Finite Group Databasebeta

← #13159171

Subgroups — #13159171

Order 1728.

836 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 152 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
1C32×#892C211yesmaximal, characteristic, complemented
2#11723516C211yesmaximal, characteristic
3#11722179C211yesmaximal, characteristic, complemented
4C3×#63634C211yesmaximal, characteristic, complemented
5#11722053C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592702—31nomaximal
9#10590424—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C32×C4×D3C2211yescharacteristic
12C2×#6291—21no—
13C33×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3—21no—
20#88804—21no—
21C32×#117—31no—
22#64139—31no—
23C3×#892D314yescomplemented
24#63909—34no—
25#63657—31no—
26#63636—31no—
27#63636—31no—
28#63636—34no—
29#63636—64no—
30#63635—31no—
31#63634D311yescharacteristic, complemented
32C3×#807—34no—
33#63574—34no—
34#63564—34no—
35(C2×D3)⋊(C3⋊C8)—34no—
36C22×C3⋊(C3⋊D3)—21no—
37C22×C32×D3D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C4⋊(C3⋊(C3⋊D3))—21no—
41C4⋊(C3⋊(C3⋊D3))—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3D411yescharacteristic
45C32×C4×D3—21no—
46C32×C4×D3—21no—
47#6291—42no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C3×C3⋊(C3⋊C8)—21no—
50#5003—94no—
51#4885—91no—
52C22×C32×C4—31no—
53C2×#638—61no—
54C2×C4⋊(C3⋊D3)—31no—
55C2×C4⋊(C3⋊D3)—34no—
56C2×C4⋊(C3⋊D3)—64no—
57C2×C3×C4×D3C2×D314yes—
58C2×#626—64no—
59C32×C4⋊C4—24no—
60C32×C4⋊C4—31no—
61C32×C4⋊C4C2×D314yes—
62C32×C4⋊C4C2×D311yescharacteristic
63C32×C22⋊C4—61no—
64C32×C42—61no—
65#3752—61no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—34no—
69C2×C3⋊(C3⋊C8)—64no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×(C2×D3)⋊C4—24no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74C3×C4×Dic3—24no—
75C2×C3×C3⋊C8—34no—
76#3722—64no—
77C22×C33C2×D411yescharacteristic
78C2×C3⋊(C3⋊D3)—41no—
79C2×C3⋊(C3⋊D3)—41no—
80C2×C3⋊(C3⋊D3)—41no—
81C2×C32×D3—21no—
82C2×C32×D3—21no—
83C2×C32×D3—21no—
84C33×C4—41no—
85C33×C4#4111yescharacteristic
86C33×C4#4111yescharacteristic
87C3×C3⋊Dic3—41no—
88C32×Dic3—21no—
89C32×Dic3—41no—
90C3×#117—34no—
91#930—94no—
92#895—91no—
93#892C3⋊D311yescharacteristic, complemented
94C3×#107—91no—
95#830—94no—
96#821—91no—
97(C2×D3)⋊C8—91no—
98#809—94no—
99#809—94no—
100#809—91no—
101#809—184no—
102#808—94no—
103#808—91no—
104#807—34no—
105C23×C32—31no—
106C22×C3⋊D3—61no—
107C22×C3⋊D3—64no—
108C22×C3⋊D3—124no—
109C22×C3×D3#6714yes—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—31no—
114C2×C32×C4—44no—
115C2×C32×C4—61no—
116C2×C32×C4—61no—
117C2×C32×C4—61no—
118C2×C32×C4C22×D314yes—
119C2×C32×C4C22×D311yescharacteristic
120#638—122no—
121C2×C3⋊Dic3—21no—
122C4⋊(C3⋊D3)—68no—
123C4⋊(C3⋊D3)—64no—
124C4⋊(C3⋊D3)—61no—
125C4⋊(C3⋊D3)—61no—
126C4⋊(C3⋊D3)—64no—
127C4⋊(C3⋊D3)—124no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3#6714yes—
131C3×C4×D3—24no—
132C3×C4×D3—24no—
133#626—128no—
134C3⋊(C3⋊C8)—21no—
135C3⋊(C3⋊C8)—61no—
136C3⋊(C3⋊C8)—64no—
137C3⋊(C3⋊C8)—124no—
138C3×C3⋊C8—64no—
139#482—271noSylow
140C2×C33#12311yescharacteristic
141C2×C33#13511yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C3⋊(C3⋊D3)—81no—
144C32×D3—41no—
145C22×C3×C4—34no—
146C2×#67—184no—
147C2×D12—94no—
148C2×D12—91no—
149C2×D12—184no—
150C2×D12—181no—
151C2×C4×D3C2×C3⋊D311yescharacteristic
152C2×C3×C8—91no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—34no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4D3214yes—
157C3×C4⋊C4C2×C3⋊D311yescharacteristic
158C3×C22⋊C4—64no—
159C3×C42—64no—
160(C2×D3)⋊C4—21no—
161(C2×D3)⋊C4—184no—
162(C2×D3)⋊C4—181no—
163C4⋊Dic3—34no—
164C4⋊Dic3C2×C3⋊D311yescharacteristic
165C4×Dic3—21no—
166C2×C3⋊C8—34no—
167C2×C3⋊C8—94no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—184no—
170C22×C32—24no—
171C22×C32—61no—
172C22×C32—61no—
173C22×C32—61no—
174C22×C32D3×D411yescharacteristic
175C22×C32C2×#6714yes—
176C2×C3⋊D3—128no—
177C2×C3⋊D3—121no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—121no—
180C2×C3⋊D3—124no—
181C2×C3⋊D3—124no—
182C2×C3⋊D3—124no—
183C2×C3⋊D3—244no—
184C2×C3⋊D3—244no—
185C2×C3×D3—24no—
186C2×C3×D3—24no—
187C2×C3×D3—24no—
188C32×C4—24no—
189C32×C4—24no—
190C32×C4—41no—
191C32×C4—44no—
192C32×C4—61no—
193C32×C4—84no—
194C32×C4—121no—
195C32×C4#23514yes—
196C32×C4#23514yes—
197C32×C4#23511yescharacteristic
198C32×C4#23911yescharacteristic
199C3⋊Dic3—41no—
200C3×Dic3—24no—
201C3×Dic3—44no—
202C3×Dic3—124no—
203#121—271no—
204#117—31no—
205#107—91no—
206#106—271no—
207#102—271no—
208#102—271no—
209C22⋊C8—271no—
210C33#48211yescharacteristic, complemented, Sylow
211C23×C3—34no—
212C22×D3—184no—
213C22×D3—181no—
214C22×D3—364no—
215C22×D3#63811yescharacteristic
216C2×C3×C4—24no—
217C2×C3×C4—21no—
218C2×C3×C4—24no—
219C2×C3×C4—34no—
220C2×C3×C4—44no—
221C2×C3×C4—64no—
222C2×C3×C4—64no—
223C2×C3×C4—64no—
224C2×C3×C4—181no—
225C2×C3×C4C2×D3214yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227#67—368no—
228C2×Dic3—21no—
229C2×Dic3—64no—
230C2×Dic3#63811yescharacteristic
231D12—188no—
232D12—184no—
233D12—184no—
234D12—181no—
235D12—181no—
236D12—364no—
237D12—362no—
238C4×D3—21no—
239C4×D3—21no—
240C3×C8—181no—
241C3⋊C8—64no—
242C3⋊C8—184no—
243C3⋊C8—181no—
244C3⋊C8—364no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—24no—
248C2×C32—121no—
249C2×C32#89411yescharacteristic
250C2×C32#91111yescharacteristic
251C2×C32#91911yescharacteristic
252C2×C32#92914yes—
253C2×C32#94914yes—
254C2×C32#95014yes—
255C3⋊D3—248no—
256C3⋊D3—241no—
257C3⋊D3—244no—
258C3×D3—44no—
259C2×D4—271no—
260C2×D4—541no—
261C22×C4—31no—
262C2×C8—91no—
263C2×C8—271no—
264C4⋊C4—31no—
265C4⋊C4—91no—
266C4⋊C4D3×C3⋊D311yescharacteristic
267C22⋊C4—61no—
268C22⋊C4—541no—
269C42—61no—
270C22×C3—24no—
271C22×C3—64no—
272C22×C3—64no—
273C22×C3—64no—
274C22×C3D3×#6714yes—
275C22×C3C2×#63811yescharacteristic
276C2×D3—21no—
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—368no—
280C2×D3—364no—
281C2×D3—361no—
282C2×D3—361no—
283C2×D3—361no—
284C2×D3—364no—
285C2×D3—364no—
286C2×D3—724no—
287C2×D3—724no—
288C3×C4—24no—
289C3×C4—24no—
290C3×C4—41no—
291C3×C4—44no—
292C3×C4—64no—
293C3×C4—84no—
294C3×C4—124no—
295C3×C4—361no—
296C3×C4#379814yes—
297C3×C4#379914yes—
298C3×C4#382811yescharacteristic
299C3×C4#382811yescharacteristic
300Dic3—21no—
301Dic3—41no—
302Dic3—124no—
303C32—24no—
304C32#488511yescharacteristic, complemented
305C32#500314yescomplemented
306C23—31no—
307C23—541no—
308D4—541no—
309D4—541no—
310D4—1082no—
311C2×C4—21no—
312C2×C4—31no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—181no—
317C2×C4C2×D3×C3⋊D311yescharacteristic
318C8—181no—
319C8—541no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—124no—
324C2×C3#6385014yes—
325C2×C3#6392514yes—
326C2×C3#6394914yes—
327C2×C3#6413811yescharacteristic
328C2×C3#6415811yescharacteristic
329C2×C3#6415911yescharacteristic
330D3—41no—
331D3—728no—
332D3—721no—
333D3—724no—
334C22—61no—
335C22—61no—
336C22—61no—
337C22—1081no—
338C22—1081no—
339C22—1081no—
340C22D3×#63811yescharacteristic
341C4—41no—
342C4—61no—
343C4—121no—
344C4—361no—
345C4#8902011yescharacteristic
346C4#8902211yescharacteristic
347C3—24no—
348C3#1059042414yescomplemented
349C3#1059270211yescharacteristic, complemented
350C2—121no—
351C2—2161no—
352C2#1172341411yescharacteristic
353C2#1172354411yescharacteristic
354C2#1172358211yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723516C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#482
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C32×#892C2semidirect
3. #11722179C2semidirect
4. C3×#63634C2semidirect
5. #11722053C2semidirect
6. #11722035C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
23. C3×#892D3semidirect
31. #63634D3semidirect
93. #892C3⋊D3semidirect
210. C33#482semidirect
304. C32#4885semidirect
305. C32#5003semidirect
348. C3#10590424semidirect
349. C3#10592702semidirect

Maximal subgroups

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

subgroupindex
C32×#8922
#117235162
#117221792
C3×#636342
#117220532
#117220352
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105927023
#105904243