Finite Group Databasebeta

← #11792777

Subgroups — #11792777

Order 1152.

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

normal subgroups only

This group has 729 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
1C22×C3×Dic12C211yesmaximal, characteristic, complemented
2C2×#63827C212yesmaximal
3C3×#5110C211yesmaximal, characteristic
4C2×#63560C212yesmaximal, complemented
5#10589418C211yesmaximal, characteristic, complemented
6#72074—31nomaximal
7#70008—31nomaximal
8C2×C3×#235C2211yescharacteristic
9C22×C3×Dic6C2212yes—
10C22×C32×C8C2211yescharacteristic
11C2×#3748C2212yes—
12C2×C3×#208C2211yescharacteristic
13C2×C3×Dic12—24no—
14C2×C3×Dic12—22no—
15C2×#3718—22no—
16#63827—24no—
17#63827—24no—
18C2×#3715—22no—
19C3×(C2×D3)⋊C8—22no—
20#63560—24no—
21C22×C3×Q16—31no—
22C2×#951—32no—
23C2×#942—32no—
24C22×Dic12D311yescharacteristic, complemented
25C3×#408—31no—
26C2×#836—32no—
27C2×#824—32no—
28#5110D311yescharacteristic
29#4552—31no—
30#4552—31no—
31#4552—61no—
32C22×C32×C4C2311yescharacteristic
33C2×C3×#67—21no—
34C22×C3×Dic3—22no—
35C3×#235—22no—
36C3×#235—22no—
37C2×C3×D12D411yescharacteristic
38C2×C3×C4×D3—21no—
39C2×C3×Dic6—22no—
40C2×C3×Dic6—22no—
41C2×C3×Dic6—22no—
42C2×C3×Dic6—22no—
43C2×C3×Dic6—22no—
44C2×C3×Dic6D411yescharacteristic
45C2×C3×Dic6D414yes—
46C2×C32×C8—22no—
47C2×C32×C8D412yes—
48#3748—22no—
49#3748D414yes—
50C2×C3⋊(C3⋊C8)—22no—
51C3×#208—22no—
52C2×C3×C3⋊C8—21no—
53C3×Dic12—28no—
54#3718—28no—
55#3715—28no—
56#2020—91noSylow
57C2×C3×#41—31no—
58C22×C3×Q8—32no—
59C2×#235C2×D311yescharacteristic
60C22×Dic6C2×D312yes—
61C2×C3×Q16—64no—
62C2×C3×Q16—62no—
63C2×C3×M16—31no—
64C22×C3×C8—21no—
65C22×C3×C8C2×D311yescharacteristic
66C22×C3×C8C2×D311yescharacteristic
67#951—64no—
68#951—64no—
69C2×#216—62no—
70C2×#216—62no—
71#942—64no—
72#942—64no—
73C2×#215—62no—
74C2×#214—62no—
75C2×#208—32no—
76C2×#208—32no—
77C2×#208—62no—
78C2×#208C2×D311yescharacteristic
79C2×Dic12—24no—
80C2×Dic12—22no—
81C3×C22⋊C8—62no—
82#836—64no—
83#824—64no—
84(C2×D3)⋊C8—22no—
85C23×C32#4111yescharacteristic
86C22×C3×D3—21no—
87C2×C32×C4C2×D411yescharacteristic
88C2×C32×C4C2×D412yes—
89C2×C32×C4C2×D412yes—
90C2×C32×C4C2×D411yescharacteristic
91C3×#67—42no—
92C2×C3×Dic3—24no—
93C2×C3×Dic3—21no—
94C2×C3×Dic3—42no—
95C2×C3×Dic3—42no—
96C3×D12—22no—
97C3×C4×D3—41no—
98C3×C4×D3—41no—
99C3×Dic6—24no—
100C3×Dic6—22no—
101C3×Dic6—24no—
102C3×Dic6—42no—
103C3×Dic6—42no—
104C32×C8—22no—
105C3⋊(C3⋊C8)—24no—
106C3×C3⋊C8—41no—
107C2×#137—92no—
108C22×Q16—31no—
109C2×#101—92no—
110#408—31no—
111#341—91no—
112C3×#41—62no—
113C3×#41—62no—
114C2×C3×Q8—34no—
115C2×C3×Q8—31no—
116C2×C3×Q8—62no—
117C2×C3×Q8—62no—
118C2×C3×Q8—62no—
119C2×C3×Q8—62no—
120C2×C3×Q8—62no—
121C2×C3×D4—31no—
122C2×C3×D4—61no—
123C22×C3×C4—21no—
124C22×C3×C4—62no—
125C22×C3×C4—61no—
126C22×C3×C4C22×D311yescharacteristic
127C22×C3×C4C22×D311yescharacteristic
128C2×#67—21no—
129C22×Dic3—22no—
130#235—22no—
131#235—22no—
132C2×D12D1211yescharacteristic
133C2×C4×D3—21no—
134C2×Dic6—22no—
135C2×Dic6—22no—
136C2×Dic6—22no—
137C2×Dic6—22no—
138C2×Dic6—22no—
139C2×Dic6D1211yescharacteristic
140C2×Dic6#6714yes—
141C3×Q16—68no—
142C3×M16—62no—
143C2×C3×C8—22no—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C2×C3×C8—42no—
147C2×C3×C8—61no—
148C2×C3×C8#6712yes—
149C2×C3×C8#6712yes—
150#216—68no—
151#216—68no—
152#215—68no—
153#214—68no—
154#208—22no—
155#208—34no—
156#208—34no—
157#208—64no—
158#208—62no—
159#208—62no—
160#208—122no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—62no—
163C2×C3⋊C8—62no—
164C2×C3⋊C8—122no—
165Dic12—28no—
166C22×C32—22no—
167C22×C32#12112yes—
168C22×C32#12111yescharacteristic
169C2×C3×D3—41no—
170C2×C3×D3—42no—
171C32×C4#12012yes—
172C32×C4#12011yescharacteristic
173C32×C4#12711yescharacteristic
174C3×Dic3—44no—
175C3×Dic3—41no—
176C2×#41—31no—
177C22×Q8—32no—
178#137—184no—
179#137—184no—
180C2×Q16—64no—
181C2×Q16—62no—
182C2×Q16—182no—
183C2×SD16—182no—
184C2×M16—31no—
185C2×M16—92no—
186C22×C8D3211yescharacteristic
187#101—184no—
188C22⋊C8—62no—
189C23×C3—21no—
190C23×C3—61no—
191C23×C3#23511yescharacteristic
192C23×C3#23711yescharacteristic
193C22×D3—21no—
194C3×Q8—64no—
195C3×Q8—62no—
196C3×Q8—64no—
197C3×Q8—122no—
198C3×Q8—122no—
199C3×D4—62no—
200C3×D4—122no—
201C2×C3×C4—21no—
202C2×C3×C4—22no—
203C2×C3×C4—21no—
204C2×C3×C4—22no—
205C2×C3×C4—64no—
206C2×C3×C4—61no—
207C2×C3×C4—122no—
208C2×C3×C4—122no—
209C2×C3×C4—121no—
210C2×C3×C4—121no—
211C2×C3×C4C2×D1211yescharacteristic
212C2×C3×C4D3×D411yescharacteristic
213C2×C3×C4D3×D412yes—
214C2×C3×C4D3×D412yes—
215C2×C3×C4D3×D412yes—
216C2×C3×C4C2×#6711yescharacteristic
217C2×C3×C4C2×#6711yescharacteristic
218C2×C3×C4C2×#6712yes—
219#67—42no—
220C2×Dic3—21no—
221C2×Dic3—24no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224D12—22no—
225C4×D3—41no—
226C4×D3—41no—
227Dic6—22no—
228Dic6—24no—
229Dic6—24no—
230Dic6—42no—
231Dic6—42no—
232C3×C8—22no—
233C3×C8—22no—
234C3×C8—42no—
235C3×C8—121no—
236C3⋊C8—41no—
237C3⋊C8—64no—
238C3⋊C8—64no—
239C3⋊C8—124no—
240C2×C32—22no—
241C2×C32#39211yescharacteristic
242C2×C32#47312yes—
243C3×D3—81no—
244#41—62no—
245#41—62no—
246C2×Q8—31no—
247C2×Q8—34no—
248C2×Q8—62no—
249C2×Q8—62no—
250C2×Q8—62no—
251C2×Q8—62no—
252C2×Q8—62no—
253C2×D4—31no—
254C2×D4—61no—
255C22×C4—62no—
256C22×C4—61no—
257C22×C4C2×D3211yescharacteristic
258Q16—68no—
259Q16—188no—
260SD16—188no—
261M16—62no—
262M16—94no—
263M16—182no—
264C2×C8—22no—
265C2×C8—61no—
266C2×C8—182no—
267C2×C8#62512yes—
268C22×C3—22no—
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—21no—
272C22×C3—42no—
273C22×C3—122no—
274C22×C3—121no—
275C22×C3#88412yes—
276C22×C3#93011yescharacteristic
277C22×C3#93811yescharacteristic
278C22×C3#93912yes—
279C2×D3—42no—
280C2×D3—41no—
281C3×C4—21no—
282C3×C4—22no—
283C3×C4—21no—
284C3×C4—124no—
285C3×C4—121no—
286C3×C4#88212yes—
287C3×C4#93712yes—
288C3×C4#93711yescharacteristic
289C3×C4#94011yescharacteristic
290C3×C4#94011yescharacteristic
291C3×C4#95311yescharacteristic
292Dic3—41no—
293Dic3—44no—
294C32#202011yescharacteristic, complemented, Sylow
295C23—61no—
296C23#379511yescharacteristic
297Q8—64no—
298Q8—62no—
299Q8—64no—
300Q8—122no—
301Q8—122no—
302D4—62no—
303D4—122no—
304C2×C4—64no—
305C2×C4—61no—
306C2×C4—122no—
307C2×C4—122no—
308C2×C4—121no—
309C2×C4—121no—
310C2×C4D3×D1211yescharacteristic
311C2×C4#380212yes—
312C2×C4C2×#62511yescharacteristic
313C2×C4D3×#6712yes—
314C8—22no—
315C8—121no—
316C8—184no—
317C2×C3—22no—
318C2×C3—22no—
319C2×C3—21no—
320C2×C3—22no—
321C2×C3—42no—
322C2×C3—241no—
323C2×C3#496011yescharacteristic
324C2×C3#514312yes—
325C2×C3#519212yes—
326C2×C3#522311yescharacteristic
327D3—81no—
328C22—22no—
329C22—121no—
330C22—122no—
331C22#6388512yes—
332C22#6390811yescharacteristic
333C4—121no—
334C4—124no—
335C4#6391411yescharacteristic
336C4#6392212yes—
337C4#6397611yescharacteristic
338C3—21no—
339C3#7000811yescharacteristic, complemented
340C3#7207411yescharacteristic, complemented
341C2—22no—
342C2—241no—
343C2#1059093012yes—
344C2#1059138911yescharacteristic
subgroupG/H
centerC22#63908
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC22×C32×C8C22
socleC22×C32#121
prime radicalC2×C3×#235C22
hypercenterC22×C8D32
hypocenterC32#2020
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C22×C3×Dic12C2semidirect
4. C2×#63560C2semidirect
5. #10589418C2semidirect
24. C22×Dic12D3semidirect
294. C32#2020semidirect
339. C3#70008semidirect
340. C3#72074semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic122
C2×#638272
C3×#51102
C2×#635602
#105894182
#720743
#700083