Finite Group Databasebeta

← #11868169

Subgroups — #11868169

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×#575C211yesmaximal, characteristic
2C2×#64160C211yesmaximal, characteristic, complemented
3C2×#64160C211yesmaximal, characteristic, complemented
4#10596290C211yesmaximal, characteristic, complemented
5#10592936C211yesmaximal, characteristic, complemented
6#10592903C214yesmaximal, complemented
7#10592899C214yesmaximal, complemented
8C2×#63651C211yesmaximal, characteristic, complemented
9#10592823C211yesmaximal, characteristic, complemented
10#84003—34nomaximal
11C2×C32×#41C2211yescharacteristic
12C2×C32×#41C2211yescharacteristic
13C22×#634C2211yescharacteristic
14C32×#137—21no—
15C32×#137C2212yes—
16C32×AGL(1,ℤ/8)—21no—
17C32×AGL(1,ℤ/8)C2212yes—
18C32×#135—22no—
19C2×C32×M16C2211yescharacteristic
20#64160—22no—
21#64160—22no—
22#64160—22no—
23#64160C2214yes—
24#64157—22no—
25#64157C2214yes—
26C2×#3756—21no—
27C2×#3756C2212yescomplemented
28C2×#3754—21no—
29C2×#3754C2212yescomplemented
30C2×#3748—22no—
31C2×#3748C2211yescharacteristic
32C2×#3748C2211yescharacteristic
33C2×#3748C2211yescharacteristic
34#63663C2212yescomplemented
35#63660C2212yescomplemented
36#63651C2214yescomplemented
37#63649C2214yescomplemented
38C3×#575D314yes—
39C2×#951—34no—
40C2×#951—34no—
41#5819—34no—
42#5237—34no—
43#5204—316no—
44#5200—316no—
45C2×#824—34no—
46#5124—34no—
47C32×#41—21no—
48C32×#41—21no—
49C32×#41—21no—
50C32×#41—21no—
51C32×#41—21no—
52C32×#41—21no—
53C32×#41D412yes—
54C32×#41C2312yes—
55C2×C32×Q8D411yescharacteristic
56C2×C32×Q8C2311yescharacteristic
57C2×C32×D4—21no—
58C2×C32×D4—21no—
59C2×C32×D4D411yescharacteristic
60C2×C32×D4C2311yescharacteristic
61C22×C32×C4—21no—
62C22×C32×C4—21no—
63C22×C32×C4C2311yescharacteristic
64C22×C3⋊Dic3—21no—
65C2×#634—21no—
66C2×#634—21no—
67C2×#634—21no—
68C2×#634—21no—
69C2×#634—21no—
70C2×#634C2312yes—
71C32×Q16—22no—
72C32×SD16—22no—
73C32×SD16—22no—
74C32×D8—22no—
75C32×M16—21no—
76C32×M16C2312yes—
77C2×C32×C8—21no—
78#3756—24no—
79#3756—24no—
80#3754—24no—
81#3754—24no—
82#3748—21no—
83#3748—21no—
84#3748—24no—
85#3748D412yes—
86#3748D412yes—
87#3748C2312yes—
88#3748C2312yes—
89C2×C3⋊(C3⋊C8)—21no—
90C2×C3⋊(C3⋊C8)—21no—
91C2×C3⋊(C3⋊C8)—24no—
92C2×C3⋊(C3⋊C8)C2312yes—
93#3069—91noSylow
94C2×C3×#41C2×D314yes—
95C2×C3×#41C2×D314yes—
96C22×Dic6—34no—
97C3×#137—24no—
98C3×#137C2×D318yes—
99C3×AGL(1,ℤ/8)—24no—
100C3×AGL(1,ℤ/8)C2×D318yes—
101C3×#135—28no—
102C2×C3×M16C2×D314yes—
103#951—316no—
104#951—68no—
105#951—68no—
106#951—68no—
107#948—316no—
108#948—68no—
109C2×#216—38no—
110C2×#216—64no—
111C2×#214—38no—
112C2×#214—64no—
113C2×#208—34no—
114C2×#208—34no—
115C2×#208—34no—
116C2×#208—68no—
117#836—38no—
118#833—38no—
119#824—316no—
120#822—316no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32#4111yescharacteristic
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8C2×D412yes—
128C32×D4—22no—
129C32×D4—22no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—41no—
134C32×D4—41no—
135C32×D4C2×D412yes—
136C2×C32×C4—21no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—21no—
140C2×C32×C4—41no—
141C2×C32×C4—41no—
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4#4111yescharacteristic
147C2×C32×C4C2411yescharacteristic
148C2×C3⋊Dic3—22no—
149C2×C3⋊Dic3—41no—
150C2×C3⋊Dic3—41no—
151#634—22no—
152#634—22no—
153#634—41no—
154#634—41no—
155C32×C8—22no—
156C3⋊(C3⋊C8)—22no—
157C3⋊(C3⋊C8)—22no—
158C3⋊(C3⋊C8)C2×D414yes—
159#575C3⋊D311yescharacteristic
160C2×#137—91no—
161C2×#137—91no—
162#568—91no—
163#473—94no—
164#471—94no—
165#430—91no—
166#413—91no—
167C2×#101—91no—
168C3×#41—24no—
169C3×#41—24no—
170C3×#41—24no—
171C3×#41—24no—
172C3×#41—24no—
173C3×#41—24no—
174C3×#41#6718yes—
175C3×#41C22×D318yes—
176C2×C3×Q8#6714yes—
177C2×C3×Q8C22×D314yes—
178C2×C3×D4—24no—
179C2×C3×D4—24no—
180C2×C3×D4#6714yes—
181C2×C3×D4C22×D314yes—
182C22×C3×C4—24no—
183C22×C3×C4—24no—
184C22×C3×C4C22×D314yes—
185C22×Dic3—64no—
186C2×Dic6—38no—
187C2×Dic6—64no—
188C2×Dic6—64no—
189C2×Dic6—64no—
190C2×Dic6—64no—
191C2×Dic6—64no—
192C3×Q16—28no—
193C3×SD16—28no—
194C3×SD16—28no—
195C3×D8—28no—
196C3×M16—24no—
197C3×M16C22×D318yes—
198C2×C3×C8—24no—
199#216—616no—
200#216—616no—
201#214—616no—
202#214—616no—
203#208—38no—
204#208—38no—
205#208—38no—
206#208—38no—
207#208—616no—
208#208—64no—
209#208—64no—
210C2×C3⋊C8—38no—
211C2×C3⋊C8—616no—
212C2×C3⋊C8—64no—
213C2×C3⋊C8—64no—
214C22×C32—22no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—81no—
218C22×C32—81no—
219C22×C32#12111yescharacteristic
220C22×C32#12111yescharacteristic
221C22×C32C2×#4111yescharacteristic
222C32×C4—22no—
223C32×C4—41no—
224C32×C4#12111yescharacteristic
225C32×C4#12111yescharacteristic
226C32×C4C22×D411yescharacteristic
227C32×C4C22×D411yescharacteristic
228C3⋊Dic3—42no—
229C2×#41C2×C3⋊D311yescharacteristic
230C2×#41C2×C3⋊D311yescharacteristic
231C22×Q8—91no—
232#137—21no—
233#137—94no—
234#137—182no—
235#137—182no—
236#137—182no—
237#137C2×C3⋊D312yes—
238AGL(1,ℤ/8)—21no—
239AGL(1,ℤ/8)C2×C3⋊D312yes—
240#135—22no—
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—91no—
250C2×M16—182no—
251C2×M16C2×C3⋊D311yescharacteristic
252#108—94no—
253#101—92no—
254#101—94no—
255#100—92no—
256C23×C3—24no—
257C23×C3—24no—
258C23×C3#23714yes—
259C3×Q8—24no—
260C3×Q8—24no—
261C3×Q8—24no—
262C3×Q8C2×#6718yes—
263C3×D4—28no—
264C3×D4—28no—
265C3×D4—24no—
266C3×D4—24no—
267C3×D4—24no—
268C3×D4—44no—
269C3×D4—44no—
270C3×D4C2×#6718yes—
271C2×C3×C4—24no—
272C2×C3×C4—28no—
273C2×C3×C4—28no—
274C2×C3×C4—24no—
275C2×C3×C4—44no—
276C2×C3×C4—44no—
277C2×C3×C4D3×D414yes—
278C2×C3×C4D3×D414yes—
279C2×C3×C4#23714yes—
280C2×C3×C4C2×#6714yes—
281C2×C3×C4C2×#6714yes—
282C2×C3×C4C23×D314yes—
283C2×Dic3—68no—
284C2×Dic3—124no—
285C2×Dic3—124no—
286Dic6—68no—
287Dic6—68no—
288Dic6—124no—
289Dic6—124no—
290C3×C8—28no—
291C3⋊C8—316no—
292C3⋊C8—68no—
293C3⋊C8—68no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32—42no—
298C2×C32—81no—
299C2×C32C2×#12111yescharacteristic
300#41—21no—
301#41—21no—
302#41—21no—
303#41—21no—
304#41—21no—
305#41—21no—
306#41#63812yes—
307#41C22×C3⋊D312yes—
308C2×Q8—92no—
309C2×Q8—181no—
310C2×Q8—181no—
311C2×Q8—181no—
312C2×Q8—181no—
313C2×Q8—181no—
314C2×Q8#63811yescharacteristic
315C2×Q8C22×C3⋊D311yescharacteristic
316C2×D4—21no—
317C2×D4—21no—
318C2×D4#63811yescharacteristic
319C2×D4C22×C3⋊D311yescharacteristic
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—181no—
323C22×C4C22×C3⋊D311yescharacteristic
324Q16—22no—
325Q16—184no—
326Q16—184no—
327SD16—22no—
328SD16—22no—
329SD16—184no—
330SD16—184no—
331D8—22no—
332M16—21no—
333M16—92no—
334M16—92no—
335M16—92no—
336M16—92no—
337M16—184no—
338M16—181no—
339M16—181no—
340M16C22×C3⋊D312yes—
341C2×C8—21no—
342C2×C8—92no—
343C2×C8—184no—
344C2×C8—181no—
345C2×C8—181no—
346C22×C3—28no—
347C22×C3—44no—
348C22×C3—44no—
349C22×C3—84no—
350C22×C3—84no—
351C22×C3#93914yes—
352C22×C3#93914yes—
353C22×C3C2×#23714yes—
354C3×C4—28no—
355C3×C4—44no—
356C3×C4#93914yes—
357C3×C4#93914yes—
358C3×C4C2×D3×D414yes—
359C3×C4C22×#6714yes—
360Dic3—128no—
361C32#306911yescharacteristic, complemented, Sylow
362C23—21no—
363C23—21no—
364C23#383011yescharacteristic
365Q8—21no—
366Q8—21no—
367Q8—21no—
368Q8—182no—
369Q8—182no—
370Q8—361no—
371Q8—361no—
372Q8C2×#63812yes—
373D4—22no—
374D4—22no—
375D4—21no—
376D4—21no—
377D4—21no—
378D4—41no—
379D4—41no—
380D4C2×#63812yes—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—182no—
388C2×C4—361no—
389C2×C4—361no—
390C2×C4D4×C3⋊D311yescharacteristic
391C2×C4D4×C3⋊D311yescharacteristic
392C2×C4#383011yescharacteristic
393C2×C4C2×#63811yescharacteristic
394C2×C4C2×#63811yescharacteristic
395C2×C4C23×C3⋊D311yescharacteristic
396C8—22no—
397C8—94no—
398C8—182no—
399C8—182no—
400C2×C3—24no—
401C2×C3—24no—
402C2×C3—24no—
403C2×C3—48no—
404C2×C3—84no—
405C2×C3C2×#93914yes—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—81no—
410C22—81no—
411C22#6414811yescharacteristic
412C22#6414811yescharacteristic
413C22C2×#383011yescharacteristic
414C4—22no—
415C4—41no—
416C4—362no—
417C4#6414811yescharacteristic
418C4#6414811yescharacteristic
419C4C2×D4×C3⋊D311yescharacteristic
420C4C22×#63811yescharacteristic
421C3#8400314yescomplemented
422C2—21no—
423C2—21no—
424C2—21no—
425C2—42no—
426C2—81no—
427C2C2×#6414811yescharacteristic
subgroupG/H
centerC2C2×#64148
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
Fitting, prime radicalC32×#575C2
socleC2×C32C2×#121
hypercenter#575C3⋊D3
hypocenterC32#3069
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
2. C2×#64160C2semidirect
3. C2×#64160C2semidirect
4. #10596290C2semidirect
5. #10592936C2semidirect
6. #10592903C2semidirect
7. #10592899C2semidirect
8. C2×#63651C2semidirect
9. #10592823C2semidirect
27. C2×#3756C22semidirect
29. C2×#3754C22semidirect
34. #63663C22semidirect
35. #63660C22semidirect
36. #63651C22semidirect
37. #63649C22semidirect
361. C32#3069semidirect
421. C3#84003semidirect

Maximal subgroups

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

subgroupindex
C32×#5752
C2×#641602
#105962902
#105929362
#105929032
#105928992
C2×#636512
#105928232
#840033