Finite Group Databasebeta

← #13160326

Subgroups — #13160326

Order 1728.

836 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 164 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×#64139C211yesmaximal, characteristic
2C32×#926C211yesmaximal, characteristic, complemented
3#11722200C211yesmaximal, characteristic, complemented
4#11722056C211yesmaximal, characteristic, complemented
5#11722053C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic, complemented
8#10592702—31nomaximal
9#10591090—34nomaximal
10C22×C33×C4C2211yescharacteristic
11C2×C3×#638—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×#217—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C32×C4×Dic3—21no—
21C32×#117—31no—
22#64139D311yescharacteristic
23C3×#930—34no—
24C3×#926D314yescomplemented
25#63657—31no—
26#63657—34no—
27#63657—64no—
28#63636—31no—
29#63636—31no—
30#63635—31no—
31#63634—31no—
32#63576—34no—
33#63574—34no—
34#63562—34no—
35#63562—34no—
36C23×C33D411yescharacteristic
37C22×C3×C3⋊D3—21no—
38C2×C33×C4—21no—
39C2×C33×C4—21no—
40C2×C33×C4D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C3×#638—42no—
43C2×C3×C3⋊Dic3—21no—
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C2×C32×Dic3—21no—
47C2×C32×Dic3—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50#5035—91no—
51#5003—94no—
52C22×C32×C4—24no—
53C22×C32×C4C2×D314yes—
54C22×C32×C4C2×D311yescharacteristic
55C2×#638—21no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×C3×#67—64no—
58C2×C3×D12—34no—
59C32×C4⋊C4—31no—
60C32×C4⋊C4—31no—
61C32×C22⋊C4—61no—
62C32×C42—61no—
63#3752—21no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)—31no—
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—34no—
68C2×C3⋊(C3⋊C8)—34no—
69C2×C3⋊(C3⋊C8)—64no—
70C2×C3⋊(C3⋊C8)—64no—
71C3×#217—24no—
72C3×(C2×D3)⋊C4—64no—
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3C2×D314yes—
75C3×C4⋊Dic3C2×D314yes—
76C3×C4×Dic3—24no—
77C22×C33—21no—
78C22×C33—21no—
79C22×C33—21no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C33×C4—21no—
85C33×C4#4111yescharacteristic
86C33×C4#4111yescharacteristic
87C3×C3⋊Dic3—41no—
88C32×Dic3—41no—
89C32×Dic3—41no—
90C3×#121—91no—
91C3×#117—34no—
92#930—34no—
93#926C3⋊D311yescharacteristic, complemented
94#832—91no—
95#832—91no—
96#830—94no—
97#830—91no—
98#830—184no—
99#809—94no—
100#809—94no—
101#808—94no—
102#808—91no—
103#807—94no—
104#807—91no—
105C23×C32—24no—
106C23×C32#6714yes—
107C23×C32#6711yescharacteristic
108C22×C3⋊D3—21no—
109C22×C3×D3—64no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—24no—
115C2×C32×C4—24no—
116C2×C32×C4—44no—
117C2×C32×C4—44no—
118C2×C32×C4—61no—
119C2×C32×C4—61no—
120C2×C32×C4#6714yes—
121C2×C32×C4#6711yescharacteristic
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124#638—42no—
125C2×C3⋊Dic3—21no—
126C4⋊(C3⋊D3)—21no—
127C4⋊(C3⋊D3)—21no—
128C3×#67—128no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—64no—
132C3×D12—64no—
133C3×D12—64no—
134C3⋊(C3⋊C8)—61no—
135C3⋊(C3⋊C8)—61no—
136C3⋊(C3⋊C8)—64no—
137C3⋊(C3⋊C8)—64no—
138C3⋊(C3⋊C8)—124no—
139C3⋊(C3⋊C8)—124no—
140#482—271noSylow
141C2×C33—41no—
142C2×C33#12311yescharacteristic
143C2×C33#13511yescharacteristic
144C2×C33AGL(1,ℤ/8)11yescharacteristic
145C3×C3⋊D3—81no—
146C2×C3×D4—91no—
147C2×C3×D4—181no—
148C22×C3×C4—24no—
149C22×C3×C4D3214yes—
150C22×C3×C4C2×C3⋊D311yescharacteristic
151C2×#67—64no—
152C2×D12—34no—
153C3×C4⋊C4—34no—
154C3×C4⋊C4—34no—
155C3×C4⋊C4—91no—
156C3×C22⋊C4—64no—
157C3×C22⋊C4—181no—
158C3×C42—64no—
159#217—21no—
160(C2×D3)⋊C4—64no—
161C4⋊Dic3—34no—
162C4⋊Dic3C2×C3⋊D311yescharacteristic
163C4⋊Dic3C2×C3⋊D311yescharacteristic
164C4×Dic3—21no—
165C2×C3⋊C8—94no—
166C2×C3⋊C8—94no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—184no—
170C2×C3⋊C8—184no—
171C22×C32—28no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32—24no—
175C22×C32—21no—
176C22×C32—24no—
177C22×C32—24no—
178C22×C32—24no—
179C22×C32—44no—
180C22×C32—44no—
181C22×C32C2×#6714yes—
182C22×C32C2×#6711yescharacteristic
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3⋊D3—41no—
186C2×C3×D3—124no—
187C2×C3×D3—124no—
188C2×C3×D3—124no—
189C32×C4—21no—
190C32×C4—24no—
191C32×C4—24no—
192C32×C4—24no—
193C32×C4—44no—
194C32×C4—121no—
195C32×C4—121no—
196C32×C4#23514yes—
197C32×C4#23514yes—
198C32×C4#23711yescharacteristic
199C32×C4#23711yescharacteristic
200C3⋊Dic3—41no—
201C3×Dic3—44no—
202C3×Dic3—44no—
203C3×Dic3—124no—
204#121—91no—
205#117—31no—
206#107—271no—
207#106—271no—
208#102—271no—
209#102—271no—
210C22⋊C8—271no—
211C33#48211yescharacteristic, complemented, Sylow
212C23×C3—24no—
213C23×C3—181no—
214C23×C3#62514yes—
215C23×C3#63811yescharacteristic
216C22×D3—64no—
217C3×D4—181no—
218C3×D4—181no—
219C3×D4—362no—
220C2×C3×C4—24no—
221C2×C3×C4—24no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—24no—
225C2×C3×C4—24no—
226C2×C3×C4—44no—
227C2×C3×C4—44no—
228C2×C3×C4—64no—
229C2×C3×C4—64no—
230C2×C3×C4—181no—
231C2×C3×C4#62514yes—
232C2×C3×C4#63811yescharacteristic
233C2×C3×C4C2×D3214yes—
234C2×C3×C4C22×C3⋊D311yescharacteristic
235#67—128no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—64no—
239D12—64no—
240D12—64no—
241C3⋊C8—184no—
242C3⋊C8—184no—
243C3⋊C8—181no—
244C3⋊C8—181no—
245C3⋊C8—364no—
246C3⋊C8—364no—
247C2×C32—24no—
248C2×C32—24no—
249C2×C32—24no—
250C2×C32—48no—
251C2×C32—44no—
252C2×C32—41no—
253C2×C32#92914yes—
254C2×C32#93211yescharacteristic
255C2×C32#93511yescharacteristic
256C2×C32#94914yes—
257C2×C32#95014yes—
258C2×C32#95011yescharacteristic
259C3⋊D3—81no—
260C3×D3—244no—
261C2×D4—91no—
262C2×D4—181no—
263C22×C4D3×C3⋊D311yescharacteristic
264C2×C8—271no—
265C2×C8—271no—
266C4⋊C4—31no—
267C4⋊C4—31no—
268C4⋊C4—91no—
269C22⋊C4—61no—
270C22⋊C4—181no—
271C42—61no—
272C22×C3—28no—
273C22×C3—24no—
274C22×C3—24no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—24no—
279C22×C3—24no—
280C22×C3—44no—
281C22×C3—44no—
282C22×C3—361no—
283C22×C3—361no—
284C22×C3—361no—
285C22×C3C2×#62514yes—
286C22×C3C2×#63811yescharacteristic
287C2×D3—124no—
288C2×D3—124no—
289C2×D3—124no—
290C3×C4—21no—
291C3×C4—24no—
292C3×C4—24no—
293C3×C4—24no—
294C3×C4—44no—
295C3×C4—124no—
296C3×C4—124no—
297C3×C4—361no—
298C3×C4#380414yes—
299C3×C4#380414yes—
300C3×C4#382811yescharacteristic
301C3×C4#382811yescharacteristic
302Dic3—41no—
303Dic3—41no—
304Dic3—124no—
305C32—24no—
306C32#500314yescomplemented
307C32#503511yescharacteristic, complemented
308C23—181no—
309C23#629011yescharacteristic
310D4—181no—
311D4—181no—
312D4—362no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—181no—
318C2×C4#629011yescharacteristic
319C2×C4C2×D3×C3⋊D311yescharacteristic
320C8—541no—
321C8—541no—
322C2×C3—24no—
323C2×C3—24no—
324C2×C3—24no—
325C2×C3—48no—
326C2×C3—44no—
327C2×C3—41no—
328C2×C3—721no—
329C2×C3#6382214yes—
330C2×C3#6382314yes—
331C2×C3#6396114yes—
332C2×C3#6413811yescharacteristic
333C2×C3#6415811yescharacteristic
334C2×C3#6415911yescharacteristic
335D3—244no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—361no—
340C22—361no—
341C22—361no—
342C22C2×#629011yescharacteristic
343C4—21no—
344C4—121no—
345C4—121no—
346C4—361no—
347C4#8903111yescharacteristic
348C4#8903111yescharacteristic
349C3—24no—
350C3#1059109014yescomplemented
351C3#1059270211yescharacteristic, complemented
352C2—41no—
353C2—721no—
354C2#1172336311yescharacteristic
355C2#1172336511yescharacteristic
356C2#1172360511yescharacteristic
subgroupG/H
centerC22C2×#6290
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC3×#64139C2
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#482
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C32×#926C2semidirect
3. #11722200C2semidirect
4. #11722056C2semidirect
5. #11722053C2semidirect
6. #11722033C2semidirect
7. #11722033C2semidirect
24. C3×#926D3semidirect
93. #926C3⋊D3semidirect
211. C33#482semidirect
306. C32#5003semidirect
307. C32#5035semidirect
350. C3#10591090semidirect
351. C3#10592702semidirect

Maximal subgroups

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

subgroupindex
C3×#641392
C32×#9262
#117222002
#117220562
#117220532
#117220332
#105927023
#105910903