
70C3×#102—21no—
71C3×#102—21no—
72C3×C22⋊C8—21no—
73C3×C4×C8—21no—
74#830—24no—
75C4⋊(C3⋊C8)—24no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78#803—21no—
79#803D412yes—
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83C4×C3⋊C8D412yes—
84#532C2×D311yescharacteristic
85C4×#41C2×D311yescharacteristic
86#482—21no—
87#477Dic311yescharacteristic
88#476Dic311yescharacteristic
89#460Dic311yescharacteristic
90#459Dic311yescharacteristic
91#449—21no—
92C4×SD16—21no—
93C4×D8—21no—
94#433—62no—
95#431—62no—
96#431C2×D311yescharacteristic
97#424—61no—
98#423—31no—
99#408—61no—
100C4×M16—31no—
101C4×M16—31no—
102C2×C4×C8—31no—
103#329—62no—
104Q8⋊C8—61no—
105D4⋊C8—62no—
106D4⋊C8—61no—
107C3×#41—41no—
108C3×#41—41no—
109C3×#41—81no—
110C3×#41—81no—
111C2×C3×Q8—21no—
112C2×C3×Q8C22⋊C411yescharacteristic
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—41no—
118C2×C3×D4C22⋊C411yescharacteristic
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—41no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C3×SD16—42no—
127C3×D8—42no—
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4C22⋊C411yescharacteristic
135C3×C4⋊C4C22⋊C411yescharacteristic
136C3×C22⋊C4—21no—
137C3×C22⋊C4—21no—
138C3×C22⋊C4—21no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42—21no—
142C3×C42—21no—
143C3×C42—21no—
144C3×C42C22×C411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148#208—22no—
149#208—41no—
150#208—41no—
151#208—41no—
152#208—81no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×#41—21no—
160C2×#41#6711yescharacteristic
161C2×SD16—21no—
162C2×D8—21no—
163C2×M16—61no—
164C2×M16—61no—
165C2×M16—61no—
166C22×C8—61no—
167C4⋊Q8C2×Dic311yescharacteristic
168#127C2×Dic311yescharacteristic
169#124—21no—
170#121—21no—
171#121—21no—
172C4×Q8C2×Dic311yescharacteristic
173C4×Q8#6711yescharacteristic
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4C2×Dic311yescharacteristic
179C4×D4#6711yescharacteristic
180#117—21no—
181#117#6711yescharacteristic
182C2×C42—21no—
183C2×C42C22×D311yescharacteristic
184#107—21no—
185#106—21no—
186C4⋊C8—64no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8C2×Dic311yescharacteristic
190C4⋊C8C2×Dic311yescharacteristic
191#103—21no—
192#102—21no—
193#102—21no—
194#102—21no—
195C22⋊C8—21no—
196C22⋊C8—64no—
197#97—32no—
198#97—61no—
199C4×C8—21no—
200C4×C8—32no—
201C4×C8—61no—
202C4×C8—61no—
203C4×C8—61no—
204C23×C3—21no—
205C23×C3—41no—
206C23×C3#12311yescharacteristic
207C3×Q8—21no—
208C3×Q8—21no—
209C3×Q8—81no—
210C3×D4—21no—
211C3×D4—21no—
212C3×D4—42no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—41no—
216C3×D4—41no—
217C3×D4—81no—
218C3×D4—82no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—42no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—81no—
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#12111yescharacteristic
242C3×C8—41no—
243C3×C8—41no—
244C3⋊C8—42no—
245C3⋊C8—42no—
246C3⋊C8—42no—
247C3⋊C8—81no—
248#41—41no—
249#41—41no—
250#41—81no—
251#41—81no—
252C2×Q8—21no—
253C2×Q8#21711yescharacteristic
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C2×D4#21711yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4#23711yescharacteristic
265C22×C4#23711yescharacteristic
266C22×C4C2×#6711yescharacteristic
267SD16—42no—
268D8—42no—
269M16—62no—
270M16—121no—
271M16—121no—
272M16—121no—
273M16—241no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—62no—
277C2×C8—62no—
278C2×C8—121no—
279C2×C8—121no—
280C2×C8—121no—
281C2×C8—121no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4#21711yescharacteristic
287C4⋊C4#21711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42D3×D411yescharacteristic
297C42D3×D411yescharacteristic
298C42C22×Dic311yescharacteristic
299C42C2×#6711yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3#38611yescharacteristic
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—41no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4—81no—
317C3×C4#42111yescharacteristic
318C3×C4#42111yescharacteristic
319C3×C4#44311yescharacteristic
320C3×C4#44411yescharacteristic
321C3×C4#47111yescharacteristic
322C3×C4#47211yescharacteristic
323C23—21no—
324C23—41no—
325C23#93511yescharacteristic
326Q8—21no—
327Q8—21no—
328Q8—81no—
329D4—21no—
330D4—21no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—81no—
337D4—82no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4D4×Dic311yescharacteristic
356C2×C4D4×Dic311yescharacteristic
357C2×C4#93711yescharacteristic
358C2×C4#93911yescharacteristic
359C2×C4#93911yescharacteristic
360C2×C4C2×#21711yescharacteristic
361C8—41no—
362C8—41no—
363C8—122no—
364C8—122no—
365C8—122no—
366C8—241no—
367C2×C3—41no—
368C2×C3—81no—
369C2×C3—161no—
370C2×C3#187611yescharacteristic
371C2×C3#189711yescharacteristic
372C2×C3#190711yescharacteristic
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22#522011yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4#519611yescharacteristic
391C4#519711yescharacteristic
392C4#519911yescharacteristic
393C4#520011yescharacteristic
394C4#523611yescharacteristic
395C4#523611yescharacteristic
396C3#921311yescharacteristic, complemented, Sylow
397C2—41no—
398C2—81no—
399C2—161no—
400C2#7171211yescharacteristic
401C2#7208711yescharacteristic
402C2#7209111yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3106C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3106D3
hypocenterC3#9213
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
2. #82513C2semidirect
3. #71802C2semidirect
4. #71797C2semidirect
5. #71641C2semidirect
6. #67743C2semidirect
7. #67739C2semidirect
396. C3#9213semidirect

Maximal subgroups

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

subgroupindex
C3×#31062
#825132
#718022
#717972
#716412
#677432
#677392
#92133