Finite Group Databasebeta

← #10691085

Subgroups — #10691085

Order 768.

516 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82526C211yesmaximal, characteristic
2#82515C211yesmaximal, characteristic, complemented
3C3×#1477C211yesmaximal, characteristic, complemented
4#70774C211yesmaximal, characteristic, complemented
5C2×#4485C211yesmaximal, characteristic, complemented
6#67745C211yesmaximal, characteristic, complemented
7#67705C211yesmaximal, characteristic, complemented
8#9174—31nomaximal, Sylow
9C3×C4×#41C2211yescharacteristic
10C2×#949—21no—
11C2×#874C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5234—21no—
15#5027—21no—
16#5027C412yes—
17#5015—21no—
18#5015C412yes—
19#5001—21no—
20#5000—22no—
21#4997—21no—
22#4926—21no—
23#4925—21no—
24#4924C2211yescharacteristic
25#4924C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27C3×Q8⋊C8—21no—
28C3×D4⋊C8—21no—
29#4485—21no—
30#4485—21no—
31#4485—22no—
32#3038—31no—
33#2925—31no—
34#1545—31no—
35C2×#331—31no—
36#1477D311yescharacteristic, complemented
37#1301—31no—
38#1292—31no—
39C2×C3×#41D411yescharacteristic
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C22×C3×C8—21no—
44C3×C4×Q8C2×C411yescharacteristic
45C3×C4×Q8D411yescharacteristic
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4C2×C411yescharacteristic
49C3×C4×D4D411yescharacteristic
50C3×#117—21no—
51C3×#117D411yescharacteristic
52C2×C3×C42—21no—
53C2×C3×C42C2311yescharacteristic
54#949—42no—
55#949—42no—
56C2×#215—22no—
57C2×#213—22no—
58C2×#208—21no—
59C2×#208—21no—
60C22×C3⋊C8—21no—
61#874—21no—
62#874—21no—
63#874—21no—
64#874C2×C412yes—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C4×C8—21no—
70#830—22no—
71#830—22no—
72#809—22no—
73#809—22no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)C2×C412yes—
78C4⋊(C3⋊C8)D411yescharacteristic
79C4⋊(C3⋊C8)D411yescharacteristic
80C4⋊(C3⋊C8)D412yes—
81#803—22no—
82C4×C3⋊C8—22no—
83C2×AGL(1,ℤ/8)—61no—
84C2×#127—31no—
85C4×#41C2×D311yescharacteristic
86#460—32no—
87#460—61no—
88#459—32no—
89#459—61no—
90#433—62no—
91#432—61no—
92#431—61no—
93#431C2×D311yescharacteristic
94#423—31no—
95#423—31no—
96C2×C4⋊C8—31no—
97C2×C4⋊C8C2×D311yescharacteristic
98#418—61no—
99#408—61no—
100#405—61no—
101#331—61no—
102#331—61no—
103#331—62no—
104Q8⋊C8—21no—
105D4⋊C8—21no—
106C23×D3—41no—
107C3×#41—21no—
108C3×#41—21no—
109C3×#41—21no—
110C3×#41—21no—
111C2×C3×Q8C22⋊C411yescharacteristic
112C2×C3×D4—21no—
113C2×C3×D4C22⋊C411yescharacteristic
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C2×D12—22no—
121C2×D12—22no—
122C2×D12—22no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—41no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—82no—
131C2×D12—81no—
132C2×D12—81no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8—81no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4C22⋊C411yescharacteristic
139C3×C4⋊C4C22⋊C411yescharacteristic
140C3×C22⋊C4—21no—
141C3×C22⋊C4—21no—
142C3×C22⋊C4—21no—
143C3×C42—21no—
144C3×C42—21no—
145C3×C42—21no—
146C3×C42C22×C411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150#215—44no—
151#213—44no—
152#208—41no—
153#208—41no—
154#208—41no—
155#208—41no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×#41D1211yescharacteristic
163C22×D4—61no—
164C22×D4—61no—
165C22×D4—61no—
166AGL(1,ℤ/8)—122no—
167AGL(1,ℤ/8)—122no—
168C2×SD16—62no—
169C2×D8—62no—
170C2×M16—61no—
171C2×M16—61no—
172C22×C8—21no—
173C22×C8—61no—
174#127—32no—
175#127—61no—
176#127—61no—
177#127—61no—
178C4×Q8C4×D311yescharacteristic
179C4×Q8#6711yescharacteristic
180C4×D4—21no—
181C4×D4—21no—
182C4×D4C4×D311yescharacteristic
183C4×D4#6711yescharacteristic
184#117—21no—
185#117D1211yescharacteristic
186C2×C42—21no—
187C2×C42C22×D311yescharacteristic
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—32no—
192C4⋊C8—31no—
193C4⋊C8—31no—
194C4⋊C8—32no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198#102—62no—
199#102—62no—
200C22⋊C8—21no—
201C22⋊C8—62no—
202C22⋊C8—62no—
203#97—62no—
204C4×C8—21no—
205C4×C8—62no—
206C23×C3—21no—
207C23×C3#12311yescharacteristic
208C22×D3—42no—
209C22×D3—82no—
210C22×D3—82no—
211C22×D3—81no—
212C22×D3—81no—
213C3×Q8—21no—
214C3×Q8—21no—
215C3×D4—21no—
216C3×D4—21no—
217C3×D4—41no—
218C3×D4—41no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—81no—
230C2×C3×C4C2×C22⋊C411yescharacteristic
231C2×C3×C4C4×D411yescharacteristic
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#13511yescharacteristic
237C2×C3×C4#13511yescharacteristic
238C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
239C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
240D12—42no—
241D12—42no—
242D12—82no—
243D12—82no—
244D12—82no—
245D12—81no—
246D12—81no—
247D12—81no—
248D12—81no—
249C3×C8—81no—
250C3×C8—81no—
251C3⋊C8—42no—
252C3⋊C8—41no—
253C3⋊C8—41no—
254C3⋊C8—42no—
255C24—121no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41—21no—
260C2×Q8(C2×D3)⋊C411yescharacteristic
261C2×D4—21no—
262C2×D4—62no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—241no—
273C2×D4—241no—
274C2×D4—242no—
275C2×D4(C2×D3)⋊C411yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4C2×D1211yescharacteristic
280C22×C4#23711yescharacteristic
281C22×C4#23711yescharacteristic
282SD16—124no—
283D8—124no—
284M16—121no—
285M16—121no—
286M16—121no—
287M16—121no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—61no—
291C2×C8—62no—
292C2×C8—61no—
293C2×C8—62no—
294C2×C8—81no—
295C2×C8—121no—
296C2×C8—121no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4(C2×D3)⋊C411yescharacteristic
300C4⋊C4(C2×D3)⋊C411yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42C2×C4×D311yescharacteristic
308C42D3×D411yescharacteristic
309C42D3×D411yescharacteristic
310C42C2×#6711yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3#38611yescharacteristic
317C22×C3#48211yescharacteristic
318C22×C3#48211yescharacteristic
319C2×D3—82no—
320C2×D3—82no—
321C2×D3—82no—
322C2×D3—162no—
323C2×D3—162no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—41no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4#41611yescharacteristic
331C3×C4#41811yescharacteristic
332C3×C4#44911yescharacteristic
333C3×C4#44911yescharacteristic
334C3×C4#47111yescharacteristic
335C3×C4#47211yescharacteristic
336C23—21no—
337C23—122no—
338C23—242no—
339C23—242no—
340C23—241no—
341C23—241no—
342C23#88611yescharacteristic
343Q8—21no—
344Q8—21no—
345D4—21no—
346D4—21no—
347D4—41no—
348D4—41no—
349D4—122no—
350D4—122no—
351D4—242no—
352D4—242no—
353D4—242no—
354D4—241no—
355D4—241no—
356D4—241no—
357D4—241no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—81no—
369C2×C4#67⋊C411yescharacteristic
370C2×C4#67⋊C411yescharacteristic
371C2×C4#88211yescharacteristic
372C2×C4#90411yescharacteristic
373C2×C4#90411yescharacteristic
374C2×C4#90811yescharacteristic
375C2×C4#90811yescharacteristic
376C2×C4C2×(C2×D3)⋊C411yescharacteristic
377C2×C4#93911yescharacteristic
378C2×C4#93911yescharacteristic
379C8—81no—
380C8—81no—
381C8—122no—
382C8—121no—
383C8—121no—
384C8—122no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—81no—
388C2×C3#187511yescharacteristic
389C2×C3#190311yescharacteristic
390C2×C3#190711yescharacteristic
391D3—162no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—242no—
398C22—242no—
399C22—242no—
400C22—482no—
401C22—482no—
402C22#473711yescharacteristic
403C22#473711yescharacteristic
404C22#495811yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4#473411yescharacteristic
412C4#473411yescharacteristic
413C4#511311yescharacteristic
414C4#513111yescharacteristic
415C4#519911yescharacteristic
416C4#520311yescharacteristic
417C3#917411yescharacteristic, complemented, Sylow
418C2—21no—
419C2—21no—
420C2—81no—
421C2—482no—
422C2#6993311yescharacteristic
423C2#7030311yescharacteristic
424C2#7040011yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1477C2
socleC22×C3#386
prime radical#82526C2
cosocleC2×C3×C42C23
hypercenter#1477D3
hypocenterC3#9174
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
2. #82515C2semidirect
3. C3×#1477C2semidirect
4. #70774C2semidirect
5. C2×#4485C2semidirect
6. #67745C2semidirect
7. #67705C2semidirect
36. #1477D3semidirect
417. C3#9174semidirect

Maximal subgroups

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

subgroupindex
#825262
#825152
C3×#14772
#707742
C2×#44852
#677452
#677052
#91743