Finite Group Databasebeta

← #11868163

Subgroups — #11868163

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
1C32×#568C211yesmaximal, characteristic, complemented
2C2×#64160C211yesmaximal, characteristic, complemented
3#10596293C211yesmaximal, characteristic
4C2×#64118C211yesmaximal, characteristic, complemented
5#10592842C214yesmaximal, complemented
6#10592840C214yesmaximal, complemented
7C2×#63651C211yesmaximal, characteristic, complemented
8#10592831C211yesmaximal, characteristic, complemented
9#10592824C211yesmaximal, characteristic, complemented
10#83816—34nomaximal
11C2×C32×#41C2211yescharacteristic
12C2×#3828C2211yescharacteristic
13C22×#634C2211yescharacteristic
14C32×#131—22no—
15C32×#131C2214yes—
16C2×C32×M16—22no—
17C2×C32×M16C2211yescharacteristic
18C2×C32×M16C2211yescharacteristic
19#64160—22no—
20#64160C2214yes—
21#64160C2212yes—
22#64159—22no—
23#64158C2212yes—
24C2×#3756—21no—
25#64151—21no—
26C2×#3754—21no—
27#64141—21no—
28C2×#3748C2211yescharacteristic
29C2×#3748C2211yescharacteristic
30#64118—22no—
31#64118—22no—
32C2×#3746C2212yescomplemented
33C2×#3744C2212yescomplemented
34#63651C2212yescomplemented
35#63651C2214yescomplemented
36#63650C2212yescomplemented
37#63646C2214yescomplemented
38C3×#568D314yescomplemented
39C2×#951—34no—
40#5822—34no—
41C2×#909—34no—
42#5143—316no—
43#5141—316no—
44C2×#824—34no—
45#5132—34no—
46#5125—34no—
47C32×#41—21no—
48C32×#41—21no—
49C32×#41D412yes—
50C32×#41C2312yes—
51C2×C32×Q8D411yescharacteristic
52C2×C32×D4—21no—
53C2×C32×D4D411yescharacteristic
54C22×C32×C4—21no—
55C22×C32×C4C2311yescharacteristic
56C2×#638—21no—
57C22×C3⋊Dic3—21no—
58#3828—21no—
59#3828—21no—
60#3828—21no—
61#3828—21no—
62C2×C4⋊(C3⋊D3)C2311yescharacteristic
63C2×C4×C3⋊D3—21no—
64C2×#634—21no—
65C2×#634—21no—
66C2×#634—21no—
67C2×#634—21no—
68C2×#634—21no—
69C2×#634C2312yes—
70C2×#634C2311yescharacteristic
71C32×M16—21no—
72C32×M16—24no—
73C32×M16D412yes—
74C32×M16D412yes—
75C32×M16C2312yes—
76C2×C32×C8—21no—
77C2×C32×C8—24no—
78C2×C32×C8C2312yes—
79#3756—22no—
80#3756—24no—
81#3755—22no—
82#3754—22no—
83#3754—24no—
84#3753—22no—
85#3748—21no—
86#3748—21no—
87#3748C2312yes—
88#3748C2312yes—
89C2×C3⋊(C3⋊C8)—21no—
90C2×C3⋊(C3⋊C8)—21no—
91#3746—24no—
92#3744—24no—
93#3069—91noSylow
94C2×C3×#41C2×D314yes—
95C2×#235—34no—
96C22×Dic6—34no—
97C3×#131—28no—
98C3×#131C2×D3116yes—
99C2×C3×M16—28no—
100C2×C3×M16C2×D314yes—
101C2×C3×M16C2×D314yes—
102#951—316no—
103#951—38no—
104#951—68no—
105#950—68no—
106#949—38no—
107C2×#216—64no—
108#942—64no—
109C2×#214—64no—
110#932—64no—
111C2×#208—34no—
112C2×#208—34no—
113#909—68no—
114#909—68no—
115C2×Dic12—38no—
116C2×#204—38no—
117#824—38no—
118#824—316no—
119#823—38no—
120#819—316no—
121C23×C32—21no—
122C23×C32#4111yescharacteristic
123C22×C3⋊D3—21no—
124C32×Q8—21no—
125C32×Q8C2×D412yes—
126C32×D4—22no—
127C32×D4—22no—
128C32×D4—21no—
129C32×D4C2×D412yes—
130C2×C32×C4—21no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
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—41no—
140#638—41no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3—41no—
144C2×C3⋊Dic3—41no—
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)—21no—
147C4×C3⋊D3—41no—
148C4×C3⋊D3—41no—
149#634—22no—
150#634—21no—
151#634—22no—
152#634—21no—
153#634—41no—
154#634—41no—
155C32×C8—22no—
156C32×C8C2×D414yes—
157C3⋊(C3⋊C8)—22no—
158C3⋊(C3⋊C8)—22no—
159#575—91no—
160C2×#137—91no—
161C2×#137—91no—
162#568C3⋊D311yescharacteristic, complemented
163#473—94no—
164#471—94no—
165#430—91no—
166#413—91no—
167C2×#101—91no—
168C3×#41—24no—
169C3×#41—24no—
170C3×#41D1218yes—
171C3×#41C22×D318yes—
172C2×C3×Q8D1214yes—
173C2×C3×D4—24no—
174C2×C3×D4D1214yes—
175C22×C3×C4—24no—
176C22×C3×C4C22×D314yes—
177C2×#67—64no—
178C22×Dic3—64no—
179#235—64no—
180#235—64no—
181#235—64no—
182#235—64no—
183C2×D12—34no—
184C2×C4×D3—64no—
185C2×Dic6—38no—
186C2×Dic6—34no—
187C2×Dic6—64no—
188C2×Dic6—64no—
189C2×Dic6—64no—
190C2×Dic6—64no—
191C2×Dic6—64no—
192C3×M16—24no—
193C3×M16—216no—
194C3×M16#6718yes—
195C3×M16#6718yes—
196C3×M16C22×D318yes—
197C2×C3×C8—24no—
198C2×C3×C8—216no—
199C2×C3×C8C22×D318yes—
200#216—68no—
201#216—616no—
202#215—68no—
203#214—68no—
204#214—616no—
205#213—68no—
206#208—38no—
207#208—38no—
208#208—64no—
209#208—64no—
210C2×C3⋊C8—64no—
211C2×C3⋊C8—64no—
212Dic12—616no—
213#204—616no—
214C22×C32—22no—
215C22×C32—41no—
216C22×C32—81no—
217C22×C32#12111yescharacteristic
218C22×C32#12111yescharacteristic
219C22×C32C2×#4111yescharacteristic
220C2×C3⋊D3—41no—
221C2×C3⋊D3—81no—
222C32×C4—22no—
223C32×C4#12111yescharacteristic
224C32×C4#12111yescharacteristic
225C32×C4C22×D411yescharacteristic
226C32×C4C22×D411yescharacteristic
227C3⋊Dic3—42no—
228C3⋊Dic3—41no—
229C2×#41—91no—
230C2×#41C2×C3⋊D311yescharacteristic
231C22×Q8—91no—
232#137—94no—
233#137—92no—
234#137—182no—
235#137—181no—
236#137—182no—
237#137—182no—
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—22no—
246#131C2×C3⋊D314yes—
247C2×M16—22no—
248C2×M16—91no—
249C2×M16—91no—
250C2×M16C2×C3⋊D311yescharacteristic
251C2×M16C2×C3⋊D311yescharacteristic
252#108—94no—
253#101—94no—
254#101—92no—
255#100—92no—
256C23×C3—24no—
257C23×C3#23514yes—
258C22×D3—64no—
259C3×Q8—24no—
260C3×Q8C2×D1218yes—
261C3×D4—28no—
262C3×D4—28no—
263C3×D4—24no—
264C3×D4C2×D1218yes—
265C2×C3×C4—24no—
266C2×C3×C4—28no—
267C2×C3×C4—28no—
268C2×C3×C4C2×D1214yes—
269C2×C3×C4C2×D1214yes—
270C2×C3×C4#23514yes—
271C2×C3×C4C2×#6714yes—
272C2×C3×C4C2×#6714yes—
273C2×C3×C4C23×D314yes—
274#67—124no—
275#67—124no—
276C2×Dic3—68no—
277C2×Dic3—64no—
278C2×Dic3—124no—
279C2×Dic3—124no—
280D12—64no—
281D12—64no—
282C4×D3—124no—
283C4×D3—124no—
284Dic6—68no—
285Dic6—64no—
286Dic6—68no—
287Dic6—64no—
288Dic6—124no—
289Dic6—124no—
290C3×C8—28no—
291C3×C8C2×#67116yes—
292C3⋊C8—68no—
293C3⋊C8—68no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—42no—
298C2×C32C2×#12111yescharacteristic
299C3⋊D3—81no—
300#41—21no—
301#41—21no—
302#41—181no—
303#41—181no—
304#41—181no—
305#41—181no—
306#41C4⋊(C3⋊D3)12yes—
307#41C22×C3⋊D312yes—
308C2×Q8—92no—
309C2×Q8—91no—
310C2×Q8—181no—
311C2×Q8—181no—
312C2×Q8—181no—
313C2×Q8—181no—
314C2×Q8—181no—
315C2×Q8C4⋊(C3⋊D3)11yescharacteristic
316C2×D4—21no—
317C2×D4—91no—
318C2×D4—181no—
319C2×D4C4⋊(C3⋊D3)11yescharacteristic
320C22×C4—21no—
321C22×C4—181no—
322C22×C4—181no—
323C22×C4C22×C3⋊D311yescharacteristic
324Q16—184no—
325Q16—182no—
326Q16—184no—
327SD16—182no—
328SD16—184no—
329SD16—182no—
330SD16—184no—
331D8—182no—
332M16—21no—
333M16—24no—
334M16—92no—
335M16—92no—
336M16—181no—
337M16—181no—
338M16#63812yes—
339M16#63812yes—
340M16C22×C3⋊D312yes—
341C2×C8—21no—
342C2×C8—24no—
343C2×C8—181no—
344C2×C8—181no—
345C2×C8C22×C3⋊D312yes—
346C22×C3—28no—
347C22×C3—44no—
348C22×C3—84no—
349C22×C3#93014yes—
350C22×C3#93014yes—
351C22×C3C2×#23514yes—
352C2×D3—124no—
353C2×D3—244no—
354C3×C4—28no—
355C3×C4#93014yes—
356C3×C4#93014yes—
357C3×C4C22×D1214yes—
358C3×C4C22×#6714yes—
359Dic3—128no—
360Dic3—124no—
361C32#306911yescharacteristic, complemented, Sylow
362C23—21no—
363C23—181no—
364C23#382811yescharacteristic
365Q8—21no—
366Q8—182no—
367Q8—181no—
368Q8—182no—
369Q8—181no—
370Q8—361no—
371Q8—361no—
372Q8C2×C4⋊(C3⋊D3)12yes—
373D4—22no—
374D4—22no—
375D4—21no—
376D4—181no—
377D4—181no—
378D4—361no—
379D4—361no—
380D4C2×C4⋊(C3⋊D3)12yes—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—181no—
385C2×C4—182no—
386C2×C4—361no—
387C2×C4—361no—
388C2×C4—361no—
389C2×C4—361no—
390C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
391C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
392C2×C4#382811yescharacteristic
393C2×C4C2×#63811yescharacteristic
394C2×C4C2×#63811yescharacteristic
395C2×C4C23×C3⋊D311yescharacteristic
396C8—22no—
397C8—182no—
398C8—182no—
399C8C2×#63814yes—
400C2×C3—24no—
401C2×C3—24no—
402C2×C3—24no—
403C2×C3—48no—
404C2×C3C2×#93014yes—
405D3—244no—
406C22—22no—
407C22—41no—
408C22—81no—
409C22—361no—
410C22—721no—
411C22#6413911yescharacteristic
412C22#6413911yescharacteristic
413C22C2×#382811yescharacteristic
414C4—22no—
415C4—361no—
416C4—362no—
417C4#6413911yescharacteristic
418C4#6413911yescharacteristic
419C4C22×C4⋊(C3⋊D3)11yescharacteristic
420C4C22×#63811yescharacteristic
421C3#8381614yescomplemented
422C2—21no—
423C2—21no—
424C2—21no—
425C2—42no—
426C2—721no—
427C2C2×#6413911yescharacteristic
subgroupG/H
centerC2C2×#64139
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#568C2
socleC2×C32C2×#121
prime radical#10596293C2
hypercenter#568C3⋊D3
hypocenterC32#3069
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
1. C32×#568C2semidirect
2. C2×#64160C2semidirect
4. C2×#64118C2semidirect
5. #10592842C2semidirect
6. #10592840C2semidirect
7. C2×#63651C2semidirect
8. #10592831C2semidirect
9. #10592824C2semidirect
32. C2×#3746C22semidirect
33. C2×#3744C22semidirect
34. #63651C22semidirect
35. #63651C22semidirect
36. #63650C22semidirect
37. #63646C22semidirect
38. C3×#568D3semidirect
162. #568C3⋊D3semidirect
361. C32#3069semidirect
421. C3#83816semidirect

Maximal subgroups

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

subgroupindex
C32×#5682
C2×#641602
#105962932
C2×#641182
#105928422
#105928402
C2×#636512
#105928312
#105928242
#838163