Finite Group Databasebeta

← #10692046

Subgroups — #10692046

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#79056C211yesmaximal, characteristic
2#79013C211yesmaximal, characteristic, complemented
3#70529C211yesmaximal, characteristic, complemented
4#70502C211yesmaximal, characteristic, complemented
5#70457C211yesmaximal, characteristic, complemented
6C3×#1292C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#9318—31nomaximal, Sylow
9C2×#908—21no—
10C2×#874C2211yescharacteristic
11C4×#235C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×#825—21no—
16#5131—21no—
17C2×#821—22no—
18C8×#67—22no—
19C4×C4⋊Dic3C2211yescharacteristic
20#4925C2211yescharacteristic
21#4848—21no—
22#4848C412yescomplemented
23#4843—21no—
24#4843C412yescomplemented
25#4834—21no—
26#4833—21no—
27D3×C4⋊C8—21no—
28#4830—21no—
29#4555—21no—
30#4485—21no—
31#4485—21no—
32#3038—31no—
33#2988—31no—
34#1982—31no—
35#1806—31no—
36#1545—31no—
37#1303—31no—
38#1292D311yescharacteristic, complemented
39C2×#235D411yescharacteristic
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C2×C3×M16—21no—
44C2×C3×M16—21no—
45C22×C3×C8D412yes—
46C2×C3×C42C2311yescharacteristic
47C4×#67—21no—
48C4×#67—21no—
49#926D411yescharacteristic
50C2×C4⋊Dic3—21no—
51C2×C4×Dic3—21no—
52#908—42no—
53#908—42no—
54C2×D24—22no—
55C2×#204—22no—
56C2×D3×C8—22no—
57#874—21no—
58#874—21no—
59#874—21no—
60#874C2×C412yes—
61C4×D12C2×C411yescharacteristic
62C4×D12D411yescharacteristic
63#872—21no—
64C42×D3—21no—
65C4×Dic6C2×C411yescharacteristic
66C4×Dic6D411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8C2×C412yes—
73C3⋊#95—21no—
74#830—21no—
75#825—42no—
76#821—22no—
77#821—22no—
78#821—44no—
79(C2×D3)⋊C8—22no—
80Dic3⋊C8—22no—
81C8×Dic3—22no—
82C4⋊(C3⋊C8)D411yescharacteristic
83C4⋊(C3⋊C8)D411yescharacteristic
84C4×C3⋊C8—21no—
85C2×AGL(1,ℤ/8)—61no—
86C2×#127—31no—
87C4×#41—31no—
88#460—32no—
89#460—61no—
90#459—32no—
91#459—61no—
92C8×Q8—61no—
93C8×D4—61no—
94C8×D4—62no—
95#431—31no—
96#431—61no—
97#423C2×D311yescharacteristic
98#423C2×D311yescharacteristic
99C2×C4⋊C8—61no—
100C2×C4⋊C8C2×D311yescharacteristic
101C2×#104—61no—
102#418—61no—
103C2×#102—62no—
104C4×C4⋊C4—31no—
105#339—61no—
106#331—61no—
107#331—61no—
108C23×D3—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C2×#67—21no—
113C22×Dic3—41no—
114#235—21no—
115#235—21no—
116#235D812yes—
117#235SD1612yes—
118C2×D12—22no—
119C2×D12—22no—
120C2×D12—22no—
121C2×D12—42no—
122C2×D12—42no—
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×D12C22⋊C411yescharacteristic
132C2×C4×D3—21no—
133C2×C4×D3—21no—
134C2×C4×D3—21no—
135C2×Dic6C22⋊C411yescharacteristic
136C3×M16—41no—
137C3×M16—41no—
138C3×M16—41no—
139C3×M16—41no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—42no—
143C2×C3×C8—41no—
144C3×C42C22×C411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42#4111yescharacteristic
148#217—21no—
149(C2×D3)⋊C4—21no—
150(C2×D3)⋊C4—21no—
151C4⋊Dic3—22no—
152C4⋊Dic3—41no—
153C4⋊Dic3C22⋊C411yescharacteristic
154C4⋊Dic3C22⋊C411yescharacteristic
155Dic3⋊C4—21no—
156Dic3⋊C4—21no—
157C4×Dic3—21no—
158C4×Dic3—21no—
159C4×Dic3—21no—
160C4×Dic3—41no—
161C4×Dic3—41no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164D24—44no—
165#204—44no—
166D3×C8—82no—
167C2×#41—31no—
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—21no—
177C22×C8—62no—
178C22×C8D1212yes—
179#127—32no—
180#127—61no—
181#127—61no—
182#127—61no—
183C4×Q8—31no—
184C4×Q8—31no—
185C4×D4—31no—
186C4×D4—31no—
187C4×D4—61no—
188C4×D4—61no—
189#117—31no—
190#117—61no—
191C2×C4⋊C4—61no—
192C2×C42—61no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—31no—
201C4⋊C8—31no—
202C4⋊C8—62no—
203C4⋊C8C4×D312yes—
204#104—122no—
205#102—62no—
206#102—62no—
207#102—124no—
208C22⋊C8—62no—
209C22⋊C8—61no—
210C4×C8—62no—
211C4×C8—61no—
212#95—61no—
213C23×C3#12111yescharacteristic
214C22×D3—21no—
215C22×D3—42no—
216C22×D3—82no—
217C22×D3—82no—
218C22×D3—81no—
219C22×D3—81no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—41no—
223C2×C3×C4C2×C22⋊C411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12311yescharacteristic
229C2×C3×C4C2×D811yescharacteristic
230C2×C3×C4C2×SD1611yescharacteristic
231C2×C3×C4#13511yescharacteristic
232C2×C3×C4#13511yescharacteristic
233#67—22no—
234#67—22no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—42no—
239C2×Dic3—81no—
240C2×Dic3—81no—
241D12—21no—
242D12—44no—
243D12—42no—
244D12—42no—
245D12—82no—
246D12—82no—
247D12—81no—
248D12—81no—
249D12—81no—
250D12—81no—
251D12#10212yescomplemented
252C4×D3—22no—
253C4×D3—22no—
254C4×D3—41no—
255C4×D3—41no—
256C4×D3—81no—
257Dic6—21no—
258Dic6#10212yescomplemented
259C3×C8—42no—
260C3×C8—42no—
261C3×C8—81no—
262C3⋊C8—41no—
263C3⋊C8—41no—
264C24—121no—
265#41—32no—
266#41—32no—
267#41—61no—
268#41—61no—
269C2×Q8—31no—
270C2×D4—31no—
271C2×D4—61no—
272C2×D4—62no—
273C2×D4—62no—
274C2×D4—62no—
275C2×D4—122no—
276C2×D4—122no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—242no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—121no—
289C22×C4C2×D1211yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4#23911yescharacteristic
292SD16—124no—
293D8—124no—
294M16—41no—
295M16—41no—
296M16—41no—
297M16—41no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—61no—
303C2×C8—61no—
304C2×C8—242no—
305C4⋊C4—31no—
306C4⋊C4—31no—
307C4⋊C4—62no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—121no—
311C22⋊C4—61no—
312C22⋊C4—61no—
313C22⋊C4—61no—
314C42—61no—
315C42—61no—
316C42—61no—
317C42—121no—
318C42—121no—
319C42C2×C4×D311yescharacteristic
320C42#23511yescharacteristic
321C42D3×D411yescharacteristic
322C42D3×D411yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3#38611yescharacteristic
326C22×C3#46511yescharacteristic
327C22×C3#46611yescharacteristic
328C2×D3—22no—
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—82no—
332C2×D3—82no—
333C2×D3—82no—
334C2×D3—162no—
335C2×D3—162no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4C2×#10211yescharacteristic
340C3×C4#41611yescharacteristic
341C3×C4#45311yescharacteristic
342C3×C4#45411yescharacteristic
343C3×C4#46511yescharacteristic
344C3×C4#46611yescharacteristic
345Dic3—22no—
346Dic3—41no—
347Dic3—41no—
348Dic3—82no—
349C23—61no—
350C23—122no—
351C23—242no—
352C23—242no—
353C23—241no—
354C23—241no—
355C23#89511yescharacteristic
356Q8—32no—
357Q8—61no—
358D4—32no—
359D4—62no—
360D4—62no—
361D4—61no—
362D4—124no—
363D4—122no—
364D4—122no—
365D4—242no—
366D4—242no—
367D4—241no—
368D4—241no—
369D4—241no—
370D4—241no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—61no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—122no—
382C2×C4—241no—
383C2×C4—241no—
384C2×C4—241no—
385C2×C4C4×D1211yescharacteristic
386C2×C4D3×C22⋊C411yescharacteristic
387C2×C4#88211yescharacteristic
388C2×C4#88411yescharacteristic
389C2×C4#89311yescharacteristic
390C2×C4#89411yescharacteristic
391C2×C4D3×D811yescharacteristic
392C2×C4D3×SD1611yescharacteristic
393C2×C4#91611yescharacteristic
394C2×C4#91911yescharacteristic
395C8—42no—
396C8—42no—
397C8—81no—
398C8—121no—
399C8—121no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3#188311yescharacteristic
403C2×C3#189311yescharacteristic
404C2×C3#190311yescharacteristic
405D3—42no—
406D3—162no—
407C22—21no—
408C22—21no—
409C22—62no—
410C22—121no—
411C22—121no—
412C22—242no—
413C22—242no—
414C22—242no—
415C22—482no—
416C22—482no—
417C22#466511yescharacteristic
418C22#486411yescharacteristic
419C22#488311yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—62no—
424C4—121no—
425C4—121no—
426C4—242no—
427C4#474711yescharacteristic
428C4#475311yescharacteristic
429C4D3×#10211yescharacteristic
430C4#477511yescharacteristic
431C4#480411yescharacteristic
432C4#480711yescharacteristic
433C3#931811yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—122no—
437C2—482no—
438C2#6872611yescharacteristic
439C2#6888311yescharacteristic
440C2#6900511yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1292C2
socleC22×C3#386
prime radical#79056C2
cosocleC2×C3×C42C23
hypercenter#1292D3
hypocenterC3#9318
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
2. #79013C2semidirect
3. #70529C2semidirect
4. #70502C2semidirect
5. #70457C2semidirect
6. C3×#1292C2semidirect
7. #67715C2semidirect
22. #4848C4semidirect
24. #4843C4semidirect
38. #1292D3semidirect
251. D12#102semidirect
258. Dic6#102semidirect
433. C3#9318semidirect

Maximal subgroups

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

subgroupindex
#790562
#790132
#705292
#705022
#704572
C3×#12922
#677152
#93183