Finite Group Databasebeta

← #13156579

Subgroups — #13156579

Order 1728.

770 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 116 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#11723678C211yesmaximal, characteristic, complemented
2C2×#88792C212yesmaximal, complemented
3C3×#63914C211yesmaximal, characteristic
4C2×C3×#3716C212yesmaximal, complemented
5C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
6#10591490—32nomaximal
7#10590884—31nomaximal
8C2×C3×#634C2212yes—
9C2×C32×D12C2212yes—
10C2×C3×#625—22no—
11C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2211yescharacteristic
14#88815—22no—
15#88792—24no—
16C3×#3716—24no—
17C2×#3754—32no—
18C3×#940—32no—
19C2×C3×#214—32no—
20C2×C3×#204—32no—
21#63914D311yescharacteristic
22#63873—31no—
23#63866—32no—
24C2×#3716D312yescomplemented
25C2×#3715—32no—
26C2×#3715—32no—
27C2×#3715—62no—
28C8×C3⋊Dic3—31no—
29Dic3×C3⋊C8—32no—
30Dic3×C3⋊C8—61no—
31C22×C32×D3—22no—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3D412yes—
35C3×#634—22no—
36C3×#634—22no—
37C32×D12—22no—
38C32×D12—22no—
39C3×#625—44no—
40C3⋊(C3⋊(C3⋊C8))D412yescomplemented
41C32×C3⋊C8D412yescomplemented
42#5175—92no—
43#5076—91no—
44C2×C32×D4—32no—
45C2×#634C2×D312yes—
46C2×C3×#67—64no—
47C2×C3×D12—22no—
48C2×C3×D12C2×D312yes—
49C2×C3×D12C2×D312yes—
50C2×C3×Dic6—32no—
51C2×C3×Dic6—32no—
52C2×C3×Dic6—62no—
53C2×#625—22no—
54C2×C32×C8—31no—
55#3754—64no—
56C4×C3⋊Dic3C2×D311yescharacteristic
57C2×C3⋊(C3⋊C8)—32no—
58C2×C3⋊(C3⋊C8)—31no—
59C2×C3⋊(C3⋊C8)—61no—
60C2×C3⋊(C3⋊C8)—62no—
61C2×C3⋊(C3⋊C8)—121no—
62C3×#214—64no—
63C3×C4×Dic3—32no—
64C3×C4×Dic3—61no—
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8C2×D312yes—
67C3×#204—64no—
68#3724—62no—
69Dic3⋊Dic3—64no—
70#3716—24no—
71#3715—64no—
72#3715—64no—
73#3715—68no—
74C22×C33C2×D411yescharacteristic
75C2×C32×D3—42no—
76C2×C32×D3—44no—
77C33×C4C2×D411yescharacteristic
78C33×C4C2×D411yescharacteristic
79C3×C3⋊Dic3—42no—
80C3×C3⋊Dic3SD1614yescomplemented
81C2×C3×SD16—92no—
82C3×#127—91no—
83#944—92no—
84C2×#215—92no—
85#940—32no—
86C2×#214—32no—
87C2×#214—92no—
88C2×#214—92no—
89C2×#214—182no—
90C2×#204—32no—
91#869—91no—
92C8×Dic3—92no—
93C8×Dic3—181no—
94C4×C3⋊C8—91no—
95C23×C32—62no—
96C22×C3×D3—22no—
97C22×C3×D3—22no—
98C22×C3×D3—42no—
99C32×D4—62no—
100C32×D4—62no—
101C2×C32×C4—21no—
102C2×C32×C4—22no—
103C2×C32×C4—41no—
104C2×C32×C4C22×D311yescharacteristic
105C2×C32×C4C22×D312yes—
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3#6712yes—
108#634—22no—
109#634—22no—
110C3×#67—128no—
111C2×C3×Dic3—34no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—122no—
116C3×D12—24no—
117C3×D12—22no—
118C3×D12—22no—
119C3×D12—22no—
120C3×D12—22no—
121C3×D12—42no—
122C3×Dic6—64no—
123C3×Dic6—62no—
124C3×Dic6—62no—
125C3×Dic6—62no—
126C3×Dic6—62no—
127C3×Dic6—122no—
128#625—44no—
129C32×C8—32no—
130C3⋊(C3⋊C8)—32no—
131C3⋊(C3⋊C8)—34no—
132C3⋊(C3⋊C8)—64no—
133C3⋊(C3⋊C8)—62no—
134C3⋊(C3⋊C8)—122no—
135C3×C3⋊C8—22no—
136C3×C3⋊C8#6714yescomplemented
137#492—271noSylow
138C2×C33#12711yescharacteristic
139C2×C33C2×SD1612yes—
140C32×D3—82no—
141C2×C3×Q8—92no—
142C2×C3×D4—32no—
143C2×C3×D4—32no—
144C2×C3×D4—62no—
145C2×C3×D4—182no—
146C2×#67—64no—
147C2×D12D3212yes—
148C2×Dic6—32no—
149C2×Dic6—32no—
150C2×Dic6—62no—
151C3×SD16—184no—
152C2×C3×C8—32no—
153C2×C3×C8—61no—
154C3×C42—91no—
155#215—184no—
156#214—64no—
157#214—184no—
158#214—184no—
159#214—188no—
160C4⋊Dic3—182no—
161Dic3⋊C4—184no—
162C4×Dic3—32no—
163C4×Dic3—61no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—92no—
166C2×C3⋊C8—181no—
167C2×C3⋊C8—182no—
168C2×C3⋊C8—361no—
169C2×C3⋊C8D3211yescharacteristic
170#204—64no—
171C22×C32—21no—
172C22×C32—22no—
173C22×C32—41no—
174C22×C32—122no—
175C22×C32—124no—
176C22×C32D3×D412yes—
177C22×C32C2×#6711yescharacteristic
178C2×C3×D3—44no—
179C2×C3×D3—44no—
180C2×C3×D3—44no—
181C2×C3×D3—42no—
182C2×C3×D3—42no—
183C2×C3×D3—84no—
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—22no—
187C32×C4—22no—
188C32×C4—41no—
189C32×C4—41no—
190C32×C4D3×D412yes—
191C32×C4D3×D411yescharacteristic
192C32×C4D3×D411yescharacteristic
193C32×C4C2×#6712yes—
194C3⋊Dic3—42no—
195C3⋊Dic3#21414yescomplemented
196C3×Dic3—38no—
197C3×Dic3—64no—
198C3×Dic3—124no—
199C3×Dic3—122no—
200C3×Dic3—122no—
201C2×SD16—92no—
202C2×SD16—272no—
203C4⋊Q8—271no—
204#127—91no—
205C4×C8—271no—
206C33#49211yescharacteristic, complemented, Sylow
207C23×C3—62no—
208C23×C3—62no—
209C23×C3—122no—
210C22×D3—22no—
211C3×Q8—182no—
212C3×Q8—182no—
213C3×D4—64no—
214C3×D4—62no—
215C3×D4—62no—
216C3×D4—62no—
217C3×D4—62no—
218C3×D4—122no—
219C3×D4—364no—
220C2×C3×C4—21no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—92no—
224C2×C3×C4—182no—
225C2×C3×C4C2×D3211yescharacteristic
226C2×C3×C4C2×D3212yes—
227#67—128no—
228C2×Dic3—34no—
229C2×Dic3—62no—
230C2×Dic3—62no—
231C2×Dic3—62no—
232C2×Dic3—122no—
233D12—22no—
234D12—22no—
235Dic6—64no—
236Dic6—62no—
237Dic6—62no—
238Dic6—62no—
239Dic6—62no—
240Dic6—122no—
241C3×C8—34no—
242C3×C8—62no—
243C3⋊C8—92no—
244C3⋊C8—94no—
245C3⋊C8—182no—
246C3⋊C8—184no—
247C3⋊C8—362no—
248C3⋊C8#62512yescomplemented
249C2×C32—22no—
250C2×C32—21no—
251C2×C32—24no—
252C2×C32—22no—
253C2×C32—41no—
254C2×C32—42no—
255C2×C32—242no—
256C2×C32D3×SD1614yes—
257C2×C32C2×#21412yes—
258C2×C32#94012yes—
259C2×C32#94011yescharacteristic
260C3×D3—84no—
261C3×D3—82no—
262C3×D3—82no—
263C2×Q8—92no—
264C2×D4—32no—
265C2×D4—182no—
266SD16—184no—
267SD16—544no—
268C2×C8—31no—
269C2×C8—271no—
270C4⋊C4—542no—
271C42—91no—
272C22×C3—21no—
273C22×C3—22no—
274C22×C3—41no—
275C22×C3—124no—
276C22×C3—124no—
277C22×C3—122no—
278C22×C3—124no—
279C22×C3—122no—
280C22×C3—244no—
281C22×C3#380211yescharacteristic
282C22×C3D3×#6712yes—
283C2×D3—44no—
284C2×D3—42no—
285C3×C4—21no—
286C3×C4—22no—
287C3×C4—22no—
288C3×C4—21no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—94no—
292C3×C4—362no—
293C3×C4#380211yescharacteristic
294C3×C4C2×#62511yescharacteristic
295C3×C4D3×#6712yes—
296C3×C4#381112yes—
297Dic3—38no—
298Dic3—64no—
299Dic3—124no—
300Dic3—122no—
301Dic3—122no—
302C32—22no—
303C32—21no—
304C32—41no—
305C32#507611yescharacteristic, complemented
306C32#517512yescomplemented
307C23—62no—
308Q8—182no—
309Q8—182no—
310D4—62no—
311D4—62no—
312D4—364no—
313C2×C4—92no—
314C2×C4—182no—
315C2×C4D3311yescharacteristic
316C8—32no—
317C8—272no—
318C2×C3—22no—
319C2×C3—21no—
320C2×C3—22no—
321C2×C3—24no—
322C2×C3—41no—
323C2×C3—42no—
324C2×C3—244no—
325C2×C3—242no—
326C2×C3—242no—
327C2×C3#6379612yes—
328C2×C3#6391411yescharacteristic
329C2×C3D3×#21414yes—
330C2×C3#6397912yes—
331D3—82no—
332C22—122no—
333C22—124no—
334C22#8895311yescharacteristic
335C4—94no—
336C4—362no—
337C4D3×#62511yescharacteristic
338C4#8895411yescharacteristic
339C3—22no—
340C3—21no—
341C3—41no—
342C3#1059088411yescharacteristic, complemented
343C3#1059149012yescomplemented
344C2—242no—
345C2#1172283912yes—
346C2#1172302911yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63914C2
hypocenterC33#492
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723678C2semidirect
2. C2×#88792C2semidirect
4. C2×C3×#3716C2semidirect
5. C3⋊C8×C3⋊Dic3C2semidirect
24. C2×#3716D3semidirect
40. C3⋊(C3⋊(C3⋊C8))D4semidirect
41. C32×C3⋊C8D4semidirect
80. C3×C3⋊Dic3SD16semidirect
136. C3×C3⋊C8#67semidirect
195. C3⋊Dic3#214semidirect
206. C33#492semidirect
248. C3⋊C8#625semidirect
305. C32#5076semidirect
306. C32#5175semidirect
342. C3#10590884semidirect
343. C3#10591490semidirect

Maximal subgroups

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

subgroupindex
#117236782
C2×#887922
C3×#639142
C2×C3×#37162
C3⋊C8×C3⋊Dic32
#105914903
#105908843