Finite Group Databasebeta

← #10691138

Subgroups — #10691138

Order 768.

548 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3036C211yesmaximal, characteristic
2#82338C211yesmaximal, characteristic, complemented
3#71814C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#67715C211yesmaximal, characteristic, complemented
7#67713C211yesmaximal, characteristic, complemented
8#9181—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C3×#463—22no—
13C3×#461C412yescomplemented
14C3×#459C412yescomplemented
15C3×#423C2211yescharacteristic
16C3×#417—21no—
17#5221—21no—
18#5211—21no—
19C4×#217—21no—
20Dic3×C4⋊C4—21no—
21#4951—22no—
22#4936C2211yescharacteristic
23C4×C4⋊Dic3C2211yescharacteristic
24#4925C2211yescharacteristic
25#4925C2211yescharacteristic
26#4537—21no—
27#4537—21no—
28#4537—22no—
29D4⋊(C3⋊C8)—22no—
30#3036D311yescharacteristic
31#2357—31no—
32#1517—31no—
33#1502—31no—
34#1478—31no—
35#1303—31no—
36#1286—31no—
37C2×C3×#41—21no—
38C22×C3×D4D411yescharacteristic
39C23×C3×C4—21no—
40C3×AGL(1,ℤ/8)—42no—
41C3×AGL(1,ℤ/8)—42no—
42C2×C3×SD16—22no—
43C2×C3×D8—22no—
44C2×C3×M16—21no—
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×#127C2×C411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4C2×C412yes—
55C3×C4×D4D412yes—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#217—22no—
60C2×#217—22no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4×Dic3—21no—
64C2×C4×Dic3—22no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8C2×C412yes—
67C3×#103—22no—
68C3×#102—22no—
69C3⋊#95—22no—
70C3⋊#95—21no—
71C3⋊#95D411yescharacteristic
72C3⋊#95D411yescharacteristic
73C3⋊#95D412yes—
74#830—21no—
75#830—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4×C3⋊C8—21no—
79C4×C3⋊C8—21no—
80C2×AGL(1,ℤ/8)—21no—
81#532C2×D311yescharacteristic
82C2×C4×D4C2×D311yescharacteristic
83#463—22no—
84#461Dic312yescomplemented
85#459Dic312yescomplemented
86#431—31no—
87#431—31no—
88#423C2×D311yescharacteristic
89#417—21no—
90#390—61no—
91#388—62no—
92#382—31no—
93#381—61no—
94C4×C4⋊C4—31no—
95C4×C4⋊C4—61no—
96C4×C22⋊C4—61no—
97#327—61no—
98#327—61no—
99#327—62no—
100D4⋊C8—62no—
101C24×C3—21no—
102C3×#41—41no—
103C3×#41—41no—
104C3×#41—41no—
105C3×#41—41no—
106C2×C3×Q8—21no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—41no—
116C2×C3×D4C22⋊C412yes—
117C22×C3×C4—21no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—41no—
121C22×C3×C4—82no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×Dic3—21no—
126C22×Dic3—22no—
127C22×Dic3—21no—
128C3×SD16—44no—
129C3×D8—44no—
130C3×M16—41no—
131C3×M16—41no—
132C2×C3×C8—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4C22⋊C412yes—
137C3×C22⋊C4—22no—
138C3×C22⋊C4—22no—
139C3×C22⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42C22×C411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144C3×C42#4111yescharacteristic
145#217—44no—
146#217—82no—
147C4⋊Dic3—42no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152C4×Dic3—82no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×#41—21no—
156C22×D4#6711yescharacteristic
157C23×C4—21no—
158AGL(1,ℤ/8)—42no—
159AGL(1,ℤ/8)—42no—
160C2×SD16—22no—
161C2×D8—22no—
162C2×M16—21no—
163C4⋊Q8C2×Dic311yescharacteristic
164#127C2×Dic311yescharacteristic
165#124—21no—
166#121—21no—
167#121—21no—
168C4×D4—21no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4C2×Dic312yes—
173C4×D4#6712yes—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4#6711yescharacteristic
177C2×C22⋊C4—21no—
178C2×C22⋊C4—62no—
179C2×C22⋊C4—62no—
180C2×C42—61no—
181C2×C42—62no—
182C2×C42C22×D311yescharacteristic
183C4⋊C8—21no—
184C4⋊C8—61no—
185C4⋊C8—61no—
186C4⋊C8C2×Dic312yes—
187#103—22no—
188#102—22no—
189C22⋊C8—61no—
190C22⋊C8—61no—
191C4×C8—61no—
192C4×C8—61no—
193#95—31no—
194#95—31no—
195#95—32no—
196#95—62no—
197#95—61no—
198C23×C3—22no—
199C23×C3—42no—
200C23×C3—42no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3#12011yescharacteristic
205C3×Q8—41no—
206C3×Q8—41no—
207C3×D4—22no—
208C3×D4—22no—
209C3×D4—41no—
210C3×D4—41no—
211C3×D4—41no—
212C3×D4—41no—
213C3×D4—81no—
214C3×D4—81no—
215C3×D4—81no—
216C3×D4—81no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—42no—
221C2×C3×C4—41no—
222C2×C3×C4—82no—
223C2×C3×C4—82no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4C2×C22⋊C411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12311yescharacteristic
232C2×C3×C4#13511yescharacteristic
233C2×C3×C4#13511yescharacteristic
234C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
235C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
236C2×Dic3—22no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—42no—
240C2×Dic3—41no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—42no—
245C3×C8—42no—
246C3⋊C8—81no—
247C3⋊C8—81no—
248C24—21no—
249#41—41no—
250#41—41no—
251#41—41no—
252#41—41no—
253C2×Q8—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—41no—
263C2×D4#21712yes—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—61no—
269C22×C4—62no—
270C22×C4—61no—
271C22×C4—82no—
272C22×C4D3×D411yescharacteristic
273C22×C4D3×D411yescharacteristic
274C22×C4C2×#6711yescharacteristic
275SD16—44no—
276D8—44no—
277M16—41no—
278M16—41no—
279C2×C8—22no—
280C2×C8—121no—
281C2×C8—121no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—122no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C4⋊C4#21712yes—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—124no—
294C22⋊C4—242no—
295C42—121no—
296C42—121no—
297C42—242no—
298C42#23711yescharacteristic
299C42#23711yescharacteristic
300C42C22×Dic311yescharacteristic
301C42C2×#6711yescharacteristic
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—42no—
305C22×C3—42no—
306C22×C3—42no—
307C22×C3—82no—
308C22×C3—82no—
309C22×C3—81no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3#38611yescharacteristic
313C22×C3#44911yescharacteristic
314C22×C3#44911yescharacteristic
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—42no—
320C3×C4—81no—
321C3×C4#41611yescharacteristic
322C3×C4#41811yescharacteristic
323C3×C4#48211yescharacteristic
324C3×C4#48211yescharacteristic
325Dic3—42no—
326Dic3—81no—
327Dic3—82no—
328C23—22no—
329C23—42no—
330C23—42no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23#93711yescharacteristic
335Q8—41no—
336Q8—41no—
337D4—22no—
338D4—22no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—81no—
344D4—81no—
345D4—81no—
346D4—81no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—62no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—122no—
365C2×C4#93211yescharacteristic
366C2×C4#93211yescharacteristic
367C2×C4D4×Dic311yescharacteristic
368C2×C4D4×Dic311yescharacteristic
369C2×C4#93511yescharacteristic
370C2×C4#93911yescharacteristic
371C2×C4#93911yescharacteristic
372C2×C4#95011yescharacteristic
373C2×C4#95011yescharacteristic
374C2×C4C2×#21711yescharacteristic
375C8—42no—
376C8—241no—
377C8—241no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—82no—
381C2×C3—161no—
382C2×C3#187511yescharacteristic
383C2×C3#190411yescharacteristic
384C2×C3#190611yescharacteristic
385C22—21no—
386C22—21no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—82no—
391C22—82no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22#503711yescharacteristic
396C22#503711yescharacteristic
397C22#522011yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—81no—
404C4—122no—
405C4—241no—
406C4—242no—
407C4#503511yescharacteristic
408C4#503511yescharacteristic
409C4#521511yescharacteristic
410C4#523311yescharacteristic
411C3#918111yescharacteristic, complemented, Sylow
412C2—21no—
413C2—21no—
414C2—82no—
415C2—161no—
416C2#7167811yescharacteristic
417C2#7208911yescharacteristic
418C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3036C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3036D3
hypocenterC3#9181
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
2. #82338C2semidirect
3. #71814C2semidirect
4. #71811C2semidirect
5. #71597C2semidirect
6. #67715C2semidirect
7. #67713C2semidirect
13. C3×#461C4semidirect
14. C3×#459C4semidirect
84. #461Dic3semidirect
85. #459Dic3semidirect
411. C3#9181semidirect

Maximal subgroups

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

subgroupindex
C3×#30362
#823382
#718142
#718112
#715972
#677152
#677132
#91813