Finite Group Databasebeta

← #11784856

Subgroups — #11784856

Order 1152.

1860 conjugacy classes of subgroups; only the 353 normal subgroups are shown.

normal subgroups only

This group has 1860 conjugacy classes of subgroups — too many to draw. Only its 353 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×C32×H(5,𝔽2)C211yesmaximal, characteristic
2C2×#63664C212yesmaximal, complemented
3#10592931C212yesmaximal, complemented
4#10592920C211yesmaximal, characteristic, complemented
5C2×#63660C211yesmaximal, characteristic, complemented
6#72140—34nomaximal
7C32×H(5,𝔽2)—24no—
8C32×H(5,𝔽2)—22no—
9C32×H(5,𝔽2)C414yescomplemented
10C2×C32×#41—22no—
11C2×C32×#41C2212yes—
12C22×C32×D4—22no—
13C22×C32×D4—22no—
14C22×C32×D4C2211yescharacteristic
15C2×#3757—22no—
16C2×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
17C2×C4×C3⋊Dic3C2211yescharacteristic, complemented
18C2×#3748C2212yescomplemented
19#63664—24no—
20#63662—22no—
21#63660—22no—
22#63659—22no—
23C2×C3×H(5,𝔽2)D314yes—
24C2×#837—38no—
25#5232—38no—
26#5221—34no—
27C2×#833—34no—
28C24×C32—22no—
29C24×C32D412yes—
30C32×#41—28no—
31C32×#41—28no—
32C32×#41—22no—
33C32×#41—22no—
34C32×#41C2×C414yes—
35C32×#41D414yes—
36C2×C32×Q8D412yes—
37C2×C32×D4—24no—
38C2×C32×D4—28no—
39C2×C32×D4—28no—
40C2×C32×D4—28no—
41C2×C32×D4—22no—
42C2×C32×D4—28no—
43C2×C32×D4—22no—
44C2×C32×D4—22no—
45C2×C32×D4C2×C412yes—
46C2×C32×D4D412yes—
47C2×C32×D4D412yes—
48C22×C32×C4—22no—
49C22×C32×C4—22no—
50C22×C32×C4C2311yescharacteristic
51C22×C3⋊Dic3—21no—
52C22×C3⋊Dic3—21no—
53#3757—44no—
54C4⋊(C3⋊Dic3)—22no—
55C4×C3⋊Dic3—22no—
56#3748—22no—
57#3748—22no—
58C2×C3⋊(C3⋊C8)—22no—
59#1792—91noSylow
60C3×H(5,𝔽2)—216no—
61C3×H(5,𝔽2)—28no—
62C3×H(5,𝔽2)Dic3116yescomplemented
63C2×C3×#41—28no—
64C2×C3×#41C2×D318yes—
65C22×C3×D4—28no—
66C22×C3×D4—28no—
67C22×C3×D4C2×D314yes—
68C2×#217—68no—
69C2×C4⋊Dic3—34no—
70C2×C4×Dic3—34no—
71C2×#208—38no—
72#837—616no—
73#835—68no—
74#833—68no—
75#832—68no—
76C23×C32—28no—
77C23×C32—22no—
78C23×C32—22no—
79C23×C32—22no—
80C23×C32—42no—
81C23×C32—42no—
82C23×C32—82no—
83C23×C32—82no—
84C23×C32C22⋊C414yes—
85C23×C32C2×D411yescharacteristic
86C32×Q8—22no—
87C32×Q8C22⋊C414yescomplemented
88C32×D4—216no—
89C32×D4—24no—
90C32×D4—24no—
91C32×D4—24no—
92C32×D4—24no—
93C32×D4—22no—
94C32×D4C22⋊C414yescomplemented
95C2×C32×C4—28no—
96C2×C32×C4—22no—
97C2×C32×C4—24no—
98C2×C32×C4—24no—
99C2×C32×C4C22×C411yescharacteristic
100C2×C32×C4C2×D412yes—
101C2×C32×C4C2×D412yes—
102C2×C32×C4C2×D411yescharacteristic
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—41no—
105C2×C3⋊Dic3—41no—
106C2×C3⋊Dic3—42no—
107C2×C3⋊Dic3—41no—
108C3⋊(C3⋊C8)—42no—
109C2×H(5,𝔽2)C3⋊D311yescharacteristic
110C2×#104—92no—
111#417—92no—
112C2×#100—91no—
113#390—91no—
114C24×C3—28no—
115C24×C3#6718yes—
116C3×#41—232no—
117C3×#41—232no—
118C3×#41—28no—
119C3×#41—28no—
120C3×#41C2×Dic3116yes—
121C3×#41#67116yes—
122C2×C3×Q8#6718yes—
123C2×C3×D4—216no—
124C2×C3×D4—232no—
125C2×C3×D4—232no—
126C2×C3×D4—232no—
127C2×C3×D4—28no—
128C2×C3×D4—232no—
129C2×C3×D4—28no—
130C2×C3×D4—28no—
131C2×C3×D4C2×Dic318yes—
132C2×C3×D4#6718yes—
133C2×C3×D4#6718yes—
134C22×C3×C4—28no—
135C22×C3×C4—28no—
136C22×C3×C4C22×D314yes—
137C22×Dic3—64no—
138C22×Dic3—64no—
139#217—1216no—
140C4⋊Dic3—68no—
141C4×Dic3—68no—
142#208—68no—
143#208—68no—
144C2×C3⋊C8—68no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—24no—
148C22×C32—24no—
149C22×C32—44no—
150C22×C32—42no—
151C22×C32—42no—
152C22×C32—42no—
153C22×C32—42no—
154C22×C32—82no—
155C22×C32—88no—
156C22×C32C2×C22⋊C411yescharacteristic
157C22×C32#12011yescharacteristic
158C22×C32#12011yescharacteristic
159C32×C4—24no—
160C32×C4C2×C22⋊C412yes—
161C32×C4#12012yes—
162C3⋊Dic3—42no—
163C3⋊Dic3—81no—
164H(5,𝔽2)—24no—
165H(5,𝔽2)—22no—
166H(5,𝔽2)C3⋊Dic314yescomplemented
167C2×#41—22no—
168C2×#41C2×C3⋊D312yes—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4C2×C3⋊D311yescharacteristic
172C2×M16—92no—
173C2×C4⋊C4—91no—
174C2×C22⋊C4—182no—
175C2×C42—91no—
176#104—184no—
177#103—182no—
178#102—182no—
179#100—182no—
180C23×C3—232no—
181C23×C3—28no—
182C23×C3—28no—
183C23×C3—28no—
184C23×C3—48no—
185C23×C3—48no—
186C23×C3—88no—
187C23×C3—88no—
188C23×C3#217116yes—
189C23×C3C2×#6714yes—
190C3×Q8—28no—
191C3×Q8#217116yescomplemented
192C3×D4—264no—
193C3×D4—216no—
194C3×D4—216no—
195C3×D4—216no—
196C3×D4—216no—
197C3×D4—28no—
198C3×D4#217116yescomplemented
199C2×C3×C4—232no—
200C2×C3×C4—28no—
201C2×C3×C4—216no—
202C2×C3×C4—216no—
203C2×C3×C4C22×Dic314yes—
204C2×C3×C4C2×#6718yes—
205C2×C3×C4C2×#6718yes—
206C2×C3×C4C2×#6714yes—
207C2×Dic3—68no—
208C2×Dic3—124no—
209C2×Dic3—124no—
210C2×Dic3—128no—
211C2×Dic3—124no—
212C3⋊C8—128no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—44no—
216C2×C32—44no—
217C2×C32#37911yescharacteristic
218C2×C32#45311yescharacteristic
219C2×C32#45511yescharacteristic
220C24—22no—
221C24#63812yes—
222#41—28no—
223#41—28no—
224#41—22no—
225#41—22no—
226#41C2×C3⋊Dic314yes—
227#41#63814yes—
228C2×Q8#63812yes—
229C2×D4—24no—
230C2×D4—28no—
231C2×D4—28no—
232C2×D4—28no—
233C2×D4—22no—
234C2×D4—28no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4C2×C3⋊Dic312yes—
238C2×D4#63812yes—
239C2×D4#63812yes—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—181no—
243C22×C4—181no—
244C22×C4C22×C3⋊D311yescharacteristic
245M16—182no—
246M16—182no—
247C2×C8—182no—
248C4⋊C4—182no—
249C22⋊C4—364no—
250C42—182no—
251C22×C3—24no—
252C22×C3—24no—
253C22×C3—216no—
254C22×C3—216no—
255C22×C3—416no—
256C22×C3—48no—
257C22×C3—48no—
258C22×C3—48no—
259C22×C3—48no—
260C22×C3—88no—
261C22×C3—832no—
262C22×C3C2×#21714yes—
263C22×C3#95314yes—
264C22×C3#95314yes—
265C3×C4—216no—
266C3×C4C2×#21718yes—
267C3×C4#95318yes—
268Dic3—128no—
269Dic3—244no—
270C32#179211yescharacteristic, complemented, Sylow
271C23—28no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23—82no—
278C23—82no—
279C23#375714yes—
280C23C2×#63811yescharacteristic
281Q8—22no—
282Q8#375714yescomplemented
283D4—216no—
284D4—24no—
285D4—24no—
286D4—24no—
287D4—24no—
288D4—22no—
289D4#375714yescomplemented
290C2×C4—28no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—182no—
295C2×C4—361no—
296C2×C4—361no—
297C2×C4—362no—
298C2×C4—361no—
299C2×C4C22×C3⋊Dic311yescharacteristic
300C2×C4C2×#63812yes—
301C2×C4C2×#63812yes—
302C2×C4C2×#63811yescharacteristic
303C8—362no—
304C2×C3—24no—
305C2×C3—24no—
306C2×C3—416no—
307C2×C3—416no—
308C2×C3#524114yes—
309C2×C3#524214yes—
310C2×C3#524714yes—
311C22—21no—
312C22—21no—
313C22—24no—
314C22—24no—
315C22—44no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—82no—
321C22—88no—
322C22C2×#375711yescharacteristic
323C22#6416211yescharacteristic
324C22#6416211yescharacteristic
325C4—24no—
326C4—362no—
327C4—721no—
328C4C2×#375712yes—
329C4#6416212yes—
330C3#7214014yescomplemented
331C2—21no—
332C2—21no—
333C2—44no—
334C2—44no—
335C2#1059294011yescharacteristic
336C2#1059294111yescharacteristic
337C2#1059294611yescharacteristic
subgroupG/H
centerC22#64162
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
Fitting, prime radicalC2×C32×H(5,𝔽2)C2
socleC22×C32#120
cosocleC22×C32×C4C23
hypercenterC2×H(5,𝔽2)C3⋊D3
hypocenterC32#1792
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C2×#63664C2semidirect
3. #10592931C2semidirect
4. #10592920C2semidirect
5. C2×#63660C2semidirect
9. C32×H(5,𝔽2)C4semidirect
16. C2×C4⋊(C3⋊Dic3)C22semidirect
17. C2×C4×C3⋊Dic3C22semidirect
18. C2×#3748C22semidirect
62. C3×H(5,𝔽2)Dic3semidirect
87. C32×Q8C22⋊C4semidirect
94. C32×D4C22⋊C4semidirect
166. H(5,𝔽2)C3⋊Dic3semidirect
191. C3×Q8#217semidirect
198. C3×D4#217semidirect
270. C32#1792semidirect
282. Q8#3757semidirect
289. D4#3757semidirect
330. C3#72140semidirect

Maximal subgroups

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

subgroupindex
C2×C32×H(5,𝔽2)2
C2×#636642
#105929312
#105929202
C2×#636602
#721403