Finite Group Databasebeta

← #10690411

Subgroups — #10690411

Order 768.

538 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83291C211yesmaximal, characteristic
2#79004C211yesmaximal, characteristic, complemented
3C3×#1480C211yesmaximal, characteristic, complemented
4#71818C211yesmaximal, characteristic, complemented
5#70789C211yesmaximal, characteristic, complemented
6#67739C211yesmaximal, characteristic, complemented
7#67704C211yesmaximal, characteristic, complemented
8#9053—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#901—21no—
11#5477C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×#837—21no—
15C2×#837—21no—
16#5110—21no—
17#5088C411yescharacteristic, complemented
18#5078C411yescharacteristic, complemented
19#5073C411yescharacteristic, complemented
20#5070C411yescharacteristic, complemented
21#5038—22no—
22C4×#213—22no—
23C4×C4⋊Dic3C2211yescharacteristic
24C4×#208C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26#4838—21no—
27#4836—22no—
28#4834—21no—
29C3×D4⋊C8—21no—
30C3×D4⋊C8—21no—
31#4470—22no—
32#3169—31no—
33#2935—31no—
34#1980—31no—
35#1502—31no—
36#1480D311yescharacteristic, complemented
37#1279—31no—
38#1278—31no—
39C2×C3×#41—21no—
40C2×#235—21no—
41C22×C3×C8—21no—
42C3×C4⋊Q8C2×C411yescharacteristic
43C3×#127C2×C411yescharacteristic
44C3×#124—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#213—22no—
51#930—21no—
52#930D412yes—
53C2×C4⋊Dic3—21no—
54C2×C4×Dic3—21no—
55C2×#208D411yescharacteristic
56C2×#208D411yescharacteristic
57C22×C3⋊C8—21no—
58#901—42no—
59#901—42no—
60C2×ΓL(1,𝔽25)—22no—
61C2×D3×C8—22no—
62#875—21no—
63#874C2×C411yescharacteristic
64C4×D12—21no—
65C4×D12D412yes—
66#869C2×C411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8D411yescharacteristic
70C3×C4⋊C8D411yescharacteristic
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73#837—42no—
74#837—42no—
75#832—22no—
76C3⋊#95—21no—
77(C2×D3)⋊C8—22no—
78(C2×D3)⋊C8—22no—
79#809—22no—
80#807—22no—
81#803—21no—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84C4×C3⋊C8C2×C411yescharacteristic
85C4×C3⋊C8C2×C411yescharacteristic
86C2×#131—61no—
87#532—31no—
88#532C2×D311yescharacteristic
89#500—31no—
90#493—31no—
91#486—31no—
92#486—31no—
93#466—62no—
94C4×D8—62no—
95#436—61no—
96#435—62no—
97C8×D4—61no—
98#431C2×D311yescharacteristic
99C2×C4⋊C8C2×D311yescharacteristic
100C2×#104—61no—
101C2×#104—61no—
102#408—61no—
103C4×M16—31no—
104C2×C4×C8—31no—
105C4×C4⋊C4—31no—
106#327—62no—
107D4⋊C8—21no—
108D4⋊C8—21no—
109C3×#41—41no—
110C3×#41—41no—
111C3×#41—41no—
112C3×#41—41no—
113C2×C3×Q8—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—41no—
118C22×C3×C4—41no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C2×#67—22no—
123C22×Dic3—41no—
124#235—41no—
125#235—41no—
126#235—41no—
127#235—41no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12C22⋊C412yes—
131C2×C4×D3—22no—
132C2×Dic6—21no—
133C2×C3×C8—21no—
134C2×C3×C8—21no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C22⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C42C22×C411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146#213—44no—
147(C2×D3)⋊C4—22no—
148(C2×D3)⋊C4—22no—
149C4⋊Dic3—21no—
150C4⋊Dic3—22no—
151C4⋊Dic3—41no—
152C4⋊Dic3C22⋊C412yes—
153C4×Dic3—41no—
154C4×Dic3—41no—
155#208—21no—
156#208—21no—
157#208—21no—
158#208D812yescomplemented
159C2×C3⋊C8—21no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—42no—
164ΓL(1,𝔽25)—44no—
165D3×C8—44no—
166C2×#41—21no—
167C2×#41—61no—
168C2×D8—62no—
169#131—122no—
170#131—122no—
171C2×M16—31no—
172C2×M16—31no—
173C2×M16—62no—
174C22×C8—21no—
175C22×C8—62no—
176C22×C8—61no—
177C4⋊Q8—31no—
178C4⋊Q8C4×D311yescharacteristic
179#127—31no—
180#127C4×D311yescharacteristic
181#124—21no—
182#124—61no—
183#121—21no—
184#121—21no—
185#121—32no—
186#121—61no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—32no—
190C4×D4—61no—
191C2×C4⋊C4—61no—
192C2×C42—61no—
193C2×C42C22×D311yescharacteristic
194#107—62no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8#6711yescharacteristic
198C4⋊C8#6711yescharacteristic
199#104—122no—
200#104—122no—
201#102—62no—
202#102—62no—
203C22⋊C8—21no—
204C22⋊C8—62no—
205C22⋊C8—62no—
206#97—61no—
207C4×C8—21no—
208C4×C8—31no—
209C4×C8—31no—
210C4×C8—61no—
211C4×C8—61no—
212#95—61no—
213C23×C3—41no—
214C23×C3#12111yescharacteristic
215C22×D3—22no—
216C3×Q8—41no—
217C3×Q8—41no—
218C3×D4—42no—
219C3×D4—41no—
220C3×D4—41no—
221C3×D4—41no—
222C3×D4—41no—
223C3×D4—81no—
224C3×D4—81no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4C2×C22⋊C411yescharacteristic
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12311yescharacteristic
237C2×C3×C4C2×D811yescharacteristic
238C2×C3×C4#13511yescharacteristic
239C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
240C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
241#67—42no—
242#67—42no—
243C2×Dic3—22no—
244C2×Dic3—42no—
245C2×Dic3—81no—
246C2×Dic3—81no—
247D12—42no—
248D12—41no—
249D12—41no—
250D12—41no—
251D12—41no—
252C4×D3—42no—
253C4×D3—42no—
254Dic6—41no—
255Dic6—41no—
256C3×C8—41no—
257C3×C8—41no—
258C3×C8—81no—
259C3⋊C8—22no—
260C3⋊C8—22no—
261C3⋊C8—41no—
262C3⋊C8—41no—
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—121no—
268#41—121no—
269#41—121no—
270#41—121no—
271C2×Q8—21no—
272C2×Q8—61no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—32no—
277C2×D4—41no—
278C2×D4—62no—
279C2×D4—61no—
280C2×D4—61no—
281C22×C4—41no—
282C22×C4—62no—
283C22×C4—121no—
284C22×C4#23511yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287D8—124no—
288M16—32no—
289M16—61no—
290M16—61no—
291M16—61no—
292M16—124no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—61no—
299C2×C8—61no—
300C2×C8—61no—
301C2×C8—61no—
302C2×C8—124no—
303C2×C8—122no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—32no—
307C4⋊C4—62no—
308C4⋊C4—61no—
309C4⋊C4—121no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C42—121no—
315C42—121no—
316C42C2×C4×D311yescharacteristic
317C42D3×D411yescharacteristic
318C42#23711yescharacteristic
319C42C2×#6711yescharacteristic
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3#38611yescharacteristic
326C22×C3#45911yescharacteristic
327C22×C3#46311yescharacteristic
328C2×D3—42no—
329C2×D3—42no—
330C2×D3—42no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—81no—
335C3×C4C4×D811yescharacteristic
336C3×C4#44211yescharacteristic
337C3×C4#44311yescharacteristic
338C3×C4#44411yescharacteristic
339C3×C4#44711yescharacteristic
340C3×C4#44911yescharacteristic
341Dic3—42no—
342Dic3—82no—
343C23—41no—
344C23—62no—
345C23#88411yescharacteristic
346Q8—41no—
347Q8—41no—
348Q8—121no—
349Q8—121no—
350D4—42no—
351D4—41no—
352D4—41no—
353D4—41no—
354D4—41no—
355D4—81no—
356D4—81no—
357D4—122no—
358D4—122no—
359D4—122no—
360D4—121no—
361D4—121no—
362D4—121no—
363D4—121no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—62no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—241no—
375C2×C4—241no—
376C2×C4D3×C22⋊C411yescharacteristic
377C2×C4#67⋊C411yescharacteristic
378C2×C4#88311yescharacteristic
379C2×C4D3×D811yescharacteristic
380C2×C4#91111yescharacteristic
381C2×C4#91411yescharacteristic
382C2×C4#91611yescharacteristic
383C2×C4C4×#6711yescharacteristic
384C2×C4#93711yescharacteristic
385C2×C4#93811yescharacteristic
386C8—41no—
387C8—41no—
388C8—62no—
389C8—62no—
390C8—81no—
391C8—121no—
392C8—121no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—161no—
396C2×C3#187611yescharacteristic
397C2×C3#187911yescharacteristic
398C2×C3#188411yescharacteristic
399D3—82no—
400C22—21no—
401C22—21no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—122no—
406C22—122no—
407C22—122no—
408C22#478611yescharacteristic
409C22#478711yescharacteristic
410C22#495611yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—81no—
415C4—122no—
416C4—242no—
417C4#475611yescharacteristic
418C4#478511yescharacteristic
419C4#514411yescharacteristic
420C4#514511yescharacteristic
421C4#515611yescharacteristic
422C4#517211yescharacteristic
423C3#905311yescharacteristic, complemented, Sylow
424C2—21no—
425C2—21no—
426C2—161no—
427C2—242no—
428C2#7001111yescharacteristic
429C2#7020811yescharacteristic
430C2#7041711yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1480C2
socleC22×C3#386
prime radical#83291C2
cosocleC2×C3×C42C23
hypercenter#1480D3
hypocenterC3#9053
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1480

Complements

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

NH ≅ G/Ntype
2. #79004C2semidirect
3. C3×#1480C2semidirect
4. #71818C2semidirect
5. #70789C2semidirect
6. #67739C2semidirect
7. #67704C2semidirect
17. #5088C4semidirect
18. #5078C4semidirect
19. #5073C4semidirect
20. #5070C4semidirect
36. #1480D3semidirect
158. #208D8semidirect
423. C3#9053semidirect

Maximal subgroups

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

subgroupindex
#832912
#790042
C3×#14802
#718182
#707892
#677392
#677042
#90533