Finite Group Databasebeta

← #13156236

Subgroups — #13156236

Order 1728.

606 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×#88810C211yesmaximal, characteristic, complemented
2#11723349C211yesmaximal, characteristic, complemented
3C3×#63825C212yesmaximal, complemented
4#11722026C212yesmaximal, complemented
5C3×#63578C211yesmaximal, characteristic, complemented
6#10591678—31nomaximal
7#10590984—32nomaximal
8C2×C3×C4⋊(C3⋊D3)C2212yes—
9C32×#235C2211yescharacteristic
10#88855C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×#208C2212yes—
13#88810—21no—
14#88810—21no—
15#88786—21no—
16D3×C3⋊(C3⋊C8)—21no—
17C3×#3717—22no—
18C3×#3714—22no—
19#64157—31no—
20C3×#949—32no—
21C3×#908—32no—
22#63825D312yescomplemented
23C2×#3714—32no—
24C2×#3713—31no—
25#63814—32no—
26#63814—61no—
27#63650—32no—
28C3×#822—32no—
29#63578D311yescharacteristic, complemented
30#63558—32no—
31#63558—32no—
32#63558—62no—
33C22×C3×C3⋊D3—22no—
34C2×C33×C4C2311yescharacteristic
35C3×C4⋊(C3⋊D3)—22no—
36C3×C4⋊(C3⋊D3)—22no—
37C32×#67—21no—
38C32×D12D411yescharacteristic
39C32×C4×D3—21no—
40C32×Dic6D411yescharacteristic
41C3⋊(C3⋊(C3⋊C8))—21no—
42C3×C3⋊(C3⋊C8)D411yescharacteristic
43C3×C3⋊(C3⋊C8)D411yescharacteristic
44C32×C3⋊C8—22no—
45#5199—92no—
46#4897—91no—
47C32×#41—31no—
48C2×C4⋊(C3⋊D3)C2×D312yes—
49C3×#235—21no—
50C3×#235C2×D312yes—
51C2×C3×D12—32no—
52C2×C3×D12—32no—
53C2×C3×D12—62no—
54C32×M16—32no—
55#3748—31no—
56#3748—32no—
57#3748—61no—
58#3748—61no—
59#3748—62no—
60#3748—121no—
61C2×C3⋊(C3⋊C8)—61no—
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C3×#215—62no—
64C3×#213—62no—
65C3×#208—22no—
66C3×#208C2×D312yes—
67C3×#208C2×D312yes—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—61no—
70C3×D24—62no—
71C3×#204—62no—
72#3717—22no—
73#3714—22no—
74#3714—62no—
75#3714—62no—
76#3713—61no—
77#3713—61no—
78#3711—62no—
79#3711—62no—
80D3×C3⋊C8—62no—
81D3×C3⋊C8—62no—
82C22×C33#4111yescharacteristic
83C2×C3×C3⋊D3—42no—
84C2×C3×C3⋊D3—82no—
85C2×C32×D3—21no—
86C33×C4C2×D411yescharacteristic
87C33×C4C2×D411yescharacteristic
88C32×Dic3—21no—
89C3×AGL(1,ℤ/8)—92no—
90#949—32no—
91#948—92no—
92#948—181no—
93C2×#213—92no—
94#908—32no—
95C2×D24—91no—
96#901—91no—
97C3×#108—91no—
98#833—92no—
99#823—92no—
100#823—92no—
101#823—182no—
102#822—32no—
103C22×C3⋊D3—22no—
104C22×C3×D3—64no—
105C22×C3×D3—62no—
106C22×C3×D3—62no—
107C32×Q8—31no—
108C32×D4—31no—
109C32×D4—61no—
110C2×C32×C4—21no—
111C2×C32×C4—22no—
112C2×C32×C4—41no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C32×C4C22×D312yes—
116C4⋊(C3⋊D3)—22no—
117C4⋊(C3⋊D3)—22no—
118C3×#67—22no—
119C3×#67—22no—
120C3×D12—21no—
121C3×D12—62no—
122C3×D12—62no—
123C3×D12—62no—
124C3×D12—62no—
125C3×D12—122no—
126C3×D12—122no—
127C3×D12#6712yes—
128C3×C4×D3—22no—
129C3×C4×D3—22no—
130C3×Dic6—21no—
131C3×Dic6#6712yes—
132C32×C8—62no—
133C3⋊(C3⋊C8)—61no—
134C3⋊(C3⋊C8)—62no—
135C3⋊(C3⋊C8)—62no—
136C3⋊(C3⋊C8)—122no—
137C3⋊(C3⋊C8)—122no—
138C3⋊(C3⋊C8)D1211yescharacteristic
139C3⋊(C3⋊C8)D1211yescharacteristic
140C3×C3⋊C8—22no—
141C3×C3⋊C8—22no—
142C3×C3⋊C8—32no—
143C3×C3⋊C8—32no—
144C3×C3⋊C8—42no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147#472—271noSylow
148C2×C33—21no—
149C2×C33#12111yescharacteristic
150C3×C3⋊D3—82no—
151C32×D3—41no—
152C3×#41—32no—
153C3×#41—61no—
154C2×C3×D4—92no—
155#235D3211yescharacteristic
156C2×D12—32no—
157C2×D12—32no—
158C2×D12—62no—
159C3×SD16—182no—
160C3×D8—182no—
161C3×M16—32no—
162C3×M16—32no—
163C3×M16—62no—
164C2×C3×C8—91no—
165#215—62no—
166#213—62no—
167#213—182no—
168#213—182no—
169#208—91no—
170#208—92no—
171#208—181no—
172#208—182no—
173#208—182no—
174#208—182no—
175#208—361no—
176#208D3212yes—
177C2×C3⋊C8—32no—
178C2×C3⋊C8—61no—
179C2×C3⋊C8—182no—
180C2×C3⋊C8—182no—
181D24—62no—
182D24—181no—
183D24—181no—
184#204—62no—
185ΓL(1,𝔽25)—181no—
186D3×C8—181no—
187C22×C32—22no—
188C22×C32—21no—
189C22×C32—41no—
190C22×C32—61no—
191C22×C32#23712yes—
192C22×C32#23911yescharacteristic
193C2×C3⋊D3—42no—
194C2×C3⋊D3—82no—
195C2×C3×D3—22no—
196C2×C3×D3—22no—
197C2×C3×D3—124no—
198C2×C3×D3—122no—
199C2×C3×D3—122no—
200C2×C3×D3—242no—
201C2×C3×D3—242no—
202C2×C3×D3—244no—
203C32×C4—22no—
204C32×C4—21no—
205C32×C4—22no—
206C32×C4—21no—
207C32×C4—41no—
208C32×C4—41no—
209C32×C4—61no—
210C32×C4C2×D1211yescharacteristic
211C32×C4D3×D412yes—
212C32×C4D3×D411yescharacteristic
213C32×C4C2×#6712yes—
214C3×Dic3—22no—
215C3×Dic3—22no—
216AGL(1,ℤ/8)—92no—
217C2×D8—271no—
218#131—271no—
219#108—91no—
220#100—272no—
221C33#47211yescharacteristic, complemented, Sylow
222C23×C3—182no—
223C22×D3—64no—
224C22×D3—62no—
225C22×D3—62no—
226C3×Q8—32no—
227C3×Q8—61no—
228C3×D4—32no—
229C3×D4—61no—
230C3×D4—62no—
231C3×D4—62no—
232C3×D4—182no—
233C3×D4—182no—
234C2×C3×C4—21no—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4C2×D3212yes—
240C2×C3×C4C2×D3211yescharacteristic
241#67—21no—
242D12—62no—
243D12—62no—
244D12—62no—
245D12—62no—
246D12—122no—
247D12—122no—
248D12#62511yescharacteristic
249C4×D3—21no—
250Dic6#62511yescharacteristic
251C3×C8—62no—
252C3×C8—62no—
253C3×C8—91no—
254C3×C8—91no—
255C3×C8—122no—
256C3⋊C8—22no—
257C3⋊C8—32no—
258C3⋊C8—32no—
259C3⋊C8—61no—
260C3⋊C8—61no—
261C3⋊C8—182no—
262C3⋊C8—182no—
263C3⋊C8—181no—
264C3⋊C8—362no—
265C3⋊C8—362no—
266C2×C32—21no—
267C2×C32—22no—
268C2×C32—22no—
269C2×C32—21no—
270C2×C32—41no—
271C2×C32—42no—
272C2×C32—41no—
273C2×C32—81no—
274C2×C32—121no—
275C2×C32#89511yescharacteristic
276C2×C32#93912yes—
277C3⋊D3—82no—
278C3×D3—42no—
279C3×D3—42no—
280C3×D3—244no—
281C3×D3—242no—
282C3×D3—242no—
283#41—31no—
284C2×D4—92no—
285SD16—182no—
286D8—182no—
287D8—541no—
288D8—541no—
289M16—32no—
290M16—271no—
291M16—541no—
292C2×C8—91no—
293C2×C8—541no—
294C22×C3—21no—
295C22×C3—22no—
296C22×C3—41no—
297C22×C3—62no—
298C22×C3—62no—
299C22×C3—362no—
300C22×C3—722no—
301C22×C3#379612yes—
302C22×C3#380511yescharacteristic
303C2×D3—21no—
304C2×D3—124no—
305C2×D3—122no—
306C2×D3—122no—
307C2×D3—242no—
308C2×D3—242no—
309C2×D3—244no—
310C3×C4—22no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—22no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4—62no—
317C3×C4—62no—
318C3×C4D3×D1212yes—
319C3×C4C2×#62511yescharacteristic
320C3×C4D3×#6712yes—
321C3×C4#381111yescharacteristic
322Dic3—21no—
323C32—22no—
324C32—21no—
325C32—41no—
326C32#489711yescharacteristic, complemented
327C32#519912yescomplemented
328C23—182no—
329Q8—31no—
330D4—31no—
331D4—61no—
332D4—182no—
333D4—182no—
334C2×C4—61no—
335C2×C4D3311yescharacteristic
336C8—62no—
337C8—91no—
338C8—91no—
339C8—541no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—42no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—81no—
348C2×C3—122no—
349C2×C3—122no—
350C2×C3—722no—
351C2×C3#6388612yes—
352C2×C3#6398011yescharacteristic
353D3—41no—
354D3—244no—
355D3—242no—
356D3—242no—
357C22—61no—
358C22—362no—
359C22—722no—
360C22#8896411yescharacteristic
361C4—61no—
362C4D3×#62511yescharacteristic
363C4#8895511yescharacteristic
364C3—22no—
365C3—21no—
366C3—41no—
367C3#1059098412yescomplemented
368C3#1059167811yescharacteristic, complemented
369C2—21no—
370C2—121no—
371C2—722no—
372C2#1172295511yescharacteristic
subgroupG/H
centerC2#11722955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC2×C33#121
hypocenterC33#472
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88810C2semidirect
2. #11723349C2semidirect
3. C3×#63825C2semidirect
4. #11722026C2semidirect
5. C3×#63578C2semidirect
22. #63825D3semidirect
29. #63578D3semidirect
221. C33#472semidirect
326. C32#4897semidirect
327. C32#5199semidirect
367. C3#10590984semidirect
368. C3#10591678semidirect

Maximal subgroups

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

subgroupindex
C2×#888102
#117233492
C3×#638252
#117220262
C3×#635782
#105916783
#105909843