Finite Group Databasebeta

← #11868002

Subgroups — #11868002

Order 1152.

1356 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#135C211yesmaximal, characteristic, complemented
2#10596293C212yesmaximal, complemented
3C2×#64110C211yesmaximal, characteristic, complemented
4#10592936C212yesmaximal, complemented
5#10592891C212yesmaximal, complemented
6#10592876C214yesmaximal, complemented
7#10592859C212yesmaximal, complemented
8C2×#63646C211yesmaximal, characteristic, complemented
9#83928—34nomaximal
10C2×C32×#41C2212yes—
11C2×#3828C2211yescharacteristic
12C32×#135—21no—
13C32×#135—21no—
14C32×#135—21no—
15C32×#135—21no—
16C2×C32×Q16C2211yescharacteristic, complemented
17C2×C32×SD16C2212yescomplemented
18C2×C32×D8C2211yescharacteristic, complemented
19C22×C32×C8C2211yescharacteristic
20#64160—22no—
21#64159—24no—
22#64158—22no—
23#64151C2214yes—
24#64141C2214yes—
25C2×#3748C2212yes—
26C2×#3748C2211yescharacteristic
27#64110—21no—
28#64110—21no—
29#64110C2212yes—
30#64110C2212yes—
31C2×#3743—21no—
32C2×C8×C3⋊D3—21no—
33#63663C2214yescomplemented
34#63660C2214yescomplemented
35#63646C2212yescomplemented
36#63646C2212yescomplemented
37C2×C3×#135D314yescomplemented
38#5822—38no—
39C2×#901—34no—
40#5237—38no—
41#5192—38no—
42#5177—316no—
43#5160—38no—
44C2×#819—34no—
45C32×#41—22no—
46C32×#41—22no—
47C32×#41—22no—
48C32×#41—22no—
49C2×C32×Q8C2312yes—
50C2×C32×D4—22no—
51C2×C32×D4C2312yes—
52C22×C32×C4—22no—
53C22×C32×C4C2311yescharacteristic
54C2×#638—21no—
55#3828—21no—
56#3828—21no—
57#3828D412yes—
58#3828C2312yes—
59C2×C4⋊(C3⋊D3)D411yescharacteristic
60C2×C4×C3⋊D3—21no—
61C2×#634D411yescharacteristic
62C32×Q16—21no—
63C32×Q16—21no—
64C32×SD16—22no—
65C32×SD16—22no—
66C32×D8—21no—
67C32×D8—21no—
68C2×C32×C8D411yescharacteristic
69C2×C32×C8D411yescharacteristic
70C2×C32×C8D411yescharacteristic
71C2×C32×C8D411yescharacteristic
72C2×C32×C8C2311yescharacteristic
73C2×C32×C8C2311yescharacteristic
74#3756—24no—
75#3755—24no—
76#3754—24no—
77#3753—24no—
78#3748—21no—
79#3748—22no—
80#3748C2312yes—
81#3748C2314yes—
82C2×C3⋊(C3⋊C8)—21no—
83C2×C3⋊(C3⋊C8)—22no—
84#3743—22no—
85#3743—22no—
86C8×C3⋊D3—22no—
87C8×C3⋊D3—22no—
88#3067—91noSylow
89C2×C3×#41C2×D318yes—
90C2×#235—34no—
91C3×#135—24no—
92C3×#135—24no—
93C3×#135—24no—
94C3×#135—24no—
95C2×C3×Q16C2×D314yescomplemented
96C2×C3×SD16C2×D318yescomplemented
97C2×C3×D8C2×D314yescomplemented
98C22×C3×C8C2×D314yes—
99#951—68no—
100#950—616no—
101#949—68no—
102#942—316no—
103#932—316no—
104C2×#208—34no—
105C2×#208—38no—
106#901—38no—
107#901—38no—
108#901—64no—
109#901—64no—
110C2×ΓL(1,𝔽25)—64no—
111C2×D3×C8—64no—
112#836—316no—
113#833—316no—
114#819—38no—
115#819—38no—
116C23×C32—22no—
117C23×C32#4111yescharacteristic
118C22×C3⋊D3—21no—
119C32×Q8—22no—
120C32×Q8—22no—
121C32×D4—22no—
122C32×D4—22no—
123C32×D4—42no—
124C32×D4—42no—
125C2×C32×C4—22no—
126C2×C32×C4—42no—
127C2×C32×C4—42no—
128C2×C32×C4C2×D411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4#4111yescharacteristic
133C2×C32×C4C2411yescharacteristic
134#638—22no—
135#638—22no—
136C2×C3⋊Dic3—21no—
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)C2×D412yes—
139C4×C3⋊D3—22no—
140C4×C3⋊D3—22no—
141#634—21no—
142#634C2×D412yes—
143C32×C8C2×D411yescharacteristic
144C32×C8C2×D411yescharacteristic
145C32×C8C2×D411yescharacteristic
146C32×C8C2×D411yescharacteristic
147C3⋊(C3⋊C8)—22no—
148C3⋊(C3⋊C8)—24no—
149#575—92no—
150C2×#135C3⋊D311yescharacteristic, complemented
151C2×#131—91no—
152#473—92no—
153#472—92no—
154#471—94no—
155C2×#108—91no—
156#413—92no—
157C3×#41—28no—
158C3×#41—28no—
159C3×#41—28no—
160C3×#41—28no—
161C2×C3×Q8C22×D318yes—
162C2×C3×D4—28no—
163C2×C3×D4C22×D318yes—
164C22×C3×C4—28no—
165C22×C3×C4C22×D314yes—
166C2×#67—64no—
167#235—38no—
168#235—38no—
169#235—64no—
170#235—64no—
171C2×D12—34no—
172C2×C4×D3—64no—
173C2×Dic6—34no—
174C3×Q16—24no—
175C3×Q16—24no—
176C3×SD16—28no—
177C3×SD16—28no—
178C3×D8—24no—
179C3×D8—24no—
180C2×C3×C8#6714yes—
181C2×C3×C8#6714yes—
182C2×C3×C8#6714yes—
183C2×C3×C8#6714yes—
184C2×C3×C8C22×D314yes—
185C2×C3×C8C22×D314yes—
186#216—616no—
187#215—616no—
188#214—616no—
189#213—616no—
190#208—316no—
191#208—38no—
192#208—68no—
193#208—64no—
194C2×C3⋊C8—68no—
195C2×C3⋊C8—64no—
196ΓL(1,𝔽25)—68no—
197ΓL(1,𝔽25)—68no—
198D3×C8—68no—
199D3×C8—68no—
200C22×C32—41no—
201C22×C32—42no—
202C22×C32—82no—
203C22×C32#12111yescharacteristic
204C22×C32#12111yescharacteristic
205C22×C32C2×#4111yescharacteristic
206C2×C3⋊D3—22no—
207C2×C3⋊D3—42no—
208C32×C4—42no—
209C32×C4#12111yescharacteristic
210C32×C4#12111yescharacteristic
211C32×C4C22×D411yescharacteristic
212C32×C4C22×D411yescharacteristic
213C3⋊Dic3—22no—
214C2×#41—91no—
215C2×#41C2×C3⋊D312yes—
216#137—94no—
217#137—182no—
218AGL(1,ℤ/8)—94no—
219AGL(1,ℤ/8)—182no—
220#135—21no—
221#135—21no—
222#135—21no—
223#135—21no—
224#135—184no—
225C2×Q16C2×C3⋊D311yescharacteristic, complemented
226C2×SD16C2×C3⋊D312yescomplemented
227C2×D8C2×C3⋊D311yescharacteristic, complemented
228#131—92no—
229#131—92no—
230#131—181no—
231#131—181no—
232C2×M16—91no—
233C2×M16—92no—
234C2×M16—181no—
235C22×C8—181no—
236C22×C8C2×C3⋊D311yescharacteristic
237#108—92no—
238#108—92no—
239#101—94no—
240#100—94no—
241C23×C3—28no—
242C23×C3#23714yes—
243C22×D3—64no—
244C3×Q8—28no—
245C3×Q8—28no—
246C3×D4—28no—
247C3×D4—28no—
248C3×D4—48no—
249C3×D4—48no—
250C2×C3×C4—28no—
251C2×C3×C4—48no—
252C2×C3×C4—48no—
253C2×C3×C4D3×D414yes—
254C2×C3×C4D3×D414yes—
255C2×C3×C4#23714yes—
256C2×C3×C4C2×#6714yes—
257C2×C3×C4C2×#6714yes—
258C2×C3×C4C23×D314yes—
259#67—68no—
260#67—68no—
261C2×Dic3—64no—
262D12—38no—
263D12—64no—
264C4×D3—68no—
265C4×D3—68no—
266Dic6—38no—
267Dic6—64no—
268C3×C8C2×#6714yes—
269C3×C8C2×#6714yes—
270C3×C8C2×#6714yes—
271C3×C8C2×#6714yes—
272C3⋊C8—616no—
273C3⋊C8—68no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—82no—
278C2×C32C2×#12111yescharacteristic
279C3⋊D3—42no—
280#41—22no—
281#41—22no—
282#41—22no—
283#41—22no—
284#41—92no—
285#41—92no—
286#41—181no—
287#41—181no—
288C2×Q8—91no—
289C2×Q8C22×C3⋊D312yes—
290C2×D4—22no—
291C2×D4—91no—
292C2×D4—181no—
293C2×D4C22×C3⋊D312yes—
294C22×C4—22no—
295C22×C4—181no—
296C22×C4C22×C3⋊D311yescharacteristic
297Q16—21no—
298Q16—21no—
299Q16—184no—
300SD16—22no—
301SD16—22no—
302SD16—184no—
303SD16—184no—
304D8—21no—
305D8—21no—
306D8—184no—
307M16—92no—
308M16—94no—
309M16—182no—
310M16—181no—
311M16—182no—
312M16—182no—
313C2×C8—182no—
314C2×C8—181no—
315C2×C8—182no—
316C2×C8—182no—
317C2×C8#63811yescharacteristic
318C2×C8#63811yescharacteristic
319C2×C8#63811yescharacteristic
320C2×C8#63811yescharacteristic
321C2×C8C22×C3⋊D311yescharacteristic
322C2×C8C22×C3⋊D311yescharacteristic
323C22×C3—44no—
324C22×C3—48no—
325C22×C3—88no—
326C22×C3#93814yes—
327C22×C3#93814yes—
328C22×C3C2×#23714yes—
329C2×D3—68no—
330C2×D3—128no—
331C3×C4—48no—
332C3×C4#93814yes—
333C3×C4#93814yes—
334C3×C4C2×D3×D414yes—
335C3×C4C22×#6714yes—
336Dic3—68no—
337C32#306711yescharacteristic, complemented, Sylow
338C23—22no—
339C23—181no—
340C23#383011yescharacteristic
341Q8—22no—
342Q8—22no—
343Q8—92no—
344Q8—181no—
345D4—22no—
346D4—22no—
347D4—42no—
348D4—42no—
349D4—92no—
350D4—182no—
351D4—182no—
352D4—181no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—181no—
357C2×C4—182no—
358C2×C4—182no—
359C2×C4D4×C3⋊D311yescharacteristic
360C2×C4D4×C3⋊D311yescharacteristic
361C2×C4#383011yescharacteristic
362C2×C4C2×#63811yescharacteristic
363C2×C4C2×#63811yescharacteristic
364C2×C4C23×C3⋊D311yescharacteristic
365C8—184no—
366C8—182no—
367C8C2×#63811yescharacteristic
368C8C2×#63811yescharacteristic
369C8C2×#63811yescharacteristic
370C8C2×#63811yescharacteristic
371C2×C3—24no—
372C2×C3—24no—
373C2×C3—24no—
374C2×C3—88no—
375C2×C3C2×#93814yes—
376D3—128no—
377C22—41no—
378C22—42no—
379C22—82no—
380C22—182no—
381C22—362no—
382C22#6414711yescharacteristic
383C22#6414711yescharacteristic
384C22C2×#383011yescharacteristic
385C4—42no—
386C4—182no—
387C4#6414711yescharacteristic
388C4#6414711yescharacteristic
389C4C2×D4×C3⋊D311yescharacteristic
390C4C22×#63811yescharacteristic
391C3#8392814yescomplemented
392C2—21no—
393C2—21no—
394C2—21no—
395C2—82no—
396C2—362no—
397C2C2×#6414711yescharacteristic
subgroupG/H
centerC4#64147
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC2×C32×#135C2
socleC2×C32C2×#121
hypercenterC2×#135C3⋊D3
hypocenterC32#3067
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×C32×#135C2semidirect
2. #10596293C2semidirect
3. C2×#64110C2semidirect
4. #10592936C2semidirect
5. #10592891C2semidirect
6. #10592876C2semidirect
7. #10592859C2semidirect
8. C2×#63646C2semidirect
16. C2×C32×Q16C22semidirect
17. C2×C32×SD16C22semidirect
18. C2×C32×D8C22semidirect
33. #63663C22semidirect
34. #63660C22semidirect
35. #63646C22semidirect
36. #63646C22semidirect
37. C2×C3×#135D3semidirect
95. C2×C3×Q16C2×D3semidirect
96. C2×C3×SD16C2×D3semidirect
97. C2×C3×D8C2×D3semidirect
150. C2×#135C3⋊D3semidirect
225. C2×Q16C2×C3⋊D3semidirect
226. C2×SD16C2×C3⋊D3semidirect
227. C2×D8C2×C3⋊D3semidirect
337. C32#3067semidirect
391. C3#83928semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1352
#105962932
C2×#641102
#105929362
#105928912
#105928762
#105928592
C2×#636462
#839283