Finite Group Databasebeta

← #11798282

Subgroups — #11798282

Order 1152.

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

normal subgroups only

This group has 481 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, complemented
2C3×#5135C211yesmaximal, characteristic, complemented
3#10590630C212yesmaximal, complemented
4#10589419C211yesmaximal, characteristic, complemented
5#10589123C212yesmaximal, complemented
6#72112—31nomaximal
7#71089—31nomaximal
8C2×C3×#235—21no—
9C2×C32×M16C2211yescharacteristic
10C3×#930—21no—
11C3×#909—22no—
12C3×#908—22no—
13C3×#904—21no—
14C3×#904C2212yes—
15C2×C3×D24C2211yescharacteristic, complemented
16#63639C2212yescomplemented
17C3×#821—21no—
18C3×#817C2211yescharacteristic, complemented
19#63547—22no—
20#63545—22no—
21C3×#575—31no—
22#5748D311yescharacteristic, complemented
23C3×#469—31no—
24#5193—32no—
25#5147—32no—
26#5135D311yescharacteristic, complemented
27#4566—32no—
28#4553—31no—
29#4553—31no—
30#4553—61no—
31#4488—32no—
32C22×C32×C4D411yescharacteristic
33C2×C3×#67—41no—
34C2×C3×#67—41no—
35C3×#235—22no—
36C3×#235—22no—
37C3×#235—41no—
38C3×#235—41no—
39C2×C3×D12—21no—
40C2×C3×D12—21no—
41C2×C3×C4×D3—41no—
42C2×C3×Dic6—21no—
43C32×M16—22no—
44C2×C32×C8D411yescharacteristic
45C2×C32×C8C2311yescharacteristic
46C3×(C2×D3)⋊C4—41no—
47C3×C4⋊Dic3—21no—
48C3×Dic12—21no—
49C3×Dic12D412yescomplemented
50C3×D24—21no—
51C3×D24—22no—
52C3×D24D412yescomplemented
53C3×#204—22no—
54C3×#204—22no—
55C3⋊(C3⋊C16)—22no—
56#2190—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×D8—31no—
64C2×C3×M16—21no—
65C2×C3×M16C2×D311yescharacteristic
66C2×C3×M16C2×D311yescharacteristic
67C3×#121—61no—
68#930—21no—
69#909—22no—
70#908—22no—
71#904—21no—
72#904C2×D312yes—
73C2×D24C2×D311yescharacteristic, complemented
74C3×#106—31no—
75C3×#102—61no—
76#828—62no—
77#827—62no—
78#826—62no—
79#826—62no—
80#821—21no—
81#817C2×D311yescharacteristic, complemented
82#812—32no—
83#812—32no—
84#812—62no—
85C23×C32D811yescharacteristic
86C22×C3×D3—41no—
87C22×C3×D3—41no—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4D811yescharacteristic
91C2×C32×C4C2×D411yescharacteristic
92C3×#67—42no—
93C3×#67—42no—
94C3×#67—82no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—41no—
97C3×D12—22no—
98C3×D12—41no—
99C3×D12—41no—
100C3×D12—41no—
101C3×C4×D3—42no—
102C3×C4×D3—42no—
103C3×Dic6—22no—
104C3×Dic6—41no—
105C32×C8—21no—
106C32×C8C2×D412yes—
107#575—31no—
108#509—92no—
109#469—31no—
110#360—92no—
111#349—91no—
112C3×#41—62no—
113C3×#41—62no—
114C3×#41—121no—
115C3×#41—121no—
116C2×C3×Q8—61no—
117C2×C3×D4—61no—
118C2×C3×D4—61no—
119C2×C3×D4—121no—
120C2×C3×D4—121no—
121C22×C3×C4—21no—
122C22×C3×C4—121no—
123C22×C3×C4#6711yescharacteristic
124C22×C3×C4#6711yescharacteristic
125C2×#67—41no—
126C2×#67—41no—
127#235—22no—
128#235—22no—
129#235—41no—
130#235—41no—
131C2×D12—21no—
132C2×D12—21no—
133C2×C4×D3—41no—
134C2×Dic6—21no—
135C3×Q16—32no—
136C3×Q16—61no—
137C3×SD16—62no—
138C3×SD16—62no—
139C3×D8—32no—
140C3×D8—61no—
141C3×D8—62no—
142C3×M16—22no—
143C3×M16—22no—
144C3×M16—42no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8#6711yescharacteristic
148C2×C3×C8#6711yescharacteristic
149C2×C3×C8C22×D311yescharacteristic
150C2×C3×C8C22×D311yescharacteristic
151C3×C4⋊C4—61no—
152C3×C22⋊C4—121no—
153(C2×D3)⋊C4—41no—
154C4⋊Dic3—21no—
155Dic12—21no—
156Dic12#6712yescomplemented
157D24—21no—
158D24—22no—
159D24#6712yescomplemented
160#204—22no—
161#204—22no—
162C3⋊C16—62no—
163C3⋊C16—62no—
164C3⋊C16—122no—
165C22×C32—21no—
166C22×C32—41no—
167C22×C32C2×D811yescharacteristic
168C2×C3×D3—42no—
169C2×C3×D3—82no—
170C2×C3×D3—81no—
171C2×C3×D3—81no—
172C2×C3×D3—81no—
173C32×C4—21no—
174C32×C4#12011yescharacteristic
175C32×C4AGL(1,ℤ/8)11yescharacteristic
176C3×Dic3—42no—
177C3×Dic3—81no—
178C2×#41—61no—
179#137—62no—
180AGL(1,ℤ/8)—62no—
181#135—32no—
182#135—61no—
183C2×D8—31no—
184C2×M16D3211yescharacteristic
185#121—61no—
186SD32—182no—
187D16—182no—
188M32—92no—
189#106—31no—
190#102—61no—
191C23×C3—21no—
192C23×C3—121no—
193C23×C3—121no—
194C23×C3#21311yescharacteristic
195C23×C3#21311yescharacteristic
196C22×D3—41no—
197C22×D3—41no—
198C3×Q8—62no—
199C3×Q8—121no—
200C3×D4—62no—
201C3×D4—121no—
202C3×D4—122no—
203C3×D4—122no—
204C3×D4—121no—
205C3×D4—121no—
206C3×D4—242no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—122no—
216C2×C3×C4—121no—
217C2×C3×C4—122no—
218C2×C3×C4—121no—
219C2×C3×C4#21311yescharacteristic
220C2×C3×C4#21311yescharacteristic
221C2×C3×C4C2×#6711yescharacteristic
222C2×C3×C4C2×#6711yescharacteristic
223#67—42no—
224#67—42no—
225#67—82no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228D12—22no—
229D12—41no—
230D12—41no—
231D12—41no—
232C4×D3—42no—
233C4×D3—42no—
234Dic6—22no—
235Dic6—41no—
236C3×C8—21no—
237C3×C8—21no—
238C3×C8—22no—
239C3×C8—41no—
240C3×C8D3×D412yes—
241C3×C8C2×#6712yes—
242C2×C32—21no—
243C2×C32—41no—
244C2×C32#44711yescharacteristic
245C3×D3—82no—
246C3×D3—161no—
247#41—62no—
248#41—62no—
249#41—121no—
250#41—121no—
251C2×Q8—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—121no—
255C2×D4—121no—
256C22×C4—121no—
257C22×C4#62511yescharacteristic
258Q16—32no—
259Q16—61no—
260SD16—62no—
261SD16—62no—
262D8—32no—
263D8—61no—
264D8—62no—
265M16—22no—
266C2×C8#62511yescharacteristic
267C2×C8C2×D3211yescharacteristic
268C4⋊C4—61no—
269C22⋊C4—121no—
270C16—182no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—81no—
278C22×C3—122no—
279C22×C3—241no—
280C22×C3—241no—
281C22×C3—242no—
282C22×C3—241no—
283C22×C3C2×#21311yescharacteristic
284C22×C3C2×#21311yescharacteristic
285C2×D3—42no—
286C2×D3—82no—
287C2×D3—81no—
288C2×D3—81no—
289C2×D3—81no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—41no—
295C3×C4—122no—
296C3×C4—241no—
297C3×C4#93211yescharacteristic
298C3×C4#93711yescharacteristic
299C3×C4#94911yescharacteristic
300C3×C4#95311yescharacteristic
301Dic3—42no—
302Dic3—81no—
303C32#219011yescharacteristic, complemented, Sylow
304C23—121no—
305C23—121no—
306C23#371311yescharacteristic
307Q8—62no—
308Q8—121no—
309D4—62no—
310D4—121no—
311D4—122no—
312D4—122no—
313D4—121no—
314D4—121no—
315D4—242no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—122no—
319C2×C4—121no—
320C2×C4—122no—
321C2×C4—121no—
322C2×C4#371311yescharacteristic
323C2×C4C2×#62511yescharacteristic
324C8—21no—
325C8D3×#6712yes—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—41no—
332C2×C3—81no—
333C2×C3—242no—
334C2×C3—481no—
335C2×C3#503611yescharacteristic
336C2×C3#521711yescharacteristic
337D3—82no—
338D3—161no—
339C22—21no—
340C22—41no—
341C22—122no—
342C22—241no—
343C22—241no—
344C22—242no—
345C22—241no—
346C22C2×#371311yescharacteristic
347C4—21no—
348C4—122no—
349C4—241no—
350C4#6382311yescharacteristic
351C4#6397611yescharacteristic
352C3—21no—
353C3#7108911yescharacteristic, complemented
354C3#7211211yescharacteristic, complemented
355C2—21no—
356C2—41no—
357C2—242no—
358C2—481no—
359C2#1059116011yescharacteristic
subgroupG/H
centerC2#10591160
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC2×C32×M16C22
socleC2×C32#447
hypercenterC2×M16D32
hypocenterC32#2190
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
1. C3×#5748C2semidirect
2. C3×#5135C2semidirect
3. #10590630C2semidirect
4. #10589419C2semidirect
5. #10589123C2semidirect
15. C2×C3×D24C22semidirect
16. #63639C22semidirect
18. C3×#817C22semidirect
22. #5748D3semidirect
26. #5135D3semidirect
49. C3×Dic12D4semidirect
52. C3×D24D4semidirect
73. C2×D24C2×D3semidirect
81. #817C2×D3semidirect
156. Dic12#67semidirect
159. D24#67semidirect
303. C32#2190semidirect
353. C3#71089semidirect
354. C3#72112semidirect

Maximal subgroups

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

subgroupindex
C3×#57482
C3×#51352
#105906302
#105894192
#105891232
#721123
#710893