Finite Group Databasebeta

← #11865022

Subgroups — #11865022

Order 1152.

2286 conjugacy classes of subgroups; only the 301 normal subgroups are shown.

normal subgroups only

This group has 2286 conjugacy classes of subgroups — too many to draw. Only its 301 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64361C211yesmaximal, characteristic, complemented
2C32×#535C211yesmaximal, characteristic, complemented
3#10596293C212yesmaximal, complemented
4C2×#63660C211yesmaximal, characteristic, complemented
5#10592776C214yesmaximal, complemented
6#10592760C214yesmaximal, complemented
7#10592704C212yesmaximal, complemented
8#83394—34nomaximal
9C2×C32×#41C2211yescharacteristic, complemented
10C22×C32×D4C2211yescharacteristic
11C22×#638—22no—
12#64361—22no—
13#64361—24no—
14#64361C2214yes—
15C2×#3830—22no—
16C2×D4×C3⋊D3—22no—
17C2×#3828C2212yes—
18C32×#127C2212yescomplemented
19C32×#124C2212yescomplemented
20C32×#121—22no—
21C32×#120—22no—
22C32×#117C2211yescharacteristic, complemented
23#64160—22no—
24#64159—24no—
25#64158—22no—
26#64151C2214yescomplemented
27#64141C2214yescomplemented
28C2×#3748C2212yescomplemented
29#63660C2214yescomplemented
30#63632C2218yescomplemented
31C2×#1004—34no—
32C3×#535D314yescomplemented
33#5822—38no—
34C2×#833—34no—
35#5077—316no—
36#5061—316no—
37#5005—38no—
38C24×C32D412yes—
39C23×C3⋊D3—22no—
40C32×#41—22no—
41C32×#41—22no—
42C2×C32×Q8C2311yescharacteristic, complemented
43C2×C32×D4—21no—
44C2×C32×D4—24no—
45C2×C32×D4—22no—
46C2×C32×D4—22no—
47C2×C32×D4D412yes—
48C2×C32×D4C2312yes—
49C2×C32×D4C2311yescharacteristic, complemented
50C22×C32×C4—21no—
51C22×C32×C4C2311yescharacteristic
52C2×#638—24no—
53C2×#638—28no—
54C2×#638—22no—
55C2×#638—28no—
56C22×C3⋊Dic3—22no—
57#3830—28no—
58#3830—28no—
59D4×C3⋊D3—28no—
60D4×C3⋊D3—28no—
61#3828—22no—
62#3828—22no—
63#3828D414yescomplemented
64#3828C2314yes—
65C2×C4⋊(C3⋊D3)D412yescomplemented
66C2×C4×C3⋊D3—22no—
67C2×#634D412yescomplemented
68C32×C4⋊C4—21no—
69C32×C22⋊C4—21no—
70C32×C22⋊C4—24no—
71C32×C42C2312yescomplemented
72#3756—24no—
73#3755—24no—
74#3754—24no—
75#3753—24no—
76#3748—22no—
77#3748C2314yescomplemented
78C2×C3⋊(C3⋊C8)—22no—
79#3019—91noSylow
80C2×C3×#41C2×D314yescomplemented
81C22×C3×D4C2×D314yes—
82C22×#67—68no—
83#1004—316no—
84#1004—616no—
85#1004—68no—
86C2×#237—68no—
87C2×D3×D4—68no—
88C2×#235—38no—
89C3×#127C2×D318yescomplemented
90C3×#124C2×D318yescomplemented
91C3×#121—28no—
92C3×#120—28no—
93C3×#117C2×D314yescomplemented
94#951—68no—
95#950—616no—
96#949—68no—
97#942—316no—
98#932—316no—
99C2×#208—38no—
100#833—316no—
101#805—332no—
102C23×C32—22no—
103C23×C32—22no—
104C23×C32—21no—
105C23×C32—82no—
106C23×C32C2×D414yes—
107C23×C32C2×D411yescharacteristic
108C22×C3⋊D3—22no—
109C22×C3⋊D3—28no—
110C22×C3⋊D3—84no—
111C32×Q8—22no—
112C32×D4—24no—
113C32×D4—24no—
114C32×D4—22no—
115C32×D4—41no—
116C32×D4—41no—
117C32×D4—84no—
118C2×C32×C4—22no—
119C2×C32×C4—21no—
120C2×C32×C4—42no—
121C2×C32×C4C2×D412yes—
122C2×C32×C4C2×D412yes—
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2411yescharacteristic
125#638—216no—
126#638—24no—
127#638—24no—
128C2×C3⋊Dic3—28no—
129C2×C3⋊Dic3—22no—
130C4⋊(C3⋊D3)—22no—
131C4⋊(C3⋊D3)C2×D414yes—
132C4×C3⋊D3—24no—
133C4×C3⋊D3—24no—
134#634—22no—
135#634C2×D414yes—
136C3⋊(C3⋊C8)—24no—
137C2×H(5,𝔽2)—91no—
138#575—92no—
139#535C3⋊D311yescharacteristic, complemented
140#455—94no—
141#453—94no—
142#421—92no—
143C2×#100—91no—
144C24×C3#6718yes—
145C23×D3—68no—
146C3×#41—28no—
147C3×#41—28no—
148C2×C3×Q8C22×D314yescomplemented
149C2×C3×D4—24no—
150C2×C3×D4—216no—
151C2×C3×D4—28no—
152C2×C3×D4—28no—
153C2×C3×D4#6718yes—
154C2×C3×D4C22×D318yes—
155C2×C3×D4C22×D314yescomplemented
156C22×C3×C4—24no—
157C22×C3×C4C22×D314yes—
158C2×#67—616no—
159C2×#67—632no—
160C2×#67—68no—
161C2×#67—632no—
162C22×Dic3—68no—
163#237—632no—
164#237—632no—
165D3×D4—632no—
166D3×D4—632no—
167#235—316no—
168#235—316no—
169#235—68no—
170#235—68no—
171C2×D12—38no—
172C2×C4×D3—68no—
173C2×Dic6—38no—
174C3×C4⋊C4—24no—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—216no—
177C3×C42C22×D318yescomplemented
178#216—616no—
179#215—616no—
180#214—616no—
181#213—616no—
182#208—316no—
183#208—68no—
184C2×C3⋊C8—68no—
185C22×C32—24no—
186C22×C32—41no—
187C22×C32—44no—
188C22×C32—44no—
189C22×C32—42no—
190C22×C32—41no—
191C22×C32—82no—
192C22×C32#12011yescharacteristic
193C22×C32#12011yescharacteristic
194C22×C32C22×D411yescharacteristic
195C2×C3⋊D3—24no—
196C2×C3⋊D3—44no—
197C2×C3⋊D3—88no—
198C32×C4—42no—
199C32×C4—41no—
200C32×C4#12012yes—
201C32×C4C22×D412yes—
202C3⋊Dic3—24no—
203H(5,𝔽2)—94no—
204H(5,𝔽2)—184no—
205H(5,𝔽2)—182no—
206C2×#41—92no—
207C2×#41—182no—
208C2×#41C2×C3⋊D311yescharacteristic, complemented
209C22×D4—182no—
210C22×D4—182no—
211C22×D4C2×C3⋊D311yescharacteristic
212#137—94no—
213#137—182no—
214AGL(1,ℤ/8)—94no—
215AGL(1,ℤ/8)—182no—
216#135—184no—
217C2×M16—92no—
218#127C2×C3⋊D312yescomplemented
219#124C2×C3⋊D312yescomplemented
220#121—22no—
221#120—22no—
222#117C2×C3⋊D311yescharacteristic, complemented
223#104—98no—
224#100—94no—
225C23×C3—28no—
226C23×C3—28no—
227C23×C3—24no—
228C23×C3—88no—
229C23×C3C2×#67116yes—
230C23×C3C2×#6714yes—
231C22×D3—68no—
232C22×D3—632no—
233C22×D3—2416no—
234C3×Q8—28no—
235C3×D4—216no—
236C3×D4—216no—
237C3×D4—28no—
238C3×D4—44no—
239C3×D4—44no—
240C3×D4—816no—
241C2×C3×C4—28no—
242C2×C3×C4—24no—
243C2×C3×C4—48no—
244C2×C3×C4D3×D418yes—
245C2×C3×C4D3×D418yes—
246C2×C3×C4C2×#6714yes—
247C2×C3×C4C23×D314yes—
248#67—664no—
249#67—616no—
250#67—616no—
251C2×Dic3—632no—
252C2×Dic3—68no—
253D12—316no—
254D12—68no—
255C4×D3—616no—
256C4×D3—616no—
257Dic6—316no—
258Dic6—68no—
259C3⋊C8—616no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—44no—
264C2×C32—81no—
265C2×C32C2×#12011yescharacteristic
266C3⋊D3—44no—
267C24—182no—
268C24#63812yes—
269#41—22no—
270#41—22no—
271#41—94no—
272#41—94no—
273#41—188no—
274#41—188no—
275#41—182no—
276#41—182no—
277C2×Q8—92no—
278C2×Q8C22×C3⋊D311yescharacteristic, complemented
279C2×D4—21no—
280C2×D4—24no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—92no—
284C2×D4—188no—
285C2×D4—188no—
286C2×D4—184no—
287C2×D4—188no—
288C2×D4—182no—
289C2×D4—188no—
290C2×D4#63812yes—
291C2×D4C22×C3⋊D312yes—
292C2×D4C22×C3⋊D311yescharacteristic, complemented
293C22×C4—21no—
294C22×C4—182no—
295C22×C4—182no—
296C22×C4C22×C3⋊D311yescharacteristic
297Q16—184no—
298SD16—184no—
299SD16—184no—
300D8—184no—
301M16—94no—
302M16—182no—
303C2×C8—182no—
304C4⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—24no—
307C42C22×C3⋊D312yescomplemented
308C22×C3—216no—
309C22×C3—44no—
310C22×C3—416no—
311C22×C3—416no—
312C22×C3—48no—
313C22×C3—44no—
314C22×C3—88no—
315C22×C3#93714yes—
316C22×C3#93714yes—
317C22×C3C22×#6714yes—
318C2×D3—616no—
319C2×D3—1216no—
320C2×D3—2432no—
321C3×C4—48no—
322C3×C4—44no—
323C3×C4#93718yes—
324C3×C4C2×D3×D418yes—
325Dic3—616no—
326C32#301911yescharacteristic, complemented, Sylow
327C23—22no—
328C23—22no—
329C23—21no—
330C23—82no—
331C23—188no—
332C23—182no—
333C23—724no—
334C23C2×#63814yes—
335C23C2×#63811yescharacteristic
336Q8—22no—
337Q8—94no—
338Q8—182no—
339D4—24no—
340D4—22no—
341D4—24no—
342D4—41no—
343D4—41no—
344D4—84no—
345D4—94no—
346D4—1816no—
347D4—184no—
348D4—184no—
349D4—182no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—188no—
354C2×C4—182no—
355C2×C4—184no—
356C2×C4—184no—
357C2×C4D4×C3⋊D312yes—
358C2×C4D4×C3⋊D312yes—
359C2×C4C2×#63811yescharacteristic
360C2×C4C23×C3⋊D311yescharacteristic
361C8—184no—
362C2×C3—24no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—416no—
366C2×C3—84no—
367C2×C3C2×#93714yes—
368D3—1216no—
369C22—24no—
370C22—41no—
371C22—44no—
372C22—44no—
373C22—42no—
374C22—41no—
375C22—82no—
376C22—184no—
377C22—364no—
378C22—728no—
379C22#6414611yescharacteristic
380C22#6414611yescharacteristic
381C22C22×#63811yescharacteristic
382C4—42no—
383C4—41no—
384C4—184no—
385C4#6414612yes—
386C4C2×D4×C3⋊D312yes—
387C3#8339414yescomplemented
388C2—21no—
389C2—21no—
390C2—21no—
391C2—44no—
392C2—81no—
393C2—364no—
394C2C2×#6414611yescharacteristic
subgroupG/H
centerC2C2×#64146
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#535C2
socleC2×C32C2×#120
hypercenter#535C3⋊D3
hypocenterC32#3019
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#64361C2semidirect
2. C32×#535C2semidirect
3. #10596293C2semidirect
4. C2×#63660C2semidirect
5. #10592776C2semidirect
6. #10592760C2semidirect
7. #10592704C2semidirect
9. C2×C32×#41C22semidirect
18. C32×#127C22semidirect
19. C32×#124C22semidirect
22. C32×#117C22semidirect
26. #64151C22semidirect
27. #64141C22semidirect
28. C2×#3748C22semidirect
29. #63660C22semidirect
30. #63632C22semidirect
32. C3×#535D3semidirect
42. C2×C32×Q8C23semidirect
49. C2×C32×D4C23semidirect
63. #3828D4semidirect
65. C2×C4⋊(C3⋊D3)D4semidirect
67. C2×#634D4semidirect
71. C32×C42C23semidirect
77. #3748C23semidirect
80. C2×C3×#41C2×D3semidirect
89. C3×#127C2×D3semidirect
90. C3×#124C2×D3semidirect
93. C3×#117C2×D3semidirect
139. #535C3⋊D3semidirect
148. C2×C3×Q8C22×D3semidirect
155. C2×C3×D4C22×D3semidirect
177. C3×C42C22×D3semidirect
208. C2×#41C2×C3⋊D3semidirect
218. #127C2×C3⋊D3semidirect
219. #124C2×C3⋊D3semidirect
222. #117C2×C3⋊D3semidirect
278. C2×Q8C22×C3⋊D3semidirect
292. C2×D4C22×C3⋊D3semidirect
307. C42C22×C3⋊D3semidirect
326. C32#3019semidirect
387. C3#83394semidirect

Maximal subgroups

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

subgroupindex
C2×#643612
C32×#5352
#105962932
C2×#636602
#105927762
#105927602
#105927042
#833943