Finite Group Databasebeta

← #11868172

Subgroups — #11868172

Order 1152.

1338 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64160C211yesmaximal, characteristic, complemented
2#10596293C211yesmaximal, characteristic
3C2×#64118C211yesmaximal, characteristic, complemented
4#10596216C211yesmaximal, characteristic, complemented
5C32×#413C211yesmaximal, characteristic, complemented
6C2×#63651C211yesmaximal, characteristic, complemented
7#10592823C211yesmaximal, characteristic, complemented
8#10592454C214yesmaximal, complemented
9#10592447C214yesmaximal, complemented
10#78721—34nomaximal
11C2×C32×#41C2211yescharacteristic
12C2×#3828C2211yescharacteristic
13C22×#634C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
15C2×C32×M16C2211yescharacteristic
16#64160—21no—
17#64160—22no—
18#64160—22no—
19#64159—22no—
20#64158—21no—
21C2×#3756C2212yescomplemented
22#64151C2212yes—
23C2×#3754C2212yescomplemented
24#64141C2212yes—
25C2×#3748C2211yescharacteristic
26C2×#3748C2211yescharacteristic
27#64118—22no—
28#64118C2214yes—
29#64116C2214yes—
30M16×C3⋊D3—22no—
31C2×#3746—21no—
32C2×#3744—21no—
33#64110—22no—
34C32×#101C2212yescomplemented
35C32×#100C2212yescomplemented
36#63651C2214yescomplemented
37#63649C2214yescomplemented
38C2×#951—34no—
39#5822—34no—
40C2×#909—34no—
41#5745—34no—
42C3×#413D314yescomplemented
43C2×#824—34no—
44#5124—34no—
45#4755—316no—
46#4748—316no—
47C32×#41—21no—
48C32×#41—21no—
49C32×#41—21no—
50C32×#41—21no—
51C2×C32×Q8C2311yescharacteristic
52C2×C32×D4—21no—
53C2×C32×D4C2311yescharacteristic
54C22×C32×C4—21no—
55C22×C32×C4C2311yescharacteristic
56C2×#638—21no—
57C22×C3⋊Dic3—21no—
58#3828—21no—
59#3828—21no—
60#3828D412yes—
61#3828C2312yes—
62C2×C4⋊(C3⋊D3)D411yescharacteristic
63C2×C4×C3⋊D3—21no—
64C2×#634—21no—
65C2×#634—21no—
66C2×#634—21no—
67C2×#634—21no—
68C2×#634—21no—
69C2×#634D411yescharacteristic
70C2×#634C2312yes—
71C32×M16—21no—
72C32×M16—21no—
73C32×M16C2312yes—
74C32×M16C2312yes—
75C2×C32×C8—21no—
76C2×C32×C8—21no—
77#3756—22no—
78#3756—24no—
79#3755—22no—
80#3754—22no—
81#3754—24no—
82#3753—22no—
83#3748—21no—
84#3748D412yes—
85#3748D412yes—
86#3748C2312yes—
87C2×C3⋊(C3⋊C8)—21no—
88C2×C3⋊(C3⋊C8)C2312yes—
89#3746—24no—
90#3744—24no—
91#3743—24no—
92C8×C3⋊D3—24no—
93#3069—91noSylow
94C2×C3×#41C2×D314yes—
95C2×#235—34no—
96C22×Dic6—34no—
97C2×C3×M16C2×D314yes—
98C2×C3×M16C2×D314yes—
99#951—68no—
100#951—68no—
101#951—64no—
102#950—68no—
103#949—64no—
104C2×#216—38no—
105#942—38no—
106C2×#214—38no—
107#932—38no—
108C2×#208—34no—
109C2×#208—34no—
110#909—316no—
111#909—68no—
112#907—316no—
113D3×M16—68no—
114C2×Dic12—64no—
115C2×#204—64no—
116#901—68no—
117C3×#101C2×D318yescomplemented
118C3×#100C2×D318yescomplemented
119#824—316no—
120#822—316no—
121C23×C32—21no—
122C23×C32#4111yescharacteristic
123C22×C3⋊D3—21no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×D4—21no—
127C32×D4—21no—
128C32×D4—41no—
129C32×D4—41no—
130C2×C32×C4—21no—
131C2×C32×C4—41no—
132C2×C32×C4—41no—
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4#4111yescharacteristic
138C2×C32×C4C2411yescharacteristic
139#638—22no—
140#638—22no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3—41no—
144C2×C3⋊Dic3—41no—
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)C2×D412yes—
147C4×C3⋊D3—22no—
148C4×C3⋊D3—22no—
149#634—21no—
150#634—22no—
151#634—22no—
152#634—41no—
153#634—41no—
154#634C2×D412yes—
155C32×C8—22no—
156C32×C8—22no—
157C3⋊(C3⋊C8)—22no—
158C3⋊(C3⋊C8)C2×D414yes—
159#575—91no—
160C2×#137—91no—
161C2×#137—91no—
162#568—91no—
163#473—94no—
164#471—94no—
165#430—91no—
166#413C3⋊D311yescharacteristic, complemented
167C2×#101—91no—
168C3×#41—24no—
169C3×#41—24no—
170C3×#41—24no—
171C3×#41—24no—
172C2×C3×Q8C22×D314yes—
173C2×C3×D4—24no—
174C2×C3×D4C22×D314yes—
175C22×C3×C4—24no—
176C22×C3×C4C22×D314yes—
177C2×#67—64no—
178C22×Dic3—64no—
179#235—38no—
180#235—38no—
181#235—64no—
182#235—64no—
183C2×D12—34no—
184C2×C4×D3—64no—
185C2×Dic6—34no—
186C2×Dic6—38no—
187C2×Dic6—64no—
188C2×Dic6—64no—
189C2×Dic6—64no—
190C2×Dic6—64no—
191C2×Dic6—64no—
192C3×M16—24no—
193C3×M16—24no—
194C3×M16C22×D318yes—
195C3×M16C22×D318yes—
196C2×C3×C8—24no—
197C2×C3×C8—24no—
198#216—616no—
199#216—68no—
200#215—68no—
201#214—616no—
202#214—68no—
203#213—68no—
204#208—38no—
205#208—38no—
206#208—38no—
207#208—64no—
208C2×C3⋊C8—38no—
209C2×C3⋊C8—64no—
210Dic12—616no—
211#204—616no—
212ΓL(1,𝔽25)—616no—
213D3×C8—616no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32—81no—
217C22×C32#12111yescharacteristic
218C22×C32#12111yescharacteristic
219C22×C32C2×#4111yescharacteristic
220C2×C3⋊D3—22no—
221C2×C3⋊D3—81no—
222C32×C4—41no—
223C32×C4#12111yescharacteristic
224C32×C4#12111yescharacteristic
225C32×C4C22×D411yescharacteristic
226C32×C4C22×D411yescharacteristic
227C3⋊Dic3—22no—
228C3⋊Dic3—42no—
229C2×#41—91no—
230C2×#41C2×C3⋊D311yescharacteristic
231C22×Q8—91no—
232#137—92no—
233#137—94no—
234#137—182no—
235#137—182no—
236#137—182no—
237#137—181no—
238AGL(1,ℤ/8)—92no—
239AGL(1,ℤ/8)—181no—
240#135—182no—
241C2×Q16—92no—
242C2×Q16—181no—
243C2×SD16—92no—
244C2×SD16—181no—
245#131—94no—
246#131—182no—
247C2×M16—91no—
248C2×M16—91no—
249C2×M16—182no—
250C2×M16C2×C3⋊D311yescharacteristic
251C2×M16C2×C3⋊D311yescharacteristic
252#108—94no—
253#101—94no—
254#101C2×C3⋊D312yescomplemented
255#100C2×C3⋊D312yescomplemented
256C23×C3—24no—
257C23×C3#23514yes—
258C22×D3—64no—
259C3×Q8—24no—
260C3×Q8—24no—
261C3×D4—24no—
262C3×D4—24no—
263C3×D4—44no—
264C3×D4—44no—
265C2×C3×C4—24no—
266C2×C3×C4—44no—
267C2×C3×C4—44no—
268C2×C3×C4#23514yes—
269C2×C3×C4D3×D414yes—
270C2×C3×C4D3×D414yes—
271C2×C3×C4D3×D414yes—
272C2×C3×C4D3×D414yes—
273C2×C3×C4C23×D314yes—
274#67—68no—
275#67—68no—
276C2×Dic3—64no—
277C2×Dic3—68no—
278C2×Dic3—124no—
279C2×Dic3—124no—
280D12—38no—
281D12—64no—
282C4×D3—68no—
283C4×D3—68no—
284Dic6—38no—
285Dic6—64no—
286Dic6—68no—
287Dic6—68no—
288Dic6—124no—
289Dic6—124no—
290C3×C8—28no—
291C3×C8—28no—
292C3⋊C8—316no—
293C3⋊C8—68no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—81no—
298C2×C32C2×#12111yescharacteristic
299C3⋊D3—42no—
300#41—21no—
301#41—21no—
302#41—21no—
303#41—21no—
304#41—92no—
305#41—92no—
306#41—181no—
307#41—181no—
308C2×Q8—92no—
309C2×Q8—91no—
310C2×Q8—181no—
311C2×Q8—181no—
312C2×Q8—181no—
313C2×Q8—181no—
314C2×Q8—181no—
315C2×Q8C22×C3⋊D311yescharacteristic
316C2×D4—21no—
317C2×D4—91no—
318C2×D4—181no—
319C2×D4C22×C3⋊D311yescharacteristic
320C22×C4—21no—
321C22×C4—181no—
322C22×C4—181no—
323C22×C4C22×C3⋊D311yescharacteristic
324Q16—184no—
325Q16—184no—
326Q16—182no—
327SD16—184no—
328SD16—184no—
329SD16—182no—
330SD16—182no—
331D8—182no—
332M16—21no—
333M16—21no—
334M16—92no—
335M16—92no—
336M16—92no—
337M16—184no—
338M16—181no—
339M16C22×C3⋊D312yes—
340M16C22×C3⋊D312yes—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—92no—
344C2×C8—184no—
345C2×C8—181no—
346C22×C3—44no—
347C22×C3—44no—
348C22×C3—84no—
349C22×C3#88414yes—
350C22×C3#88414yes—
351C22×C3C2×#23514yes—
352C2×D3—68no—
353C2×D3—244no—
354C3×C4—44no—
355C3×C4#88414yes—
356C3×C4#88414yes—
357C3×C4C2×D3×D414yes—
358C3×C4C2×D3×D414yes—
359Dic3—68no—
360Dic3—128no—
361C32#306911yescharacteristic, complemented, Sylow
362C23—21no—
363C23—181no—
364C23#382811yescharacteristic
365Q8—21no—
366Q8—21no—
367Q8—92no—
368Q8—181no—
369Q8—182no—
370Q8—182no—
371Q8—361no—
372Q8—361no—
373D4—21no—
374D4—21no—
375D4—41no—
376D4—41no—
377D4—92no—
378D4—182no—
379D4—182no—
380D4—181no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—181no—
385C2×C4—182no—
386C2×C4—182no—
387C2×C4—182no—
388C2×C4—361no—
389C2×C4—361no—
390C2×C4#382811yescharacteristic
391C2×C4D4×C3⋊D311yescharacteristic
392C2×C4D4×C3⋊D311yescharacteristic
393C2×C4D4×C3⋊D311yescharacteristic
394C2×C4D4×C3⋊D311yescharacteristic
395C2×C4C23×C3⋊D311yescharacteristic
396C8—22no—
397C8—22no—
398C8—94no—
399C8—182no—
400C2×C3—24no—
401C2×C3—24no—
402C2×C3—24no—
403C2×C3—84no—
404C2×C3C2×#88414yes—
405D3—128no—
406C22—41no—
407C22—41no—
408C22—81no—
409C22—182no—
410C22—721no—
411C22#6409311yescharacteristic
412C22#6409311yescharacteristic
413C22C2×#382811yescharacteristic
414C4—41no—
415C4—182no—
416C4—362no—
417C4#6409311yescharacteristic
418C4#6409311yescharacteristic
419C4C2×D4×C3⋊D311yescharacteristic
420C4C2×D4×C3⋊D311yescharacteristic
421C3#7872114yescomplemented
422C2—21no—
423C2—21no—
424C2—21no—
425C2—81no—
426C2—362no—
427C2C2×#6409311yescharacteristic
subgroupG/H
centerC2C2×#64093
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#413C2
socleC2×C32C2×#121
prime radical#10596293C2
hypercenter#413C3⋊D3
hypocenterC32#3069
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. C2×#64160C2semidirect
3. C2×#64118C2semidirect
4. #10596216C2semidirect
5. C32×#413C2semidirect
6. C2×#63651C2semidirect
7. #10592823C2semidirect
8. #10592454C2semidirect
9. #10592447C2semidirect
21. C2×#3756C22semidirect
23. C2×#3754C22semidirect
34. C32×#101C22semidirect
35. C32×#100C22semidirect
36. #63651C22semidirect
37. #63649C22semidirect
42. C3×#413D3semidirect
117. C3×#101C2×D3semidirect
118. C3×#100C2×D3semidirect
166. #413C3⋊D3semidirect
254. #101C2×C3⋊D3semidirect
255. #100C2×C3⋊D3semidirect
361. C32#3069semidirect
421. C3#78721semidirect

Maximal subgroups

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

subgroupindex
C2×#641602
#105962932
C2×#641182
#105962162
C32×#4132
C2×#636512
#105928232
#105924542
#105924472
#787213