Finite Group Databasebeta

← #10692339

Subgroups — #10692339

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
1#82526C211yesmaximal, characteristic
2#82523C211yesmaximal, characteristic, complemented
3C3×#1489C211yesmaximal, characteristic, complemented
4#70502C211yesmaximal, characteristic, complemented
5#69876C211yesmaximal, characteristic, complemented
6C3⋊#1309C211yesmaximal, characteristic, complemented
7#67669C211yesmaximal, characteristic, complemented
8#9360—31nomaximal, Sylow
9C3×C4×#41C2211yescharacteristic
10C2×#949C412yes—
11C2×#874C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14C2×#823—22no—
15#5027—21no—
16#5027—21no—
17Q8×C3⋊C8—21no—
18#5015—21no—
19#5015—21no—
20#5012—21no—
21#5012—22no—
22#5001C412yes—
23#4925—21no—
24#4925C2211yescharacteristic
25#4924—21no—
26#4924C2211yescharacteristic
27#4924C2211yescharacteristic
28C3×Q8⋊C8—21no—
29C3×D4⋊C8—21no—
30#4485—21no—
31#4485—21no—
32#3038—31no—
33#2991—31no—
34#1965—31no—
35#1545—31no—
36#1489D311yescharacteristic, complemented
37#1309—31no—
38#1299—31no—
39C2×C3×#41C2×C411yescharacteristic
40C22×D12—21no—
41C22×D12—21no—
42C22×D12C2×C412yes—
43C2×C3×M16—21no—
44C2×C3×M16—21no—
45C3×C4×Q8D411yescharacteristic
46C3×C4×Q8D411yescharacteristic
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4D411yescharacteristic
50C3×C4×D4D411yescharacteristic
51C3×#117—21no—
52C3×#117C2×C411yescharacteristic
53C2×C3×C42—21no—
54C2×C3×C42C2311yescharacteristic
55#949—44no—
56C2×#215—22no—
57C2×#213—22no—
58C2×#208—22no—
59C2×#208C2×C412yes—
60C2×#208D412yes—
61#874—21no—
62#874—21no—
63#874—21no—
64#874—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×#97—21no—
68C3×C4×C8—21no—
69#830—22no—
70#830—21no—
71#823—44no—
72#809—22no—
73#809—22no—
74C4⋊(C3⋊C8)—22no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)D411yescharacteristic
80C4⋊(C3⋊C8)D411yescharacteristic
81C4×C3⋊C8—22no—
82C4×C3⋊C8—21no—
83C2×AGL(1,ℤ/8)—32no—
84C2×#127—31no—
85C4×#41C2×D311yescharacteristic
86#460—61no—
87#460—61no—
88#459—61no—
89#459—61no—
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#418—32no—
100C2×#100—62no—
101C4×M16C2×D311yescharacteristic
102#331—61no—
103#331—61no—
104Q8⋊C8—21no—
105D4⋊C8—21no—
106C23×D3—22no—
107C3×#41—22no—
108C3×#41—41no—
109C2×C3×Q8C22⋊C411yescharacteristic
110C2×C3×D4—21no—
111C2×C3×D4C22⋊C411yescharacteristic
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4C22×C411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C2×D12—22no—
119C2×D12—22no—
120C2×D12—42no—
121C2×D12—42no—
122C2×D12—42no—
123C2×D12—42no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—81no—
129C2×D12—81no—
130C3×M16—22no—
131C3×M16—41no—
132C3×M16—81no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4C22⋊C411yescharacteristic
138C3×C4⋊C4C22⋊C411yescharacteristic
139C3×C22⋊C4—21no—
140C3×C22⋊C4—21no—
141C3×C22⋊C4—21no—
142C3×C42—21no—
143C3×C42—21no—
144C3×C42—21no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#215—44no—
150#213—44no—
151#208—42no—
152#208—42no—
153#208—82no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—21no—
158C2×#41C4×D311yescharacteristic
159C22×D4—32no—
160C22×D4—61no—
161C22×D4—61no—
162AGL(1,ℤ/8)—124no—
163C2×SD16—62no—
164C2×D8—62no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—32no—
168C2×M16—32no—
169C2×M16—62no—
170#127—61no—
171#127—61no—
172#127—61no—
173#127—61no—
174C4×Q8D1211yescharacteristic
175C4×Q8#6711yescharacteristic
176C4×D4—21no—
177C4×D4—21no—
178C4×D4D1211yescharacteristic
179C4×D4#6711yescharacteristic
180#117—21no—
181#117C4×D311yescharacteristic
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—62no—
194#102—62no—
195#100—124no—
196C22⋊C8—62no—
197C22⋊C8—61no—
198#97—21no—
199C4×C8—21no—
200C4×C8—62no—
201C4×C8—61no—
202C23×C3—21no—
203C23×C3C4×D411yescharacteristic
204C22×D3—42no—
205C22×D3—44no—
206C22×D3—82no—
207C22×D3—82no—
208C3×Q8—21no—
209C3×Q8#10012yescomplemented
210C3×D4—21no—
211C3×D4—21no—
212C3×D4—42no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4C2×C22⋊C411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12311yescharacteristic
233D12—42no—
234D12—42no—
235D12—82no—
236D12—82no—
237D12—82no—
238D12—81no—
239D12—81no—
240D12—81no—
241D12—81no—
242C3×C8—42no—
243C3×C8—81no—
244C3⋊C8—42no—
245C3⋊C8—42no—
246C3⋊C8—41no—
247C3⋊C8—41no—
248C24—62no—
249#41—22no—
250#41—41no—
251C2×Q8(C2×D3)⋊C411yescharacteristic
252C2×D4—21no—
253C2×D4—62no—
254C2×D4—62no—
255C2×D4—122no—
256C2×D4—122no—
257C2×D4—122no—
258C2×D4—122no—
259C2×D4—121no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—121no—
263C2×D4—241no—
264C2×D4—241no—
265C2×D4(C2×D3)⋊C411yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4C2×C4×D311yescharacteristic
270C22×C4D3×D411yescharacteristic
271C22×C4#23711yescharacteristic
272SD16—124no—
273D8—124no—
274M16—22no—
275M16—41no—
276M16—81no—
277M16—122no—
278M16—122no—
279M16—242no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—61no—
283C2×C8—61no—
284C2×C8—62no—
285C2×C8—62no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4(C2×D3)⋊C411yescharacteristic
289C4⋊C4(C2×D3)⋊C411yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42C2×D1211yescharacteristic
297C42D3×D411yescharacteristic
298C42#23711yescharacteristic
299C42C2×#6711yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3#38611yescharacteristic
307C2×D3—82no—
308C2×D3—82no—
309C2×D3—82no—
310C2×D3—164no—
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—124no—
325C23—122no—
326C23—242no—
327C23—242no—
328C23#67⋊C411yescharacteristic
329Q8—21no—
330Q8#82312yescomplemented
331D4—21no—
332D4—21no—
333D4—42no—
334D4—122no—
335D4—122no—
336D4—242no—
337D4—242no—
338D4—242no—
339D4—241no—
340D4—241no—
341D4—241no—
342D4—241no—
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×C4#67⋊C411yescharacteristic
358C2×C4#88211yescharacteristic
359C2×C4#88611yescharacteristic
360C2×C4C2×(C2×D3)⋊C411yescharacteristic
361C2×C4#93911yescharacteristic
362C2×C4#93911yescharacteristic
363C8—42no—
364C8—81no—
365C8—122no—
366C8—122no—
367C8—121no—
368C8—121no—
369C2×C3—41no—
370C2×C3—81no—
371C2×C3#188311yescharacteristic
372C2×C3#190011yescharacteristic
373C2×C3#190711yescharacteristic
374D3—162no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—242no—
382C22—242no—
383C22—242no—
384C22—484no—
385C22#495811yescharacteristic
386C4—21no—
387C4—22no—
388C4—21no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4#482211yescharacteristic
393C4#482711yescharacteristic
394C4C2×#82311yescharacteristic
395C4#513211yescharacteristic
396C4#519911yescharacteristic
397C4#520311yescharacteristic
398C3#936011yescharacteristic, complemented, Sylow
399C2—41no—
400C2—81no—
401C2—482no—
402C2#6998511yescharacteristic
403C2#7030311yescharacteristic
404C2#7036611yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1489C2
socleC22×C3#386
prime radical#82526C2
cosocleC2×C3×C42C23
hypercenter#1489D3
hypocenterC3#9360
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1489

Complements

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

NH ≅ G/Ntype
2. #82523C2semidirect
3. C3×#1489C2semidirect
4. #70502C2semidirect
5. #69876C2semidirect
6. C3⋊#1309C2semidirect
7. #67669C2semidirect
36. #1489D3semidirect
209. C3×Q8#100semidirect
330. Q8#823semidirect
398. C3#9360semidirect

Maximal subgroups

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

subgroupindex
#825262
#825232
C3×#14892
#705022
#698762
C3⋊#13092
#676692
#93603