Finite Group Databasebeta

← #11788892

Subgroups — #11788892

Order 1152.

847 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 847 conjugacy classes of subgroups — too many to draw. Only its 124 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
1#10596276C211yesmaximal, characteristic
2#10594780C211yesmaximal, characteristic, complemented
3C2×C3×#833C212yesmaximal, complemented
4#10591078C211yesmaximal, characteristic, complemented
5#10589171C212yesmaximal, complemented
6#70089—32nomaximal
7C22×C32×D4C2211yescharacteristic
8C2×#3828C2211yescharacteristic
9#64162C414yes—
10#64149—21no—
11#64147—22no—
12#64145—21no—
13#64135C2211yescharacteristic
14C2×C3×#208C2212yes—
15C2×C3×#208C2212yes—
16#63818—22no—
17#63817—24no—
18C3×#833—24no—
19#63561—21no—
20#63559—21no—
21#5805—32no—
22#5805—61no—
23#5745—32no—
24C2×C3×#100—32no—
25C2×#833D312yescomplemented
26#5132—32no—
27#4559—32no—
28#4556—32no—
29C24×C32C2×C412yes—
30C2×C32×D4—21no—
31C2×C32×D4—21no—
32C2×C32×D4—22no—
33C2×C32×D4—22no—
34C22×C32×C4C2311yescharacteristic
35C2×#638—24no—
36C2×#638C2×C412yes—
37#3828—22no—
38#3828—22no—
39C2×C4⋊(C3⋊D3)D411yescharacteristic
40C2×C4×C3⋊D3D412yes—
41C2×#634D411yescharacteristic
42#3757—24no—
43#3757C2×C412yes—
44C4⋊(C3⋊Dic3)—21no—
45C4×C3⋊Dic3—21no—
46C3×#208—22no—
47C3×#208—22no—
48C3×#208—22no—
49C3×#208D414yescomplemented
50C2×C3×C3⋊C8—22no—
51C2×C3×C3⋊C8—22no—
52#3712—24no—
53#3710—24no—
54#1914—91noSylow
55C22×C3×D4—21no—
56C22×C3×D4C2×D312yes—
57C2×#235—32no—
58C2×#235—61no—
59C2×C3×M16—32no—
60C2×C3×M16—32no—
61#953—38no—
62#953—64no—
63#940—62no—
64#940—62no—
65#938—64no—
66#938—64no—
67#936—62no—
68#936—62no—
69#926—32no—
70#926—61no—
71C2×#208C2×D312yes—
72C2×#208C2×D312yes—
73#907—64no—
74D3×M16—64no—
75#901—64no—
76C3×#100—64no—
77#833—24no—
78#824—62no—
79#823—62no—
80C23×C32—22no—
81C23×C32—24no—
82C23×C32—82no—
83C23×C32C22×C411yescharacteristic
84C22×C3⋊D3C22⋊C412yes—
85C32×D4—41no—
86C32×D4—42no—
87C32×D4—41no—
88C2×C32×C4C2×D411yescharacteristic
89C2×C32×C4C2×D411yescharacteristic
90C2×C32×C4C2×D412yes—
91C2×C32×C4#4112yes—
92#638—24no—
93#638—84no—
94C2×C3⋊Dic3—22no—
95C2×C3⋊Dic3C22⋊C412yes—
96C4⋊(C3⋊D3)—22no—
97C4×C3⋊D3—22no—
98C4×C3⋊D3—22no—
99#634—22no—
100C3×C3⋊C8—24no—
101C3×C3⋊C8—42no—
102#568—91no—
103#535—91no—
104#413—91no—
105C2×#100—32no—
106#344—92no—
107C24×C3—22no—
108C24×C3C4×D314yes—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4—22no—
114C2×C3×D4—22no—
115C2×C3×D4—22no—
116C2×C3×D4—22no—
117C2×C3×D4—42no—
118C2×C3×D4—42no—
119C22×C3×C4—21no—
120C22×C3×C4C22×D312yes—
121C2×#67—34no—
122C2×#67—62no—
123C2×#67—68no—
124C2×#67—68no—
125#235—62no—
126#235—62no—
127#235—62no—
128#235—62no—
129#235—122no—
130#235—122no—
131C2×D12—32no—
132C2×D12—61no—
133C2×C4×D3—34no—
134C2×C4×D3—62no—
135C2×Dic6—32no—
136C2×Dic6—61no—
137C3×M16—34no—
138C3×M16—62no—
139C3×M16—62no—
140C3×M16—62no—
141C2×C3×C8—62no—
142C2×C3×C8—62no—
143#217—34no—
144#217—62no—
145#217—68no—
146#217—68no—
147C4⋊Dic3—62no—
148C4⋊Dic3—62no—
149C4×Dic3—62no—
150C4×Dic3—62no—
151#208—22no—
152#208—22no—
153#208—22no—
154#208#6714yescomplemented
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157ΓL(1,𝔽25)—68no—
158D3×C8—68no—
159C22×C32—41no—
160C22×C32—44no—
161C22×C32—42no—
162C22×C32—84no—
163C22×C32C2×C22⋊C411yescharacteristic
164C22×C32C4×D412yes—
165C2×C3⋊D3—22no—
166C2×C3⋊D3—42no—
167C32×C4#12011yescharacteristic
168C32×C4#12112yes—
169C32×C4#12311yescharacteristic
170C3⋊Dic3—22no—
171C3⋊Dic3—42no—
172C2×#41—91no—
173C22×D4D3211yescharacteristic
174#131—184no—
175C2×M16—32no—
176C2×M16—32no—
177C2×M16—182no—
178#127—181no—
179#124—181no—
180#121—182no—
181#120—94no—
182#117—91no—
183#101—181no—
184#100—64no—
185#100—181no—
186C23×C3—24no—
187C23×C3—24no—
188C23×C3—24no—
189C23×C3—24no—
190C23×C3—21no—
191C23×C3—44no—
192C23×C3—84no—
193C23×C3—84no—
194C23×C3C2×C4×D312yes—
195C22×D3—34no—
196C22×D3—62no—
197C3×D4—44no—
198C3×D4—42no—
199C3×D4—42no—
200C3×D4—42no—
201C3×D4—42no—
202C3×D4—42no—
203C3×D4—42no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4#23512yes—
209C2×C3×C4D3×D412yes—
210C2×C3×C4D3×D412yes—
211C2×C3×C4D3×D412yes—
212C2×C3×C4#23712yes—
213C2×C3×C4C2×#6712yes—
214#67—64no—
215#67—64no—
216#67—124no—
217#67—248no—
218#67—248no—
219C2×Dic3—34no—
220C2×Dic3—62no—
221C2×Dic3—64no—
222C2×Dic3—64no—
223D12—62no—
224D12—62no—
225D12—122no—
226C4×D3—64no—
227C4×D3—64no—
228C4×D3—122no—
229C4×D3—122no—
230Dic6—62no—
231Dic6—62no—
232Dic6—122no—
233C3×C8—64no—
234C3×C8—122no—
235C3⋊C8—24no—
236C3⋊C8—42no—
237C2×C32—21no—
238C2×C32—22no—
239C2×C32—82no—
240C2×C32#38611yescharacteristic
241C3⋊D3—42no—
242C24#62412yes—
243#41—182no—
244#41—182no—
245C2×Q8—91no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—92no—
251C2×D4—91no—
252C2×D4—184no—
253C22×C4—92no—
254C22×C4C2×D3211yescharacteristic
255M16—34no—
256M16—62no—
257M16—62no—
258M16—62no—
259M16—184no—
260C2×C8—62no—
261C2×C8—62no—
262C2×C8—184no—
263C4⋊C4—181no—
264C22⋊C4—92no—
265C22⋊C4—184no—
266C42—181no—
267C22×C3—22no—
268C22×C3—21no—
269C22×C3—42no—
270C22×C3—42no—
271C22×C3—48no—
272C22×C3—48no—
273C22×C3—44no—
274C22×C3—44no—
275C22×C3—84no—
276C22×C3—84no—
277C22×C3—88no—
278C22×C3D3×C22⋊C412yes—
279C22×C3#67⋊C412yes—
280C22×C3C4×#6712yes—
281C2×D3—64no—
282C2×D3—122no—
283C2×D3—124no—
284C2×D3—124no—
285C3×C4—22no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4#88312yes—
289C3×C4#88412yes—
290C3×C4#93712yes—
291C3×C4#93812yes—
292Dic3—64no—
293Dic3—122no—
294Dic3—124no—
295Dic3—124no—
296C32#191411yescharacteristic, complemented, Sylow
297C23—22no—
298C23—24no—
299C23—82no—
300C23—92no—
301C23C2×#62411yescharacteristic
302Q8—182no—
303D4—42no—
304D4—41no—
305D4—41no—
306D4—184no—
307D4—182no—
308D4—724no—
309C2×C4—92no—
310C2×C4—182no—
311C2×C4—182no—
312C2×C4—182no—
313C2×C4#380412yes—
314C2×C4D3×#6712yes—
315C2×C4#381111yescharacteristic
316C2×C4#381111yescharacteristic
317C8—64no—
318C8—122no—
319C2×C3—22no—
320C2×C3—22no—
321C2×C3—22no—
322C2×C3—22no—
323C2×C3—21no—
324C2×C3—42no—
325C2×C3—84no—
326C2×C3—84no—
327C2×C3#495612yes—
328D3—124no—
329D3—242no—
330C22—41no—
331C22—44no—
332C22—42no—
333C22—84no—
334C22—182no—
335C22—362no—
336C22#6395212yes—
337C22#6397411yescharacteristic
338C4—182no—
339C4—362no—
340C4#6395812yes—
341C4#6397511yescharacteristic
342C4#6398311yescharacteristic
343C3—21no—
344C3#7008912yescomplemented
345C2—21no—
346C2—22no—
347C2—82no—
348C2—362no—
349C2#1059172211yescharacteristic
subgroupG/H
centerC2#10591722
commutatorC23×C32C22×C4
FrattiniC22×C4C2×D32
FittingC22×C32×D4C22
socleC2×C32#386
prime radical#10596276C2
cosocleC22×C32×C4C23
hypercenterC22×D4D32
hypocenterC32#1914
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. #10594780C2semidirect
3. C2×C3×#833C2semidirect
4. #10591078C2semidirect
5. #10589171C2semidirect
25. C2×#833D3semidirect
49. C3×#208D4semidirect
154. #208#67semidirect
296. C32#1914semidirect
344. C3#70089semidirect

Maximal subgroups

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

subgroupindex
#105962762
#105947802
C2×C3×#8332
#105910782
#105891712
#700893