Finite Group Databasebeta

← #10692767

Subgroups — #10692767

Order 768.

436 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3180C211yesmaximal, characteristic
2#83128C211yesmaximal, characteristic, complemented
3#71814C211yesmaximal, characteristic, complemented
4#71798C211yesmaximal, characteristic, complemented
5#71756C211yesmaximal, characteristic, complemented
6#67662C211yesmaximal, characteristic, complemented
7#67659C211yesmaximal, characteristic, complemented
8#9432—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C3×#529C2211yescharacteristic
11C3×#501C411yescharacteristic, complemented
12C3×#496C411yescharacteristic, complemented
13C3×#490C411yescharacteristic, complemented
14C3×#489C411yescharacteristic, complemented
15C3×#466—21no—
16C3×#465—21no—
17C3×#442—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#4540—21no—
31#4540—21no—
32#4538—21no—
33#4537—21no—
34#4537—21no—
35#4537—21no—
36#3180D311yescharacteristic
37#2968—31no—
38#1530—31no—
39#1517—31no—
40#1512—31no—
41#1302—31no—
42#1295—31no—
43C2×C3×#41—21no—
44C2×C3×SD16—21no—
45C2×C3×D8—21no—
46C22×C3×C8—21no—
47C3×C4⋊Q8C2×C411yescharacteristic
48C3×#127C2×C411yescharacteristic
49C3×#126—21no—
50C3×#125C2×C411yescharacteristic
51C3×#124—21no—
52C3×#124C2×C411yescharacteristic
53C3×#123—21no—
54C3×C22⋊Q8D411yescharacteristic
55C3×#121—21no—
56C3×#121—21no—
57C3×#121D411yescharacteristic
58C3×C4×Q8D411yescharacteristic
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C3×C4×D4—21no—
62C3×C4×D4D411yescharacteristic
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×#102—21no—
73C3×#102—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#532C2×D311yescharacteristic
97#529C2×D311yescharacteristic
98#501Dic311yescharacteristic, complemented
99#496Dic311yescharacteristic, complemented
100#490Dic311yescharacteristic, complemented
101#489Dic311yescharacteristic, complemented
102#466—21no—
103#465—21no—
104#446—61no—
105#442—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#329—61no—
118#329—61no—
119#328—61no—
120#327—61no—
121#327—61no—
122#327—61no—
123C3×#41—81no—
124C3×#41—81no—
125C2×C3×Q8—21no—
126C2×C3×Q8C22⋊C411yescharacteristic
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—41no—
132C2×C3×D4C22⋊C411yescharacteristic
133C22×C3×C4—21no—
134C22×C3×C4—41no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×Dic3—41no—
139C3×SD16—42no—
140C3×D8—42no—
141C2×C3×C8—21no—
142C2×C3×C8—21no—
143C2×C3×C8—42no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4C22⋊C411yescharacteristic
151C3×C4⋊C4C22⋊C411yescharacteristic
152C3×C22⋊C4—21no—
153C3×C22⋊C4—21no—
154C3×C22⋊C4—21no—
155C3×C22⋊C4—21no—
156C3×C22⋊C4—21no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C42—21no—
160C3×C42C22×C411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42#4111yescharacteristic
164C4⋊Dic3—81no—
165#208—42no—
166C2×C3⋊C8—21no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—42no—
171C2×C3⋊C8—42no—
172C2×C3⋊C8—41no—
173C2×#41—21no—
174C2×SD16—21no—
175C2×D8—21no—
176C2×M16—61no—
177C22×C8—21no—
178C22×C8—61no—
179C22×C8—61no—
180C4⋊Q8C2×Dic311yescharacteristic
181#127C2×Dic311yescharacteristic
182#126—21no—
183#125C2×Dic311yescharacteristic
184#124—21no—
185#124C2×Dic311yescharacteristic
186#123—21no—
187C22⋊Q8#6711yescharacteristic
188#121—21no—
189#121—21no—
190#121#6711yescharacteristic
191C4×Q8#6711yescharacteristic
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4#6711yescharacteristic
196#117—21no—
197C2×C4⋊C4—61no—
198C2×C42C22×D311yescharacteristic
199#107—21no—
200#106—21no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204C4⋊C8—61no—
205#103—21no—
206#102—21no—
207#102—21no—
208#102—21no—
209C22⋊C8—31no—
210C22⋊C8—31no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214#97—21no—
215#97—31no—
216#97—31no—
217#97—61no—
218#97—61no—
219#97—61no—
220#97C2×Dic311yescharacteristic
221#97C2×Dic311yescharacteristic
222C4×C8—61no—
223C4×C8—61no—
224#95—61no—
225#95—61no—
226C23×C3—21no—
227C23×C3—41no—
228C23×C3#12111yescharacteristic
229C3×Q8—41no—
230C3×Q8—81no—
231C3×D4—41no—
232C3×D4—41no—
233C3×D4—41no—
234C3×D4—41no—
235C3×D4—41no—
236C3×D4—81no—
237C3×D4—81no—
238C3×D4—81no—
239C3×D4—81no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—82no—
251C2×C3×C4C2×C22⋊C411yescharacteristic
252C2×C3×C4C4×D411yescharacteristic
253C2×C3×C4C4×D411yescharacteristic
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12111yescharacteristic
256C2×C3×C4#12311yescharacteristic
257C2×Dic3—81no—
258C2×Dic3—81no—
259C2×Dic3—81no—
260C3×C8—22no—
261C3×C8—41no—
262C3⋊C8—22no—
263C3⋊C8—41no—
264C3⋊C8—41no—
265C3⋊C8—41no—
266C3⋊C8—81no—
267#41—81no—
268#41—81no—
269C2×Q8—21no—
270C2×Q8#21711yescharacteristic
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4#21711yescharacteristic
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—121no—
280C22×C4D3×D411yescharacteristic
281C22×C4#23711yescharacteristic
282C22×C4C2×#6711yescharacteristic
283SD16—42no—
284D8—42no—
285M16—122no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—42no—
289C2×C8—61no—
290C2×C8—61no—
291C2×C8—61no—
292C2×C8—61no—
293C2×C8—122no—
294C2×C8—122no—
295C2×C8—121no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—241no—
303C4⋊C4#21711yescharacteristic
304C4⋊C4#21711yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C42—21no—
313C42D3×D411yescharacteristic
314C42#23711yescharacteristic
315C42C22×Dic311yescharacteristic
316C42C2×#6711yescharacteristic
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3#38611yescharacteristic
326C22×C3#47111yescharacteristic
327C22×C3#47211yescharacteristic
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—41no—
337C3×C4—81no—
338Dic3—161no—
339C23—21no—
340C23—41no—
341C23#93911yescharacteristic
342Q8—41no—
343Q8—81no—
344D4—41no—
345D4—41no—
346D4—41no—
347D4—41no—
348D4—41no—
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—82no—
364C2×C4—241no—
365C2×C4—241no—
366C2×C4—241no—
367C2×C4D4×Dic311yescharacteristic
368C2×C4D4×Dic311yescharacteristic
369C2×C4#93511yescharacteristic
370C2×C4#93711yescharacteristic
371C2×C4#93911yescharacteristic
372C2×C4C2×#21711yescharacteristic
373C8—22no—
374C8—41no—
375C8—62no—
376C8—121no—
377C8—121no—
378C8—121no—
379C8—241no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—81no—
383C2×C3—161no—
384C2×C3#188411yescharacteristic
385C2×C3#190611yescharacteristic
386C2×C3#190711yescharacteristic
387C22—21no—
388C22—21no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22#519911yescharacteristic
396C22#520011yescharacteristic
397C22#522011yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—81no—
408C4—481no—
409C3#943211yescharacteristic, complemented, Sylow
410C2—21no—
411C2—21no—
412C2—81no—
413C2—161no—
414C2#7181711yescharacteristic
415C2#7209111yescharacteristic
416C2#7211411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3180C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3180D3
hypocenterC3#9432
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3180

Complements

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

NH ≅ G/Ntype
2. #83128C2semidirect
3. #71814C2semidirect
4. #71798C2semidirect
5. #71756C2semidirect
6. #67662C2semidirect
7. #67659C2semidirect
11. C3×#501C4semidirect
12. C3×#496C4semidirect
13. C3×#490C4semidirect
14. C3×#489C4semidirect
98. #501Dic3semidirect
99. #496Dic3semidirect
100. #490Dic3semidirect
101. #489Dic3semidirect
409. C3#9432semidirect

Maximal subgroups

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

subgroupindex
C3×#31802
#831282
#718142
#717982
#717562
#676622
#676592
#94323