Finite Group Databasebeta

← #10692050

Subgroups — #10692050

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#83165C211yesmaximal, characteristic
2#79013C211yesmaximal, characteristic, complemented
3C3×#1982C211yesmaximal, characteristic, complemented
4#71759C211yesmaximal, characteristic, complemented
5#69494C211yesmaximal, characteristic, complemented
6#67705C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9318—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×#932—21no—
11C4×#235C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×#102—22no—
14C3×C4×C4⋊C4C2211yescharacteristic
15#5215—21no—
16C8×#67—22no—
17#5075—21no—
18#5075C412yescomplemented
19#5072—21no—
20#5072C412yescomplemented
21C2×#805—21no—
22#4925C2211yescharacteristic
23#4924C2211yescharacteristic
24#4924C2211yescharacteristic
25#4834—21no—
26#4833—21no—
27D3×C4⋊C8—21no—
28#4830—21no—
29#4544—21no—
30#4544—21no—
31#4523—21no—
32#3038—31no—
33#2988—31no—
34#1982D311yescharacteristic, complemented
35#1806—31no—
36#1545—31no—
37#1303—31no—
38#1292—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C2×#235D411yescharacteristic
43C22×C3×C8D412yes—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127—21no—
47C3×#127C2×C412yes—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#214—22no—
52#932—42no—
53#932—42no—
54C2×#213—22no—
55C4×#67—21no—
56C4×#67—21no—
57#926D411yescharacteristic
58C2×#208—21no—
59C2×#208—21no—
60C2×D3×C8—22no—
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×#102—44no—
70C3×#95—21no—
71#832—22no—
72#832—22no—
73#830—21no—
74(C2×D3)⋊C8—22no—
75Dic3⋊C8—22no—
76C8×Dic3—22no—
77#805—42no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)C2×C412yes—
82C4⋊(C3⋊C8)D411yescharacteristic
83C4⋊(C3⋊C8)D411yescharacteristic
84C4×C3⋊C8—21no—
85C2×AGL(1,ℤ/8)—61no—
86C2×#127C2×D311yescharacteristic
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#423—31no—
98#423—31no—
99C2×C4⋊C8—61no—
100C2×C4⋊C8C2×D311yescharacteristic
101C2×#104—61no—
102#418—61no—
103C2×#102—22no—
104C4×C4⋊C4C2×D311yescharacteristic
105#339—61no—
106#331—61no—
107#331—61no—
108C24×C3—41no—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—22no—
112C2×C3×D4—42no—
113C2×C3×D4—42no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—82no—
122C22×C3×C4—41no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C2×#67—21no—
127#235—21no—
128#235—21no—
129#235D812yes—
130#235SD1612yes—
131C2×D12C22⋊C411yescharacteristic
132C2×C4×D3—21no—
133C2×C4×D3—21no—
134C2×C4×D3—21no—
135C2×Dic6C22⋊C411yescharacteristic
136C2×C3×C8—22no—
137C2×C3×C8—42no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—41no—
140C3×C42—41no—
141C3×C42—41no—
142C3×C42C22×C411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146#217—21no—
147#214—44no—
148#213—44no—
149(C2×D3)⋊C4—21no—
150(C2×D3)⋊C4—21no—
151C4⋊Dic3C22⋊C411yescharacteristic
152C4⋊Dic3C22⋊C411yescharacteristic
153Dic3⋊C4—21no—
154Dic3⋊C4—21no—
155C4×Dic3—21no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158#208—41no—
159#208—41no—
160#208—41no—
161#208—41no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—41no—
166D3×C8—82no—
167C2×#41—31no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171AGL(1,ℤ/8)—122no—
172AGL(1,ℤ/8)—122no—
173C2×SD16—62no—
174C2×D8—62no—
175C2×M16—61no—
176C2×M16—61no—
177C22×C8—62no—
178C22×C8#6712yes—
179#127—21no—
180#127—21no—
181#127—21no—
182#127C4×D312yes—
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—21no—
192C2×C42—21no—
193C2×C42—61no—
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—122no—
205#102—44no—
206#102—62no—
207#102—62no—
208C22⋊C8—62no—
209C22⋊C8—61no—
210C4×C8—62no—
211C4×C8—61no—
212#95—21no—
213C23×C3—42no—
214C23×C3—82no—
215C23×C3—82no—
216C23×C3—81no—
217C23×C3—81no—
218C23×C3#12111yescharacteristic
219C22×D3—21no—
220C3×D4—44no—
221C3×D4—42no—
222C3×D4—42no—
223C3×D4—82no—
224C3×D4—82no—
225C3×D4—81no—
226C3×D4—81no—
227C3×D4—81no—
228C3×D4—81no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4C2×C22⋊C411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12311yescharacteristic
241C2×C3×C4C2×D811yescharacteristic
242C2×C3×C4C2×SD1611yescharacteristic
243C2×C3×C4#13511yescharacteristic
244C2×C3×C4#13511yescharacteristic
245#67—22no—
246#67—22no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250D12—21no—
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—41no—
262C3⋊C8—41no—
263C3⋊C8—81no—
264C24—41no—
265#41—32no—
266#41—32no—
267#41—61no—
268#41—61no—
269C2×Q8—31no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—31no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—61no—
284C2×D4—82no—
285C22×C4—41no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4D3×D411yescharacteristic
290C22×C4#23711yescharacteristic
291C22×C4C2×#6711yescharacteristic
292SD16—124no—
293D8—124no—
294M16—121no—
295M16—121no—
296M16—121no—
297M16—121no—
298C2×C8—22no—
299C2×C8—42no—
300C2×C8—62no—
301C2×C8—61no—
302C2×C8—61no—
303C2×C8—121no—
304C2×C8—242no—
305C4⋊C4—22no—
306C4⋊C4—31no—
307C4⋊C4—31no—
308C4⋊C4—41no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C22⋊C4—61no—
312C22⋊C4—61no—
313C22⋊C4—61no—
314C42—41no—
315C42—41no—
316C42—61no—
317C42—61no—
318C42—61no—
319C42C2×C4×D311yescharacteristic
320C42#23511yescharacteristic
321C42D3×D411yescharacteristic
322C42D3×D411yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3—82no—
328C22×C3—162no—
329C22×C3—162no—
330C22×C3#38611yescharacteristic
331C22×C3#46511yescharacteristic
332C22×C3#46611yescharacteristic
333C2×D3—22no—
334C2×D3—41no—
335C2×D3—41no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—82no—
340C3×C4C2×#10211yescharacteristic
341C3×C4#41611yescharacteristic
342C3×C4#45311yescharacteristic
343C3×C4#45411yescharacteristic
344C3×C4#46511yescharacteristic
345C3×C4#46611yescharacteristic
346Dic3—22no—
347Dic3—41no—
348Dic3—41no—
349C23—42no—
350C23—61no—
351C23—82no—
352C23—82no—
353C23—81no—
354C23—81no—
355C23#93811yescharacteristic
356Q8—32no—
357Q8—61no—
358D4—32no—
359D4—44no—
360D4—42no—
361D4—42no—
362D4—62no—
363D4—62no—
364D4—61no—
365D4—82no—
366D4—82no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—81no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—61no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—241no—
385C2×C4D3×C22⋊C411yescharacteristic
386C2×C4#67⋊C411yescharacteristic
387C2×C4#88311yescharacteristic
388C2×C4#88411yescharacteristic
389C2×C4D3×D811yescharacteristic
390C2×C4#91211yescharacteristic
391C2×C4D3×SD1611yescharacteristic
392C2×C4#91611yescharacteristic
393C2×C4C4×#6711yescharacteristic
394C2×C4#93711yescharacteristic
395C8—42no—
396C8—122no—
397C8—121no—
398C8—121no—
399C8—241no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—162no—
403C2×C3#188311yescharacteristic
404C2×C3#189311yescharacteristic
405C2×C3#190311yescharacteristic
406D3—42no—
407C22—21no—
408C22—21no—
409C22—62no—
410C22—82no—
411C22—82no—
412C22—82no—
413C22—121no—
414C22—121no—
415C22—162no—
416C22—162no—
417C22#495611yescharacteristic
418C22#515711yescharacteristic
419C22#517111yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—62no—
424C4—82no—
425C4—121no—
426C4—121no—
427C4D3×#10211yescharacteristic
428C4#477211yescharacteristic
429C4#477511yescharacteristic
430C4#477811yescharacteristic
431C4#506211yescharacteristic
432C4#507711yescharacteristic
433C3#931811yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—122no—
437C2—162no—
438C2#6966311yescharacteristic
439C2#7020211yescharacteristic
440C2#7032911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1982C2
socleC22×C3#386
prime radical#83165C2
cosocleC2×C3×C42C23
hypercenter#1982D3
hypocenterC3#9318
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. #79013C2semidirect
3. C3×#1982C2semidirect
4. #71759C2semidirect
5. #69494C2semidirect
6. #67705C2semidirect
7. #67620C2semidirect
18. #5075C4semidirect
20. #5072C4semidirect
34. #1982D3semidirect
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
#831652
#790132
C3×#19822
#717592
#694942
#677052
#676202
#93183