Finite Group Databasebeta

← #10692343

Subgroups — #10692343

Order 768.

538 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3038C211yesmaximal, characteristic
2#82523C211yesmaximal, characteristic, complemented
3#71766C211yesmaximal, characteristic, complemented
4#71759C211yesmaximal, characteristic, complemented
5#71641C211yesmaximal, characteristic, complemented
6C3⋊#1309C211yesmaximal, characteristic, complemented
7#67710C211yesmaximal, characteristic, complemented
8#9360—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)C412yes—
10C2×C3×#127C2211yescharacteristic
11C3×C4×#41C2211yescharacteristic
12C3×#460—21no—
13C3×#460—21no—
14C3×#459—21no—
15C3×#459—21no—
16C3×#423C2211yescharacteristic
17C3×#418C412yes—
18C2×#833—22no—
19Q8×C3⋊C8—21no—
20#5012—21no—
21#5012—22no—
22#4925—21no—
23#4925C2211yescharacteristic
24#4924—21no—
25#4924C2211yescharacteristic
26#4924C2211yescharacteristic
27C4×#208C2211yescharacteristic
28#4544—21no—
29#4544—21no—
30#4535—21no—
31D4⋊(C3⋊C8)—21no—
32#3038D311yescharacteristic
33#2991—31no—
34#1965—31no—
35#1545—31no—
36#1489—31no—
37#1309—31no—
38#1299—31no—
39C2×C3×#41C2×C411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4C2×C412yes—
43C3×AGL(1,ℤ/8)—44no—
44C2×C3×SD16—22no—
45C2×C3×D8—22no—
46C2×C3×M16C2×C412yes—
47C3×#127—21no—
48C3×#127—21no—
49C3×#127—21no—
50C3×#127—21no—
51C3×C4×Q8D411yescharacteristic
52C3×C4×Q8D411yescharacteristic
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4D411yescharacteristic
56C3×C4×D4D411yescharacteristic
57C3×#117—21no—
58C3×#117C2×C411yescharacteristic
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61C2×#208—22no—
62C2×#208—21no—
63C2×#208—21no—
64C2×#208D412yes—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×#102—22no—
68C3×#102—22no—
69#833—44no—
70#830—22no—
71#830—21no—
72C4⋊(C3⋊C8)—22no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D411yescharacteristic
78C4⋊(C3⋊C8)D411yescharacteristic
79#803—21no—
80C4×C3⋊C8—22no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8—21no—
83C2×AGL(1,ℤ/8)Dic312yes—
84C2×#127C2×D311yescharacteristic
85C4×#41C2×D311yescharacteristic
86#460—21no—
87#460—21no—
88#459—21no—
89#459—21no—
90C8×Q8—61no—
91#436—62no—
92#436—61no—
93#431—31no—
94#431—61no—
95#423—31no—
96#423—31no—
97#423—61no—
98#423C2×D311yescharacteristic
99#418Dic312yes—
100C2×#100—62no—
101C4×M16—31no—
102#331—61no—
103#331—61no—
104Q8⋊C8—61no—
105D4⋊C8—61no—
106C24×C3—22no—
107C3×#41—22no—
108C3×#41—41no—
109C2×C3×Q8C22⋊C411yescharacteristic
110C2×C3×D4—21no—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4—42no—
114C2×C3×D4—42no—
115C2×C3×D4—42no—
116C2×C3×D4—42no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—81no—
122C2×C3×D4—81no—
123C2×C3×D4C22⋊C411yescharacteristic
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4C22×C411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C3×SD16—44no—
131C3×D8—44no—
132C3×M16—42no—
133C2×C3×C8—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4C22⋊C411yescharacteristic
137C3×C4⋊C4C22⋊C411yescharacteristic
138C3×C22⋊C4—21no—
139C3×C22⋊C4—21no—
140C3×C22⋊C4—21no—
141C3×C42—21no—
142C3×C42—21no—
143C3×C42—21no—
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42#4111yescharacteristic
148#208—22no—
149#208—41no—
150#208—42no—
151#208—82no—
152#208—81no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×#41C2×Dic311yescharacteristic
159C22×D4—21no—
160C22×D4—21no—
161C22×D4C2×Dic312yes—
162AGL(1,ℤ/8)—44no—
163C2×SD16—22no—
164C2×D8—22no—
165C2×M16—32no—
166C2×M16—62no—
167C2×M16—61no—
168C2×M16—61no—
169C2×M16C2×Dic312yes—
170#127—21no—
171#127—21no—
172#127—21no—
173#127—21no—
174C4×Q8#6711yescharacteristic
175C4×Q8#6711yescharacteristic
176C4×D4—21no—
177C4×D4—21no—
178C4×D4#6711yescharacteristic
179C4×D4#6711yescharacteristic
180#117—21no—
181#117C2×Dic311yescharacteristic
182C2×C42—21no—
183C2×C42C22×D311yescharacteristic
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—31no—
187C4⋊C8—31no—
188C4⋊C8—62no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#102—22no—
194#102—22no—
195#100—124no—
196C22⋊C8—62no—
197C22⋊C8—61no—
198#97—61no—
199C4×C8—62no—
200C4×C8—61no—
201C4×C8—61no—
202C23×C3—21no—
203C23×C3—44no—
204C23×C3—42no—
205C23×C3—82no—
206C23×C3—82no—
207C23×C3C4×D411yescharacteristic
208C3×Q8—21no—
209C3×Q8#10012yescomplemented
210C3×D4—21no—
211C3×D4—21no—
212C3×D4—42no—
213C3×D4—42no—
214C3×D4—42no—
215C3×D4—82no—
216C3×D4—82no—
217C3×D4—82no—
218C3×D4—81no—
219C3×D4—81no—
220C3×D4—81no—
221C3×D4—81no—
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—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—42no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4C2×C22⋊C411yescharacteristic
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12311yescharacteristic
242C3×C8—42no—
243C3⋊C8—42no—
244C3⋊C8—42no—
245C3⋊C8—41no—
246C3⋊C8—41no—
247C3⋊C8—81no—
248C24—22no—
249#41—22no—
250#41—41no—
251C2×Q8#21711yescharacteristic
252C2×D4—21no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—81no—
264C2×D4—81no—
265C2×D4#21711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4D3×D411yescharacteristic
270C22×C4#23711yescharacteristic
271C22×C4C22×Dic311yescharacteristic
272SD16—44no—
273D8—44no—
274M16—42no—
275M16—62no—
276M16—121no—
277M16—122no—
278M16—242no—
279M16—241no—
280C2×C8—22no—
281C2×C8—61no—
282C2×C8—61no—
283C2×C8—62no—
284C2×C8—121no—
285C2×C8—121no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4#21711yescharacteristic
289C4⋊C4#21711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42D3×D411yescharacteristic
297C42#23711yescharacteristic
298C42C2×#6711yescharacteristic
299C42C2×#6711yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—82no—
307C22×C3—82no—
308C22×C3—82no—
309C22×C3—164no—
310C22×C3#38611yescharacteristic
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—22no—
314C3×C4—21no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4C2×#10011yescharacteristic
318C3×C4#41311yescharacteristic
319C3×C4#45311yescharacteristic
320C3×C4#45411yescharacteristic
321C3×C4#47111yescharacteristic
322C3×C4#47211yescharacteristic
323C23—21no—
324C23—44no—
325C23—42no—
326C23—82no—
327C23—82no—
328C23D4×Dic311yescharacteristic
329Q8—21no—
330Q8#83312yescomplemented
331D4—21no—
332D4—21no—
333D4—42no—
334D4—42no—
335D4—42no—
336D4—82no—
337D4—82no—
338D4—82no—
339D4—81no—
340D4—81no—
341D4—81no—
342D4—81no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4D4×Dic311yescharacteristic
358C2×C4#93511yescharacteristic
359C2×C4#93711yescharacteristic
360C2×C4#93911yescharacteristic
361C2×C4#93911yescharacteristic
362C2×C4C2×#21711yescharacteristic
363C8—42no—
364C8—122no—
365C8—122no—
366C8—121no—
367C8—121no—
368C8—241no—
369C2×C3—41no—
370C2×C3—81no—
371C2×C3—162no—
372C2×C3#188311yescharacteristic
373C2×C3#190011yescharacteristic
374C2×C3#190711yescharacteristic
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22—164no—
385C22#522011yescharacteristic
386C4—21no—
387C4—22no—
388C4—21no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4#519811yescharacteristic
393C4#519911yescharacteristic
394C4#520011yescharacteristic
395C4#520111yescharacteristic
396C4C2×#83311yescharacteristic
397C4#523711yescharacteristic
398C3#936011yescharacteristic, complemented, Sylow
399C2—41no—
400C2—81no—
401C2—162no—
402C2#7168311yescharacteristic
403C2#7208811yescharacteristic
404C2#7209111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3038C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3038D3
hypocenterC3#9360
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
2. #82523C2semidirect
3. #71766C2semidirect
4. #71759C2semidirect
5. #71641C2semidirect
6. C3⋊#1309C2semidirect
7. #67710C2semidirect
209. C3×Q8#100semidirect
330. Q8#833semidirect
398. C3#9360semidirect

Maximal subgroups

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

subgroupindex
C3×#30382
#825232
#717662
#717592
#716412
C3⋊#13092
#677102
#93603