Finite Group Databasebeta

← #10713356

Subgroups — #10713356

Order 768.

446 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83786C211yesmaximal, characteristic
2C2×C3×#350C211yesmaximal, characteristic, complemented
3C2×#4561C211yesmaximal, characteristic, complemented
4C2×#4515C211yesmaximal, characteristic, complemented
5Dic3×M32C211yesmaximal, characteristic, complemented
6C2×#4508C211yesmaximal, characteristic, complemented
7C2×Dic3⋊C16C211yesmaximal, characteristic, complemented
8#13193—31nomaximal, Sylow
9C2×C3×#131C2211yescharacteristic
10#5832C411yescharacteristic
11C2×#901C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C22×C3×C16C2211yescharacteristic
14#5234C411yescharacteristic
15#5129C411yescharacteristic
16#5128—21no—
17#5128—21no—
18#5118C411yescharacteristic
19C8×#67—21no—
20C8×#67—21no—
21C2×C8×Dic3C2211yescharacteristic
22C2×#812C2211yescharacteristic
23C22×C3⋊C16C2211yescharacteristic
24C3×#350—22no—
25#4561—22no—
26#4515—22no—
27#4508—22no—
28#4502—21no—
29Dic3⋊C16—21no—
30Dic3⋊C16—21no—
31Dic3×C16—21no—
32#2931—31no—
33C2×C4⋊C16—31no—
34C2×#350—31no—
35C2×#350—31no—
36C2×#350—31no—
37C2×#350D311yescharacteristic, complemented
38C4×M32—31no—
39C2×C3×#41C2×C411yescharacteristic
40C2×#235C2×C411yescharacteristic
41C3×#131—21no—
42C3×#131—21no—
43C3×#131—21no—
44C3×#131—21no—
45C2×C3×M16—21no—
46C2×C3×M16C2×C411yescharacteristic
47C22×C3×C8—21no—
48C22×C3×C8C2311yescharacteristic
49#947C811yescharacteristic
50#944C811yescharacteristic
51#940C811yescharacteristic
52#939—21no—
53#939—21no—
54#936C811yescharacteristic
55C4×#67—21no—
56C4×#67—21no—
57C2×C4×Dic3C2×C411yescharacteristic
58C2×#208C2×C411yescharacteristic
59C22×C3⋊C8C2×C411yescharacteristic
60#901—21no—
61#901—21no—
62#901—21no—
63#901—21no—
64C2×ΓL(1,𝔽25)—21no—
65C2×D3×C8—21no—
66C3×M32—21no—
67C3×M32D412yescomplemented
68C2×C3×C16—21no—
69C2×C3×C16—21no—
70C2×C3×C16—21no—
71C2×C3×C16—21no—
72#830—21no—
73#830—21no—
74(C2×D3)⋊C8—21no—
75(C2×D3)⋊C8—21no—
76Dic3⋊C8—21no—
77Dic3⋊C8—21no—
78C8×Dic3D411yescharacteristic
79C8×Dic3D411yescharacteristic
80C8×Dic3D411yescharacteristic
81C8×Dic3D411yescharacteristic
82#812—21no—
83#812D412yescomplemented
84C2×C3⋊C16—21no—
85C2×C3⋊C16—21no—
86C2×C3⋊C16—21no—
87C2×C3⋊C16—21no—
88C2×#131—31no—
89C2×#131C2×D311yescharacteristic
90#532—31no—
91C2×M32—31no—
92C2×M32C2×D311yescharacteristic
93C22×C16—31no—
94C22×C16C2×D311yescharacteristic
95#436—61no—
96#436—61no—
97C8×D4—61no—
98C8×D4—61no—
99#423—31no—
100#408—31no—
101#408—31no—
102C2×C4×C8—31no—
103C4⋊C16—61no—
104C4⋊C16—61no—
105#350—22no—
106#350—62no—
107#350—62no—
108#350—62no—
109#346—61no—
110C4×C16—61no—
111C3×#41—21no—
112C3×#41—21no—
113C3×#41—21no—
114C3×#41—21no—
115C2×C3×Q8C2×C811yescharacteristic
116C2×C3×D4—21no—
117C2×C3×D4C2×C811yescharacteristic
118C22×C3×C4—21no—
119C22×C3×C4C22×C411yescharacteristic
120C2×#67—21no—
121C2×#67—21no—
122C2×#67—21no—
123C22×Dic3M1611yescharacteristic
124C22×Dic3M1611yescharacteristic
125#235—21no—
126#235—21no—
127#235—21no—
128#235—21no—
129C2×D12C2×C811yescharacteristic
130C2×C4×D3—21no—
131C2×Dic6C2×C811yescharacteristic
132C3×M16—21no—
133C3×M16—21no—
134C3×M16—41no—
135C3×M16—41no—
136C2×C3×C8—21no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C2×C3×C8C2×D411yescharacteristic
140C2×C3×C8C2×D411yescharacteristic
141C2×C3×C8C2×D411yescharacteristic
142C2×C3×C8C2×D411yescharacteristic
143C2×C3×C8#4111yescharacteristic
144C2×C3×C8#4111yescharacteristic
145#217—21no—
146#217—21no—
147(C2×D3)⋊C4—21no—
148(C2×D3)⋊C4—21no—
149Dic3⋊C4—21no—
150Dic3⋊C4—21no—
151C4×Dic3C22⋊C411yescharacteristic
152C4×Dic3C22⋊C411yescharacteristic
153C4×Dic3C2×C811yescharacteristic
154C4×Dic3C2×C811yescharacteristic
155#208—21no—
156#208—21no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8C22⋊C411yescharacteristic
161C2×C3⋊C8C22⋊C411yescharacteristic
162ΓL(1,𝔽25)—41no—
163ΓL(1,𝔽25)—41no—
164D3×C8—41no—
165D3×C8—41no—
166C3×C16—22no—
167C3×C16—41no—
168C3⋊C16—22no—
169C3⋊C16—41no—
170C2×#41—31no—
171C2×#41C4×D311yescharacteristic
172#131—21no—
173#131—21no—
174#131—21no—
175#131—21no—
176#131—61no—
177#131—61no—
178#131—61no—
179#131—61no—
180C2×M16—21no—
181C2×M16—31no—
182C2×M16—61no—
183C2×M16C4×D311yescharacteristic
184C22×C8—21no—
185C22×C8—31no—
186C22×C8—61no—
187C22×C8C22×D311yescharacteristic
188C4⋊Q8—31no—
189#127—31no—
190#124—31no—
191#124—31no—
192#121—61no—
193#121—61no—
194C4×D4—61no—
195C4×D4—61no—
196C2×C42—31no—
197M32—21no—
198M32—32no—
199M32—61no—
200M32#6712yescomplemented
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—61no—
206C2×C16—61no—
207C2×C16—61no—
208C2×C16—61no—
209C4⋊C8—61no—
210C4⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C4×C8—31no—
216C4×C8—31no—
217C4×C8—31no—
218C4×C8—31no—
219C23×C3—21no—
220C23×C3C2×M1611yescharacteristic
221C22×D3—21no—
222C3×Q8—21no—
223C3×Q8—21no—
224C3×D4—21no—
225C3×D4—21no—
226C3×D4—41no—
227C3×D4—41no—
228C2×C3×C4—21no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4C2×C22⋊C411yescharacteristic
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4C22×C811yescharacteristic
235C2×C3×C4#13111yescharacteristic
236C2×C3×C4#13111yescharacteristic
237#67—42no—
238#67—41no—
239#67—41no—
240#67—42no—
241C2×Dic3—22no—
242C2×Dic3—21no—
243C2×Dic3—41no—
244C2×Dic3C22⋊C811yescharacteristic
245C2×Dic3C22⋊C811yescharacteristic
246C2×Dic3C22⋊C811yescharacteristic
247C2×Dic3C22⋊C811yescharacteristic
248D12—21no—
249D12—21no—
250C4×D3—41no—
251C4×D3—41no—
252Dic6—21no—
253Dic6—21no—
254C3×C8—41no—
255C3×C8#12111yescharacteristic
256C3×C8#12111yescharacteristic
257C3×C8#12411yescharacteristic
258C3×C8#12711yescharacteristic
259C3⋊C8—21no—
260C3⋊C8—21no—
261C3⋊C8—41no—
262#41—21no—
263#41—21no—
264#41—21no—
265#41—21no—
266#41—61no—
267#41—61no—
268#41—61no—
269#41—61no—
270C2×Q8—31no—
271C2×Q8D3×C811yescharacteristic
272C2×D4—21no—
273C2×D4—31no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C2×D4D3×C811yescharacteristic
278C22×C4—21no—
279C22×C4—31no—
280C22×C4—31no—
281C22×C4—61no—
282C22×C4C2×C4×D311yescharacteristic
283M16—21no—
284M16—21no—
285M16—41no—
286M16—41no—
287M16—61no—
288M16—61no—
289M16—121no—
290M16—121no—
291C2×C8—21no—
292C2×C8—31no—
293C2×C8—31no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—61no—
297C2×C8—61no—
298C2×C8—61no—
299C2×C8—121no—
300C2×C8—121no—
301C2×C8#23511yescharacteristic
302C2×C8D3×D411yescharacteristic
303C2×C8D3×D411yescharacteristic
304C2×C8D3×D411yescharacteristic
305C2×C8#23711yescharacteristic
306C2×C8C2×#6711yescharacteristic
307C4⋊C4—61no—
308C4⋊C4—61no—
309C22⋊C4—61no—
310C22⋊C4—61no—
311C22⋊C4—61no—
312C22⋊C4—61no—
313C42—31no—
314C42—31no—
315C42—31no—
316C42—31no—
317C16—22no—
318C16—41no—
319C16—62no—
320C16—121no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3C2×C22⋊C811yescharacteristic
327C22×C3#43611yescharacteristic
328C22×C3#43611yescharacteristic
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—41no—
332C3×C4—41no—
333C3×C4#39011yescharacteristic
334C3×C4#40811yescharacteristic
335C3×C4C8×D411yescharacteristic
336C3×C4C8×D411yescharacteristic
337Dic3—21no—
338Dic3—21no—
339Dic3—21no—
340Dic3—41no—
341Dic3#35012yescomplemented
342C23—21no—
343C23—61no—
344C23D3×M1611yescharacteristic
345Q8—21no—
346Q8—21no—
347Q8—61no—
348Q8—61no—
349D4—21no—
350D4—21no—
351D4—41no—
352D4—41no—
353D4—61no—
354D4—61no—
355D4—122no—
356D4—122no—
357D4—121no—
358D4—121no—
359C2×C4—21no—
360C2×C4—31no—
361C2×C4—31no—
362C2×C4—31no—
363C2×C4—31no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—62no—
367C2×C4—61no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4D3×C22⋊C411yescharacteristic
372C2×C4#67⋊C411yescharacteristic
373C2×C4C2×D3×C811yescharacteristic
374C2×C4#90111yescharacteristic
375C2×C4#90711yescharacteristic
376C2×C4C4×#6711yescharacteristic
377C8—41no—
378C8—61no—
379C8—61no—
380C8—121no—
381C8#88411yescharacteristic
382C8#88511yescharacteristic
383C8#93911yescharacteristic
384C8#94011yescharacteristic
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—81no—
388C2×C3#195411yescharacteristic
389C2×C3C2×#35011yescharacteristic
390C2×C3#211711yescharacteristic
391D3—81no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—121no—
398C22—121no—
399C22—121no—
400C22D3×C22⋊C811yescharacteristic
401C22#472911yescharacteristic
402C22#512811yescharacteristic
403C4—32no—
404C4—41no—
405C4—61no—
406C4—61no—
407C4—61no—
408C4—121no—
409C4#67⋊C811yescharacteristic
410C4#472711yescharacteristic
411C4#495711yescharacteristic
412C4C8×#6711yescharacteristic
413C3#1319311yescharacteristic, complemented, Sylow
414C2—21no—
415C2—21no—
416C2—81no—
417C2—241no—
418C2D3×#35011yescharacteristic
419C2#6912511yescharacteristic
420C2#6991611yescharacteristic
subgroupG/H
centerC2×C8D3×D4
commutatorC2×C3×C4C22×C8
FrattiniC22×C8C22×D3
FittingC2×C3×#350C2
socleC22×C3C2×C22⋊C8
prime radical#5832C4
cosocleC22×C3×C8C23
hypercenterC2×#350D3
hypocenterC3#13193
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
2. C2×C3×#350C2semidirect
3. C2×#4561C2semidirect
4. C2×#4515C2semidirect
5. Dic3×M32C2semidirect
6. C2×#4508C2semidirect
7. C2×Dic3⋊C16C2semidirect
37. C2×#350D3semidirect
67. C3×M32D4semidirect
83. #812D4semidirect
200. M32#67semidirect
341. Dic3#350semidirect
413. C3#13193semidirect

Maximal subgroups

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

subgroupindex
#837862
C2×C3×#3502
C2×#45612
C2×#45152
Dic3×M322
C2×#45082
C2×Dic3⋊C162
#131933