Finite Group Databasebeta

← #10690097

Subgroups — #10690097

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×#3172C211yesmaximal, characteristic
2#83133C211yesmaximal, characteristic
3#71840C211yesmaximal, characteristic, complemented
4#71827C211yesmaximal, characteristic, complemented
5#71754C211yesmaximal, characteristic
6#67669C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#8998—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#529C2211yescharacteristic
11C3×#501C411yescharacteristic, complemented
12C3×#497C411yescharacteristic, complemented
13C3×#491C411yescharacteristic, complemented
14C3×#489C411yescharacteristic, complemented
15C3×#470—21no—
16C3×#468—21no—
17C3×C4×Q16—21no—
18C3×C4×SD16—21no—
19C3×C4×M16C2211yescharacteristic
20#5024—21no—
21#5012—21no—
22#5011—21no—
23#5011—21no—
24#5000—21no—
25#4999—21no—
26#4932C2211yescharacteristic
27#4926C2211yescharacteristic
28#4925C2211yescharacteristic
29#4924C2211yescharacteristic
30#4538—21no—
31#4537—21no—
32#4535—21no—
33#4535—21no—
34#4532—21no—
35#4532—21no—
36#3172D311yescharacteristic
37#2995—31no—
38#1995—31no—
39#1500—31no—
40#1497—31no—
41#1300—31no—
42#1299—31no—
43C2×C3×Q16—21no—
44C2×C3×SD16—21no—
45C2×C3×M16—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—
67C2×#208D411yescharacteristic
68C2×#208D411yescharacteristic
69C22×C3⋊C8—21no—
70C3×#107—21no—
71C3×#106—21no—
72C3×#103—21no—
73C3×#103—21no—
74C3×#103—21no—
75C3×#102—21no—
76C3×#97C2×C411yescharacteristic
77C3×#97C2×C411yescharacteristic
78C3×C4×C8—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)—21no—
90C4⋊(C3⋊C8)D411yescharacteristic
91C4⋊(C3⋊C8)D411yescharacteristic
92#803—21no—
93#803—21no—
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#470—21no—
103#468—21no—
104#446—61no—
105C4×Q16—21no—
106C4×SD16—21no—
107#436—61no—
108#435—61no—
109#435—61no—
110#433—61no—
111#432—31no—
112#431—31no—
113#424—61no—
114#423—31no—
115C4×M16C2×D311yescharacteristic
116#376—31no—
117#339—61no—
118#339—61no—
119#328—61no—
120#327—61no—
121Q8⋊C8—61no—
122Q8⋊C8—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—
135C3×M16—42no—
136C2×C3×C8—21no—
137C2×C3×C8—21no—
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—22no—
163C4×Dic3—41no—
164#208—41no—
165#208—41no—
166#208—81no—
167C2×C3⋊C8—21no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—21no—
171C2×C3⋊C8—41no—
172C2×C3⋊C8—81no—
173C2×Q16—21no—
174C2×SD16—21no—
175C2×M16—21no—
176C2×M16—31no—
177C2×M16—31no—
178C2×M16—61no—
179C22×C8—61no—
180C4⋊Q8C2×Dic311yescharacteristic
181C4⋊Q8C2×Dic311yescharacteristic
182#126—21no—
183#125—21no—
184#125C2×Dic311yescharacteristic
185#124C2×Dic311yescharacteristic
186#123—21no—
187C22⋊Q8—21no—
188C22⋊Q8—21no—
189C22⋊Q8#6711yescharacteristic
190#121#6711yescharacteristic
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8#6711yescharacteristic
194C4×D4—21no—
195C4×D4#6711yescharacteristic
196#117—21no—
197#117—21no—
198C2×C42—61no—
199C2×C42C22×D311yescharacteristic
200#107—21no—
201#106—21no—
202C4⋊C8—31no—
203C4⋊C8—31no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208C4⋊C8—61no—
209#103—21no—
210#103—21no—
211#103—21no—
212#102—21no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217#97—61no—
218#97—61no—
219#97C2×Dic311yescharacteristic
220#97C2×Dic311yescharacteristic
221C4×C8—21no—
222C4×C8—61no—
223C4×C8—61no—
224#95—61no—
225#95—61no—
226C23×C3—21no—
227C23×C3#12111yescharacteristic
228C3×Q8—41no—
229C3×Q8—41no—
230C3×Q8—41no—
231C3×Q8—42no—
232C3×D4—41no—
233C3×D4—81no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4C2×C22⋊C411yescharacteristic
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4C4×D411yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12311yescharacteristic
251C2×Dic3—42no—
252C2×Dic3—81no—
253C2×Dic3—81no—
254C3×C8—22no—
255C3×C8—41no—
256C3⋊C8—41no—
257C3⋊C8—41no—
258C3⋊C8—41no—
259C3⋊C8—41no—
260C3⋊C8—81no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8#21711yescharacteristic
264C2×D4—21no—
265C2×D4#21711yescharacteristic
266C22×C4—21no—
267C22×C4—121no—
268C22×C4D3×D411yescharacteristic
269C22×C4#23711yescharacteristic
270C22×C4C2×#6711yescharacteristic
271Q16—42no—
272SD16—42no—
273M16—42no—
274M16—121no—
275M16—121no—
276M16—241no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—61no—
280C2×C8—61no—
281C2×C8—61no—
282C2×C8—61no—
283C2×C8—121no—
284C2×C8—241no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4#21711yescharacteristic
296C4⋊C4#21711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—41no—
303C42—21no—
304C42—41no—
305C42—62no—
306C42—121no—
307C42D3×D411yescharacteristic
308C42#23711yescharacteristic
309C42C22×Dic311yescharacteristic
310C42C2×#6711yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3#38611yescharacteristic
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—41no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4#45511yescharacteristic
329C3×C4#45611yescharacteristic
330Dic3—82no—
331C23—21no—
332C23#93911yescharacteristic
333Q8—41no—
334Q8—41no—
335Q8—41no—
336Q8—42no—
337D4—41no—
338D4—81no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—122no—
351C2×C4—241no—
352C2×C4—241no—
353C2×C4D4×Dic311yescharacteristic
354C2×C4D4×Dic311yescharacteristic
355C2×C4#93511yescharacteristic
356C2×C4#93711yescharacteristic
357C2×C4#93911yescharacteristic
358C2×C4C2×#21711yescharacteristic
359C8—22no—
360C8—41no—
361C8—121no—
362C8—121no—
363C8—121no—
364C8—121no—
365C8—241no—
366C2×C3—41no—
367C2×C3—81no—
368C2×C3#188111yescharacteristic
369C2×C3#188211yescharacteristic
370C2×C3#188411yescharacteristic
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22#522011yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—81no—
387C4—81no—
388C4—242no—
389C4#520211yescharacteristic
390C4#520511yescharacteristic
391C3#899811yescharacteristic, complemented, Sylow
392C2—41no—
393C2—81no—
394C2#7181711yescharacteristic
395C2#7210011yescharacteristic
396C2#7211611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3172C2
socleC22×C3#386
prime radicalC3×#121D4
cosocleC2×C3×C42C23
hypercenter#3172D3
hypocenterC3#8998
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3172

Complements

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

NH ≅ G/Ntype
3. #71840C2semidirect
4. #71827C2semidirect
6. #67669C2semidirect
7. #67665C2semidirect
11. C3×#501C4semidirect
12. C3×#497C4semidirect
13. C3×#491C4semidirect
14. C3×#489C4semidirect
98. #501Dic3semidirect
99. #497Dic3semidirect
100. #491Dic3semidirect
101. #489Dic3semidirect
391. C3#8998semidirect

Maximal subgroups

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

subgroupindex
C3×#31722
#831332
#718402
#718272
#717542
#676692
#676652
#89983