Finite Group Databasebeta

← #11793864

Subgroups — #11793864

Order 1152.

575 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C3×#5748C211yesmaximal, characteristic
2C3×#5118C211yesmaximal, characteristic, complemented
3C2×#63569C212yesmaximal, complemented
4#10591076C212yesmaximal, complemented
5#10589426C211yesmaximal, characteristic, complemented
6#72117—31nomaximal
7#70531—31nomaximal
8C2×C3×#235C2212yes—
9C2×C32×M16C2211yescharacteristic
10C2×#3748C2212yes—
11C2×C3×C4×Dic3C2211yescharacteristic
12C2×C3×#208C2211yescharacteristic
13C3×#909—22no—
14C3×#908—22no—
15C3×#904—22no—
16C3×Dic3⋊C8—22no—
17#63569—22no—
18#63569—22no—
19#63560—22no—
20#63558—22no—
21C3×#575—31no—
22#5748D311yescharacteristic
23C3×#423—31no—
24#5237—32no—
25C2×#837—32no—
26#5132—32no—
27#5118D311yescharacteristic, complemented
28C2×#805—32no—
29C3⋊#338—31no—
30C3⋊#338—31no—
31C3⋊#338—61no—
32C22×C32×C4C2311yescharacteristic
33C2×C3×#67—22no—
34C22×C3×Dic3—21no—
35C3×#235—22no—
36C3×#235—22no—
37C3×#235—22no—
38C3×#235—22no—
39C2×C3×D12D412yes—
40C2×C3×C4×D3—22no—
41C2×C3×Dic6D412yes—
42C32×M16—22no—
43C2×C32×C8D412yes—
44#3748—22no—
45#3748—22no—
46C2×C3⋊(C3⋊C8)—22no—
47C3×C4×Dic3—21no—
48C3×C4×Dic3D411yescharacteristic
49C3×C4×Dic3D411yescharacteristic
50C3×#208—22no—
51C2×C3×C3⋊C8—21no—
52C3×Dic12—22no—
53C3×D24—22no—
54C3×#204—24no—
55#2050—91noSylow
56C2×C3×#41—32no—
57C2×#235C2×D312yes—
58C3×#137—62no—
59C3×AGL(1,ℤ/8)—62no—
60C3×#135—62no—
61C2×C3×M16—21no—
62C2×C3×M16—31no—
63C2×C3×M16C2×D311yescharacteristic
64C2×C3×M16C2×D311yescharacteristic
65C2×C3×C42—31no—
66C2×C4×Dic3C2×D311yescharacteristic
67C2×#208—32no—
68C2×#208—32no—
69C2×#208—62no—
70C2×#208C2×D311yescharacteristic
71#909—22no—
72#908—22no—
73#904—22no—
74C3×C4⋊C8—62no—
75#837—62no—
76#837—62no—
77#836—62no—
78#833—62no—
79#824—62no—
80#823—62no—
81Dic3⋊C8—22no—
82#805—62no—
83#805—62no—
84C23×C32#4111yescharacteristic
85C22×C3×D3—22no—
86C2×C32×C4C2×D411yescharacteristic
87C2×C32×C4C2×D412yes—
88C2×C32×C4C2×D411yescharacteristic
89C2×C32×C4#4112yes—
90C3×#67—42no—
91C3×#67—42no—
92C2×C3×Dic3—21no—
93C2×C3×Dic3—21no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—42no—
96C3×D12—22no—
97C3×D12—22no—
98C3×C4×D3—42no—
99C3×C4×D3—42no—
100C3×Dic6—22no—
101C3×Dic6—22no—
102C32×C8—22no—
103C3⋊(C3⋊C8)—42no—
104C3×C3⋊C8—41no—
105#575—31no—
106#423—31no—
107C2×#104—92no—
108#413—92no—
109#338—91no—
110C3×#41—62no—
111C3×#41—62no—
112C3×#41—62no—
113C3×#41—62no—
114C2×C3×Q8—32no—
115C2×C3×D4—32no—
116C2×C3×D4—62no—
117C22×C3×C4—21no—
118C22×C3×C4—61no—
119C22×C3×C4—62no—
120C22×C3×C4C22×D311yescharacteristic
121C22×C3×C4C22×D311yescharacteristic
122C2×#67—22no—
123C22×Dic3—21no—
124#235—22no—
125#235—22no—
126#235—22no—
127#235—22no—
128C2×D12#6712yes—
129C2×C4×D3—22no—
130C2×Dic6#6712yes—
131C3×Q16—62no—
132C3×SD16—64no—
133C3×D8—62no—
134C3×M16—22no—
135C3×M16—22no—
136C3×M16—42no—
137C3×M16—62no—
138C2×C3×C8—22no—
139C2×C3×C8—61no—
140C2×C3×C8#6712yes—
141C2×C3×C8#6712yes—
142C3×C42—31no—
143C3×C42—31no—
144C3×C42—61no—
145C4×Dic3—21no—
146C4×Dic3D1211yescharacteristic
147C4×Dic3D1211yescharacteristic
148#208—22no—
149#208—62no—
150#208—62no—
151#208—62no—
152#208—62no—
153#208—122no—
154#208—122no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—64no—
157C2×C3⋊C8—62no—
158C2×C3⋊C8—62no—
159Dic12—22no—
160D24—22no—
161#204—24no—
162C22×C32—41no—
163C22×C32#12111yescharacteristic
164C22×C32#12312yes—
165C2×C3×D3—42no—
166C2×C3×D3—82no—
167C32×C4#12011yescharacteristic
168C32×C4#12112yes—
169C32×C4#12411yescharacteristic
170C3×Dic3—22no—
171C3×Dic3—41no—
172C3×Dic3—42no—
173C2×#41—32no—
174#137—62no—
175AGL(1,ℤ/8)—62no—
176#135—62no—
177C2×M16—31no—
178C2×M16—92no—
179C2×M16D3211yescharacteristic
180C2×C42—31no—
181C4⋊C8—62no—
182#104—182no—
183#104—182no—
184#101—182no—
185#100—182no—
186C23×C3—21no—
187C23×C3—62no—
188C23×C3#23511yescharacteristic
189C23×C3#23711yescharacteristic
190C22×D3—22no—
191C3×Q8—62no—
192C3×Q8—62no—
193C3×D4—62no—
194C3×D4—62no—
195C3×D4—122no—
196C3×D4—122no—
197C2×C3×C4—22no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—22no—
201C2×C3×C4—61no—
202C2×C3×C4—62no—
203C2×C3×C4—61no—
204C2×C3×C4—122no—
205C2×C3×C4—122no—
206C2×C3×C4—122no—
207C2×C3×C4C2×D1211yescharacteristic
208C2×C3×C4D3×D411yescharacteristic
209C2×C3×C4D3×D412yes—
210C2×C3×C4#23712yes—
211C2×C3×C4#23912yes—
212C2×C3×C4C2×#6711yescharacteristic
213C2×C3×C4C2×#6711yescharacteristic
214C2×C3×C4C2×#6712yes—
215#67—42no—
216#67—42no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—22no—
220C2×Dic3—42no—
221D12—22no—
222D12—22no—
223C4×D3—42no—
224C4×D3—42no—
225Dic6—22no—
226Dic6—22no—
227C3×C8—22no—
228C3×C8—22no—
229C3×C8—42no—
230C3×C8—121no—
231C3⋊C8—41no—
232C3⋊C8—124no—
233C3⋊C8—122no—
234C3⋊C8—122no—
235C2×C32—21no—
236C2×C32—22no—
237C2×C32#39411yescharacteristic
238C3×D3—82no—
239#41—62no—
240#41—62no—
241#41—62no—
242#41—62no—
243C2×Q8—32no—
244C2×D4—32no—
245C2×D4—62no—
246C22×C4—61no—
247C22×C4—62no—
248C22×C4C2×D3211yescharacteristic
249Q16—62no—
250SD16—64no—
251D8—62no—
252M16—22no—
253M16—62no—
254M16—182no—
255M16—182no—
256C2×C8—61no—
257C2×C8—182no—
258C2×C8#62512yes—
259C42—31no—
260C42—31no—
261C42—61no—
262C22×C3—22no—
263C22×C3—21no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—81no—
267C22×C3—122no—
268C22×C3—242no—
269C22×C3#89412yes—
270C22×C3#93011yescharacteristic
271C22×C3#93512yes—
272C22×C3#93811yescharacteristic
273C2×D3—42no—
274C2×D3—82no—
275C3×C4—21no—
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—62no—
279C3×C4—121no—
280C3×C4—122no—
281C3×C4#89512yes—
282C3×C4#93611yescharacteristic
283C3×C4#93711yescharacteristic
284C3×C4#93912yes—
285C3×C4#94711yescharacteristic
286C3×C4#95311yescharacteristic
287Dic3—22no—
288Dic3—41no—
289Dic3—42no—
290C32#205011yescharacteristic, complemented, Sylow
291C23—62no—
292C23#379511yescharacteristic
293Q8—62no—
294Q8—62no—
295D4—62no—
296D4—62no—
297D4—122no—
298D4—122no—
299C2×C4—61no—
300C2×C4—62no—
301C2×C4—61no—
302C2×C4—122no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4#379612yes—
306C2×C4D3×D1211yescharacteristic
307C2×C4C2×#62511yescharacteristic
308C2×C4D3×#6712yes—
309C8—22no—
310C8—121no—
311C8—362no—
312C2×C3—21no—
313C2×C3—22no—
314C2×C3—21no—
315C2×C3—22no—
316C2×C3—21no—
317C2×C3—41no—
318C2×C3—42no—
319C2×C3—242no—
320C2×C3#499111yescharacteristic
321C2×C3#522211yescharacteristic
322D3—82no—
323C22—41no—
324C22—122no—
325C22—242no—
326C22#6388412yes—
327C22#6390811yescharacteristic
328C4—62no—
329C4—121no—
330C4—122no—
331C4#6386311yescharacteristic
332C4#6388612yes—
333C4#6397611yescharacteristic
334C3—21no—
335C3#7053111yescharacteristic, complemented
336C3#7211711yescharacteristic, complemented
337C2—21no—
338C2—22no—
339C2—242no—
340C2#1059130011yescharacteristic
subgroupG/H
centerC4#63863
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC2×C32#394
prime radicalC3×#5748C2
hypercenterC2×M16D32
hypocenterC32#2050
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C3×#5118C2semidirect
3. C2×#63569C2semidirect
4. #10591076C2semidirect
5. #10589426C2semidirect
27. #5118D3semidirect
290. C32#2050semidirect
335. C3#70531semidirect
336. C3#72117semidirect

Maximal subgroups

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

subgroupindex
C3×#57482
C3×#51182
C2×#635692
#105910762
#105894262
#721173
#705313