Finite Group Databasebeta

← #10690090

Subgroups — #10690090

Order 768.

452 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3171C211yesmaximal, characteristic
2#83133C211yesmaximal, characteristic, complemented
3#71816C211yesmaximal, characteristic, complemented
4#71803C211yesmaximal, characteristic, complemented
5#71754C211yesmaximal, characteristic, complemented
6#67669C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#8997—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C3×#529C2211yescharacteristic
11C3×#501C411yescharacteristic, complemented
12C3×#496C411yescharacteristic, complemented
13C3×#490C411yescharacteristic, complemented
14C3×#489C411yescharacteristic, complemented
15C3×#469—21no—
16C3×#468—21no—
17C3×C4×SD16—21no—
18C3×C4×D8—21no—
19C3×C4×M16C2211yescharacteristic
20C2×#837—21no—
21C2×#837—21no—
22#5024—21no—
23#5012—21no—
24#5011—21no—
25#5011—21no—
26#5000—21no—
27#4999—21no—
28#4932C2211yescharacteristic
29#4926C2211yescharacteristic
30#4925C2211yescharacteristic
31#4924C2211yescharacteristic
32#4538—21no—
33#4537—21no—
34D4⋊(C3⋊C8)—21no—
35D4⋊(C3⋊C8)—21no—
36#3171D311yescharacteristic
37#2995—31no—
38#1963—31no—
39#1500—31no—
40#1496—31no—
41#1300—31no—
42#1299—31no—
43C2×C3×#41—21no—
44C2×C3×SD16—21no—
45C2×C3×D8—21no—
46C2×C3×M16—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—
67C2×#208D411yescharacteristic
68C2×#208D411yescharacteristic
69C22×C3⋊C8—21no—
70C3×#107—21no—
71C3×#106—21no—
72C3×#103—21no—
73C3×#102—21no—
74C3×#102—21no—
75C3×#102—21no—
76C3×#97C2×C411yescharacteristic
77C3×#97C2×C411yescharacteristic
78C3×C4×C8—21no—
79#837—42no—
80#837—42no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83#830—21no—
84#830—21no—
85#830—21no—
86#830—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)—21no—
91C4⋊(C3⋊C8)—21no—
92C4⋊(C3⋊C8)D411yescharacteristic
93C4⋊(C3⋊C8)D411yescharacteristic
94#803—21no—
95#803—21no—
96C4×C3⋊C8—21no—
97C4×C3⋊C8—21no—
98#532C2×D311yescharacteristic
99#529C2×D311yescharacteristic
100#501Dic311yescharacteristic, complemented
101#496Dic311yescharacteristic, complemented
102#490Dic311yescharacteristic, complemented
103#489Dic311yescharacteristic, complemented
104#469—21no—
105#468—21no—
106#446—61no—
107C4×SD16—21no—
108C4×D8—21no—
109#436—61no—
110#435—61no—
111#435—61no—
112#433—61no—
113#432—31no—
114#431—31no—
115#424—61no—
116#423—31no—
117C2×#104—61no—
118C2×#104—61no—
119C4×M16C2×D311yescharacteristic
120#376—31no—
121#328—61no—
122#327—61no—
123D4⋊C8—61no—
124D4⋊C8—61no—
125C3×#41—41no—
126C3×#41—41no—
127C3×#41—41no—
128C3×#41—41no—
129C2×C3×Q8—21no—
130C2×C3×Q8C22⋊C411yescharacteristic
131C2×C3×D4—21no—
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4—41no—
136C2×C3×D4C22⋊C411yescharacteristic
137C22×C3×C4—21no—
138C22×C3×C4—41no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×Dic3—41no—
143C3×SD16—42no—
144C3×D8—42no—
145C3×M16—42no—
146C2×C3×C8—21no—
147C2×C3×C8—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4C22⋊C411yescharacteristic
155C3×C4⋊C4C22⋊C411yescharacteristic
156C3×C22⋊C4—21no—
157C3×C22⋊C4—21no—
158C3×C22⋊C4—21no—
159C3×C22⋊C4—21no—
160C3×C22⋊C4—21no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C42—21no—
164C3×C42C22×C411yescharacteristic
165C3×C42C2×D411yescharacteristic
166C3×C42C2×D411yescharacteristic
167C3×C42#4111yescharacteristic
168C4×Dic3—22no—
169C4×Dic3—41no—
170#208—41no—
171#208—41no—
172#208—81no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—81no—
179C2×#41—21no—
180C2×SD16—21no—
181C2×D8—21no—
182C2×M16—21no—
183C2×M16—31no—
184C2×M16—31no—
185C2×M16—61no—
186C22×C8—61no—
187C4⋊Q8C2×Dic311yescharacteristic
188#127C2×Dic311yescharacteristic
189#126—21no—
190#125C2×Dic311yescharacteristic
191#124—21no—
192#124C2×Dic311yescharacteristic
193#123—21no—
194C22⋊Q8#6711yescharacteristic
195#121—21no—
196#121—21no—
197#121#6711yescharacteristic
198C4×Q8#6711yescharacteristic
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4#6711yescharacteristic
203#117—21no—
204C2×C42—61no—
205C2×C42C22×D311yescharacteristic
206#107—21no—
207#106—21no—
208C4⋊C8—31no—
209C4⋊C8—31no—
210C4⋊C8—61no—
211C4⋊C8—61no—
212C4⋊C8—61no—
213C4⋊C8—61no—
214C4⋊C8—61no—
215#104—122no—
216#104—122no—
217#103—21no—
218#102—21no—
219#102—21no—
220#102—21no—
221C22⋊C8—61no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224C22⋊C8—61no—
225#97—61no—
226#97—61no—
227#97C2×Dic311yescharacteristic
228#97C2×Dic311yescharacteristic
229C4×C8—21no—
230C4×C8—61no—
231C4×C8—61no—
232#95—61no—
233#95—61no—
234C23×C3—21no—
235C23×C3—41no—
236C23×C3#12111yescharacteristic
237C3×Q8—41no—
238C3×Q8—41no—
239C3×Q8—41no—
240C3×D4—41no—
241C3×D4—41no—
242C3×D4—41no—
243C3×D4—42no—
244C3×D4—41no—
245C3×D4—41no—
246C3×D4—81no—
247C3×D4—81no—
248C3×D4—81no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—81no—
260C2×C3×C4—81no—
261C2×C3×C4C2×C22⋊C411yescharacteristic
262C2×C3×C4C4×D411yescharacteristic
263C2×C3×C4C4×D411yescharacteristic
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12111yescharacteristic
266C2×C3×C4#12311yescharacteristic
267C2×Dic3—42no—
268C2×Dic3—81no—
269C2×Dic3—81no—
270C3×C8—22no—
271C3×C8—41no—
272C3⋊C8—41no—
273C3⋊C8—41no—
274C3⋊C8—41no—
275C3⋊C8—41no—
276C3⋊C8—81no—
277#41—41no—
278#41—41no—
279#41—41no—
280#41—41no—
281C2×Q8—21no—
282C2×Q8#21711yescharacteristic
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—41no—
288C2×D4#21711yescharacteristic
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—121no—
292C22×C4D3×D411yescharacteristic
293C22×C4#23711yescharacteristic
294C22×C4C2×#6711yescharacteristic
295SD16—42no—
296D8—42no—
297M16—42no—
298M16—121no—
299M16—121no—
300M16—241no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—61no—
304C2×C8—61no—
305C2×C8—61no—
306C2×C8—61no—
307C2×C8—121no—
308C2×C8—241no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4#21711yescharacteristic
316C4⋊C4#21711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C42—21no—
325C42—62no—
326C42—121no—
327C42D3×D411yescharacteristic
328C42#23711yescharacteristic
329C42C22×Dic311yescharacteristic
330C42C2×#6711yescharacteristic
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3—81no—
340C22×C3#38611yescharacteristic
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—41no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4—81no—
350C3×C4#45311yescharacteristic
351C3×C4#45411yescharacteristic
352Dic3—82no—
353C23—21no—
354C23—41no—
355C23#93911yescharacteristic
356Q8—41no—
357Q8—41no—
358Q8—41no—
359D4—41no—
360D4—41no—
361D4—41no—
362D4—41no—
363D4—41no—
364D4—42no—
365D4—81no—
366D4—81no—
367D4—81no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—122no—
381C2×C4—241no—
382C2×C4—241no—
383C2×C4D4×Dic311yescharacteristic
384C2×C4D4×Dic311yescharacteristic
385C2×C4#93511yescharacteristic
386C2×C4#93711yescharacteristic
387C2×C4#93911yescharacteristic
388C2×C4C2×#21711yescharacteristic
389C8—22no—
390C8—41no—
391C8—121no—
392C8—121no—
393C8—121no—
394C8—121no—
395C8—241no—
396C2×C3—41no—
397C2×C3—81no—
398C2×C3—161no—
399C2×C3#188111yescharacteristic
400C2×C3#188311yescharacteristic
401C2×C3#188411yescharacteristic
402C22—21no—
403C22—21no—
404C22—21no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22#522011yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—81no—
421C4—242no—
422C4#519811yescharacteristic
423C4#520111yescharacteristic
424C3#899711yescharacteristic, complemented, Sylow
425C2—41no—
426C2—81no—
427C2—161no—
428C2#7181711yescharacteristic
429C2#7208811yescharacteristic
430C2#7211611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3171C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3171D3
hypocenterC3#8997
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3171

Complements

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

NH ≅ G/Ntype
2. #83133C2semidirect
3. #71816C2semidirect
4. #71803C2semidirect
5. #71754C2semidirect
6. #67669C2semidirect
7. #67665C2semidirect
11. C3×#501C4semidirect
12. C3×#496C4semidirect
13. C3×#490C4semidirect
14. C3×#489C4semidirect
100. #501Dic3semidirect
101. #496Dic3semidirect
102. #490Dic3semidirect
103. #489Dic3semidirect
424. C3#8997semidirect

Maximal subgroups

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

subgroupindex
C3×#31712
#831332
#718162
#718032
#717542
#676692
#676652
#89973