Finite Group Databasebeta

← #10692045

Subgroups — #10692045

Order 768.

580 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82526C211yesmaximal, characteristic
2#82520C211yesmaximal, characteristic, complemented
3C3×#1806C211yesmaximal, characteristic, complemented
4#70502C211yesmaximal, characteristic, complemented
5#69487C211yesmaximal, characteristic, complemented
6#67711C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9318—31nomaximal, Sylow
9C3×C4×#41C2211yescharacteristic
10C2×#949—21no—
11C2×#874C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×#104—21no—
14C3×C4×C4⋊C4C2211yescharacteristic
15#5027—21no—
16#5027C412yescomplemented
17Q8×C3⋊C8—21no—
18#5015—21no—
19#5015C412yescomplemented
20D4×C3⋊C8—21no—
21D4×C3⋊C8—22no—
22#5001—21no—
23C2×#809—22no—
24#4925—21no—
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27C2×C4⋊(C3⋊C8)—21no—
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29C3×#339—21no—
30#4485—21no—
31#4485—21no—
32#3038—31no—
33#2988—31no—
34#1982—31no—
35#1806D311yescharacteristic, complemented
36#1545—31no—
37#1303—31no—
38#1292—31no—
39C2×C3×#41D411yescharacteristic
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C2×C3×M16—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×C4⋊C4—21no—
53C2×C3×C42—21no—
54C2×C3×C42—21no—
55C2×C3×C42C2311yescharacteristic
56#949—42no—
57#949—42no—
58C2×#215—22no—
59C2×#213—22no—
60C2×#208—21no—
61C22×C3⋊C8—22no—
62C22×C3⋊C8D412yes—
63#874—21no—
64#874—21no—
65#874—21no—
66#874C2×C412yes—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×#104—42no—
70C3×#95—21no—
71#830—22no—
72#830—21no—
73#809—22no—
74#809—22no—
75#809—44no—
76C4⋊(C3⋊C8)—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)C2×C412yes—
81C4⋊(C3⋊C8)D411yescharacteristic
82C4⋊(C3⋊C8)D411yescharacteristic
83C4×C3⋊C8—22no—
84C4×C3⋊C8—21no—
85C2×AGL(1,ℤ/8)—61no—
86C2×#127—31no—
87C4×#41C2×D311yescharacteristic
88#460—32no—
89#460—61no—
90#459—32no—
91#459—61no—
92C8×Q8—61no—
93C8×D4—62no—
94C8×D4—61no—
95#431—31no—
96#431—61no—
97#423—31no—
98#423C2×D311yescharacteristic
99C2×C4⋊C8—31no—
100C2×C4⋊C8—61no—
101C2×#104—21no—
102#418—61no—
103C2×#102—62no—
104C4×C4⋊C4C2×D311yescharacteristic
105#339—21no—
106#331—61no—
107#331—61no—
108C23×D3—41no—
109C3×#41—21no—
110C3×#41—21no—
111C3×#41D812yes—
112C3×#41SD1612yes—
113C2×C3×Q8C22⋊C411yescharacteristic
114C2×C3×D4—21no—
115C2×C3×D4C22⋊C411yescharacteristic
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—41no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C2×D12—22no—
124C2×D12—22no—
125C2×D12—22no—
126C2×D12—42no—
127C2×D12—42no—
128C2×D12—41no—
129C2×D12—41no—
130C2×D12—41no—
131C2×D12—41no—
132C2×D12—41no—
133C2×D12—41no—
134C2×D12—41no—
135C2×D12—82no—
136C3×M16—41no—
137C3×M16—41no—
138C2×C3×C8—41no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4C22⋊C411yescharacteristic
144C3×C4⋊C4C22⋊C411yescharacteristic
145C3×C22⋊C4—21no—
146C3×C22⋊C4—21no—
147C3×C22⋊C4—21no—
148C3×C42—21no—
149C3×C42—21no—
150C3×C42—21no—
151C3×C42—41no—
152C3×C42—41no—
153C3×C42C22×C411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4111yescharacteristic
157#215—44no—
158#213—44no—
159#208—41no—
160#208—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—42no—
166C2×C3⋊C8—82no—
167C2×#41#6711yescharacteristic
168C22×D4—61no—
169C22×D4—61no—
170C22×D4—61no—
171AGL(1,ℤ/8)—122no—
172AGL(1,ℤ/8)—122no—
173C2×SD16—62no—
174C2×D8—62no—
175C2×M16—21no—
176C2×M16—61no—
177C22×C8—32no—
178C22×C8—62no—
179#127—32no—
180#127—61no—
181#127—61no—
182#127—61no—
183C4×Q8C4×D311yescharacteristic
184C4×Q8D1211yescharacteristic
185C4×D4—21no—
186C4×D4—21no—
187C4×D4C4×D311yescharacteristic
188C4×D4D1211yescharacteristic
189#117—21no—
190#117#6711yescharacteristic
191C2×C4⋊C4—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42C22×D311yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—32no—
198C4⋊C8—31no—
199C4⋊C8—31no—
200C4⋊C8—62no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204#104—42no—
205#102—62no—
206#102—62no—
207#102—124no—
208C22⋊C8—62no—
209C22⋊C8—61no—
210C4×C8—62no—
211C4×C8—61no—
212#95—21no—
213C23×C3—21no—
214C23×C3#12111yescharacteristic
215C22×D3—42no—
216C22×D3—82no—
217C22×D3—82no—
218C22×D3—81no—
219C22×D3—81no—
220C3×Q8—21no—
221C3×Q8#10212yescomplemented
222C3×D4—22no—
223C3×D4—22no—
224C3×D4—21no—
225C3×D4#10212yescomplemented
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—81no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4C2×C22⋊C411yescharacteristic
241C2×C3×C4C4×D411yescharacteristic
242C2×C3×C4C4×D411yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12311yescharacteristic
246C2×C3×C4C2×D811yescharacteristic
247C2×C3×C4C2×SD1611yescharacteristic
248C2×C3×C4#13511yescharacteristic
249C2×C3×C4#13511yescharacteristic
250D12—44no—
251D12—42no—
252D12—42no—
253D12—82no—
254D12—82no—
255D12—81no—
256D12—81no—
257D12—81no—
258D12—81no—
259C3×C8—81no—
260C3⋊C8—42no—
261C3⋊C8—42no—
262C3⋊C8—41no—
263C3⋊C8—41no—
264C24—121no—
265#41—21no—
266#41—21no—
267#41#21312yes—
268#41#21512yes—
269C2×Q8(C2×D3)⋊C411yescharacteristic
270C2×D4—21no—
271C2×D4—62no—
272C2×D4—62no—
273C2×D4—62no—
274C2×D4—122no—
275C2×D4—122no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—242no—
284C2×D4(C2×D3)⋊C411yescharacteristic
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4D3×D411yescharacteristic
290C22×C4#23711yescharacteristic
291C22×C4C2×#6711yescharacteristic
292SD16—124no—
293D8—124no—
294M16—41no—
295M16—41no—
296M16—121no—
297M16—121no—
298C2×C8—41no—
299C2×C8—61no—
300C2×C8—61no—
301C2×C8—62no—
302C2×C8—62no—
303C2×C8—122no—
304C2×C8—242no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—41no—
309C4⋊C4(C2×D3)⋊C411yescharacteristic
310C4⋊C4(C2×D3)⋊C411yescharacteristic
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—41no—
318C42—41no—
319C42C2×C4×D311yescharacteristic
320C42C2×D1211yescharacteristic
321C42D3×D411yescharacteristic
322C42#23711yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—22no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3#38611yescharacteristic
329C22×C3#46511yescharacteristic
330C22×C3#46611yescharacteristic
331C2×D3—82no—
332C2×D3—82no—
333C2×D3—82no—
334C2×D3—162no—
335C2×D3—162no—
336C3×C4—21no—
337C3×C4—22no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—41no—
341C3×C4—41no—
342C3×C4—82no—
343C3×C4C2×#10211yescharacteristic
344C3×C4#41611yescharacteristic
345C3×C4#45311yescharacteristic
346C3×C4#45411yescharacteristic
347C3×C4#46511yescharacteristic
348C3×C4#46611yescharacteristic
349C23—21no—
350C23—122no—
351C23—242no—
352C23—242no—
353C23—241no—
354C23—241no—
355C23#93911yescharacteristic
356Q8—21no—
357Q8#80912yescomplemented
358D4—22no—
359D4—22no—
360D4—21no—
361D4—124no—
362D4—122no—
363D4—122no—
364D4—242no—
365D4—242no—
366D4—241no—
367D4—241no—
368D4—241no—
369D4—241no—
370D4#80912yescomplemented
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4#67⋊C411yescharacteristic
386C2×C4#67⋊C411yescharacteristic
387C2×C4#88211yescharacteristic
388C2×C4#88611yescharacteristic
389C2×C4C2×(C2×D3)⋊C411yescharacteristic
390C2×C4C2×#21311yescharacteristic
391C2×C4#93911yescharacteristic
392C2×C4C2×#21511yescharacteristic
393C2×C4#95011yescharacteristic
394C2×C4#95011yescharacteristic
395C8—81no—
396C8—122no—
397C8—122no—
398C8—121no—
399C8—121no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—42no—
403C2×C3#188311yescharacteristic
404C2×C3#189311yescharacteristic
405C2×C3#190311yescharacteristic
406D3—162no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—242no—
413C22—242no—
414C22—242no—
415C22—482no—
416C22—482no—
417C22#495811yescharacteristic
418C22#503811yescharacteristic
419C22#504811yescharacteristic
420C4—21no—
421C4—21no—
422C4—22no—
423C4—21no—
424C4—41no—
425C4—41no—
426C4—82no—
427C4#482211yescharacteristic
428C4#482711yescharacteristic
429C4C2×#80911yescharacteristic
430C4#500211yescharacteristic
431C4#503811yescharacteristic
432C4#504811yescharacteristic
433C3#931811yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—42no—
437C2—482no—
438C2#6951511yescharacteristic
439C2#7036611yescharacteristic
440C2#7040011yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1806C2
socleC22×C3#386
prime radical#82526C2
cosocleC2×C3×C42C23
hypercenter#1806D3
hypocenterC3#9318
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #82520C2semidirect
3. C3×#1806C2semidirect
4. #70502C2semidirect
5. #69487C2semidirect
6. #67711C2semidirect
7. #67620C2semidirect
16. #5027C4semidirect
19. #5015C4semidirect
35. #1806D3semidirect
221. C3×Q8#102semidirect
225. C3×D4#102semidirect
357. Q8#809semidirect
370. D4#809semidirect
433. C3#9318semidirect

Maximal subgroups

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

subgroupindex
#825262
#825202
C3×#18062
#705022
#694872
#677112
#676202
#93183