Finite Group Databasebeta

← #11798312

Subgroups — #11798312

Order 1152.

487 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 84 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
2#10592836C211yesmaximal, characteristic, complemented
3#10590638C212yesmaximal, complemented
4C3×#4553C211yesmaximal, characteristic, complemented
5#10589124C212yesmaximal, complemented
6#72113—31nomaximal
7#69121—31nomaximal
8C2×C3×#235—21no—
9C2×C32×M16C2211yescharacteristic
10#64133—21no—
11C2×#3746C2211yescharacteristic, complemented
12C3×#909—22no—
13C3×#908—22no—
14C3×#904—21no—
15C3×#904C2212yes—
16#63644C2211yescharacteristic, complemented
17#63643—21no—
18C3×#812C2212yescomplemented
19#63551—22no—
20#63548—22no—
21C3×#575—31no—
22#5748D311yescharacteristic
23#5195—32no—
24#5137—31no—
25#5137—31no—
26#5137—61no—
27#4909—32no—
28C3×#349—31no—
29#4566—32no—
30#4553D311yescharacteristic, complemented
31#4518—32no—
32C22×C32×C4D411yescharacteristic
33C2×#634—21no—
34C2×C3×#67—41no—
35C3×#235—22no—
36C3×#235—22no—
37C3×#235—41no—
38C3×#235—41no—
39C2×C3×D12—21no—
40C2×C3×C4×D3—41no—
41C2×C3×Dic6—21no—
42C32×M16—22no—
43C2×C32×C8D411yescharacteristic
44C2×C32×C8C2311yescharacteristic
45#3757—41no—
46C4⋊(C3⋊Dic3)—21no—
47#3750—41no—
48#3746—22no—
49C3×Dic12—21no—
50C3×Dic12D412yes—
51C3×D24—21no—
52C3×D24D412yes—
53C3×#204—22no—
54C3×#204—22no—
55C3×C3⋊C16—22no—
56#2191—91noSylow
57C2×C3×#41—61no—
58C2×#235—21no—
59C3×#137—62no—
60C3×AGL(1,ℤ/8)—62no—
61C3×#135—32no—
62C3×#135—61no—
63C2×C3×M16—21no—
64C2×C3×M16C2×D311yescharacteristic
65C2×C3×M16C2×D311yescharacteristic
66#924—62no—
67#924—61no—
68#924—61no—
69#909—22no—
70#908—22no—
71C2×Dic12—31no—
72C2×Dic12—31no—
73C2×Dic12—61no—
74#904—21no—
75#904C2×D312yes—
76C3×M32—32no—
77#829—62no—
78#827—62no—
79#817—31no—
80#817—31no—
81#817—61no—
82#816—62no—
83#816—61no—
84#816—61no—
85#812C2×D312yescomplemented
86Dic24—62no—
87#800—62no—
88C23×C32D811yescharacteristic
89C22×C3×D3—41no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4D811yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94C2×C3⋊Dic3—41no—
95C2×C3⋊Dic3—41no—
96#634—41no—
97#634—41no—
98C3×#67—42no—
99C3×#67—42no—
100C2×C3×Dic3—41no—
101C3×D12—22no—
102C3×D12—41no—
103C3×C4×D3—42no—
104C3×C4×D3—42no—
105C3×Dic6—22no—
106C3×Dic6—41no—
107C32×C8—21no—
108C32×C8C2×D412yes—
109#575—31no—
110#510—92no—
111#470—91no—
112#360—92no—
113#349—31no—
114C3×#41—62no—
115C3×#41—62no—
116C3×#41—121no—
117C3×#41—121no—
118C2×C3×Q8—61no—
119C2×C3×D4—61no—
120C2×C3×D4—121no—
121C22×C3×C4—21no—
122C22×C3×C4—121no—
123C22×C3×C4D1211yescharacteristic
124C22×C3×C4#6711yescharacteristic
125C2×#67—41no—
126#235—22no—
127#235—22no—
128#235—41no—
129#235—41no—
130C2×D12—21no—
131C2×C4×D3—41no—
132C2×Dic6—21no—
133C2×Dic6—62no—
134C2×Dic6—61no—
135C2×Dic6—61no—
136C3×Q16—32no—
137C3×Q16—61no—
138C3×SD16—62no—
139C3×SD16—62no—
140C3×D8—32no—
141C3×D8—61no—
142C3×M16—22no—
143C3×M16—22no—
144C3×M16—42no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8D1211yescharacteristic
148C2×C3×C8#6711yescharacteristic
149C2×C3×C8C22×D311yescharacteristic
150C2×C3×C8C22×D311yescharacteristic
151#217—122no—
152#217—121no—
153#217—121no—
154C4⋊Dic3—62no—
155C4⋊Dic3—61no—
156C4⋊Dic3—61no—
157Dic3⋊C4—122no—
158Dic3⋊C4—121no—
159Dic3⋊C4—121no—
160Dic12—21no—
161Dic12—62no—
162Dic12—62no—
163Dic12—122no—
164Dic12#6712yes—
165D24—21no—
166D24#6712yes—
167#204—22no—
168#204—22no—
169C3×C16—62no—
170C3⋊C16—22no—
171C22×C32—21no—
172C22×C32—41no—
173C22×C32C2×D811yescharacteristic
174C2×C3×D3—42no—
175C2×C3×D3—82no—
176C32×C4—21no—
177C32×C4#12011yescharacteristic
178C32×C4AGL(1,ℤ/8)11yescharacteristic
179C3⋊Dic3—81no—
180C3⋊Dic3—81no—
181C3×Dic3—42no—
182C2×#41—61no—
183#137—62no—
184AGL(1,ℤ/8)—62no—
185#135—32no—
186#135—61no—
187C2×Q16—91no—
188C2×M16D3211yescharacteristic
189C22⋊Q8—181no—
190Q32—182no—
191SD32—182no—
192M32—32no—
193#106—91no—
194#103—181no—
195C23×C3—21no—
196C23×C3—121no—
197C23×C3D2411yescharacteristic
198C23×C3#21311yescharacteristic
199C22×D3—41no—
200C3×Q8—62no—
201C3×Q8—121no—
202C3×D4—62no—
203C3×D4—121no—
204C3×D4—122no—
205C3×D4—122no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—122no—
215C2×C3×C4—121no—
216C2×C3×C4—122no—
217C2×C3×C4D2411yescharacteristic
218C2×C3×C4#21311yescharacteristic
219C2×C3×C4C2×D1211yescharacteristic
220C2×C3×C4C2×#6711yescharacteristic
221#67—42no—
222#67—42no—
223C2×Dic3—41no—
224C2×Dic3—122no—
225C2×Dic3—122no—
226C2×Dic3—121no—
227C2×Dic3—121no—
228C2×Dic3—121no—
229C2×Dic3—121no—
230D12—22no—
231D12—41no—
232C4×D3—42no—
233C4×D3—42no—
234Dic6—22no—
235Dic6—41no—
236Dic6—122no—
237Dic6—122no—
238Dic6—121no—
239Dic6—121no—
240Dic6—121no—
241Dic6—121no—
242C3×C8—21no—
243C3×C8—21no—
244C3×C8—22no—
245C3×C8—41no—
246C3×C8D3×D412yes—
247C3×C8C2×#6712yes—
248C2×C32—21no—
249C2×C32—41no—
250C2×C32#44711yescharacteristic
251C3×D3—82no—
252#41—62no—
253#41—62no—
254#41—121no—
255#41—121no—
256C2×Q8—61no—
257C2×Q8—181no—
258C2×D4—61no—
259C2×D4—121no—
260C22×C4—121no—
261C22×C4#62611yescharacteristic
262Q16—32no—
263Q16—61no—
264Q16—182no—
265SD16—62no—
266SD16—62no—
267D8—32no—
268D8—61no—
269M16—22no—
270C2×C8#62611yescharacteristic
271C2×C8C2×D3211yescharacteristic
272C4⋊C4—181no—
273C4⋊C4—361no—
274C22⋊C4—361no—
275C16—62no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—42no—
283C22×C3—122no—
284C22×C3—242no—
285C22×C3C2×D2411yescharacteristic
286C22×C3C2×#21311yescharacteristic
287C2×D3—42no—
288C2×D3—82no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—41no—
294C3×C4—122no—
295C3×C4#88211yescharacteristic
296C3×C4#90811yescharacteristic
297C3×C4#93211yescharacteristic
298C3×C4#95311yescharacteristic
299Dic3—42no—
300Dic3—242no—
301Dic3—242no—
302Dic3—241no—
303Dic3—241no—
304Dic3—241no—
305Dic3—241no—
306C32#219111yescharacteristic, complemented, Sylow
307C23—121no—
308C23#371411yescharacteristic
309Q8—62no—
310Q8—121no—
311Q8—361no—
312Q8—361no—
313D4—62no—
314D4—121no—
315D4—122no—
316D4—122no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—122no—
320C2×C4—121no—
321C2×C4—122no—
322C2×C4—361no—
323C2×C4—361no—
324C2×C4#371411yescharacteristic
325C2×C4C2×#62611yescharacteristic
326C8—21no—
327C8D3×#6712yes—
328C2×C3—22no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—41no—
333C2×C3—41no—
334C2×C3—81no—
335C2×C3—242no—
336C2×C3#473211yescharacteristic
337C2×C3#521711yescharacteristic
338D3—82no—
339C22—21no—
340C22—41no—
341C22—122no—
342C22—242no—
343C22C2×#371411yescharacteristic
344C4—21no—
345C4—122no—
346C4—721no—
347C4—721no—
348C4#6383011yescharacteristic
349C4#6397711yescharacteristic
350C3—21no—
351C3#6912111yescharacteristic, complemented
352C3#7211311yescharacteristic, complemented
353C2—21no—
354C2—41no—
355C2—242no—
356C2#1059116211yescharacteristic
subgroupG/H
centerC2#10591162
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC2×C32×M16C22
socleC2×C32#447
prime radicalC3×#5748C2
hypercenterC2×M16D32
hypocenterC32#2191
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #10592836C2semidirect
3. #10590638C2semidirect
4. C3×#4553C2semidirect
5. #10589124C2semidirect
11. C2×#3746C22semidirect
16. #63644C22semidirect
18. C3×#812C22semidirect
30. #4553D3semidirect
85. #812C2×D3semidirect
306. C32#2191semidirect
351. C3#69121semidirect
352. C3#72113semidirect

Maximal subgroups

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

subgroupindex
C3×#57482
#105928362
#105906382
C3×#45532
#105891242
#721133
#691213