Finite Group Databasebeta

← #13160287

Subgroups — #13160287

Order 1728.

880 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 172 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#11723967C211yesmaximal, characteristic
2C3×#63657C211yesmaximal, characteristic
3C32×#830C211yesmaximal, characteristic
4#11722050C211yesmaximal, characteristic, complemented
5#11722050C211yesmaximal, characteristic, complemented
6#11722021C211yesmaximal, characteristic, complemented
7#11722018C211yesmaximal, characteristic, complemented
8#10592419—31nomaximal
9#10591062—34nomaximal
10C22×C33×C4C2211yescharacteristic
11#88864C411yescharacteristic
12#88864C411yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C411yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C411yescharacteristic
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#64135—31no—
22#64135—34no—
23#64135—64no—
24C32×C22⋊C8—31no—
25#63657D311yescharacteristic
26#63642—31no—
27#63641—31no—
28#63641—31no—
29C8×C3⋊Dic3—31no—
30C3×#830—34no—
31C3×#830D314yes—
32#63572—34no—
33#63572—34no—
34#63555—34no—
35#63553—34no—
36C23×C33C2×C411yescharacteristic
37C2×C33×C4—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2×C411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
42C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
43C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
44C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
45C3×C3⋊(C3⋊C8)—21no—
46C3×C3⋊(C3⋊C8)—21no—
47C32×C3⋊C8—21no—
48C32×C3⋊C8—21no—
49#4720—94no—
50#4720—91no—
51C22×C32×C4—24no—
52C22×C32×C4C2×D314yes—
53C22×C32×C4C2×D311yescharacteristic
54C2×C32×C8—31no—
55C2×C32×C8—31no—
56#3757—31no—
57#3757—31no—
58#3757—34no—
59#3757—34no—
60#3757—64no—
61#3757—64no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—34no—
66C4⋊(C3⋊Dic3)—64no—
67C4⋊(C3⋊Dic3)—64no—
68C4×C3⋊Dic3—31no—
69C4×C3⋊Dic3—31no—
70C4×C3⋊Dic3—34no—
71C4×C3⋊Dic3—34no—
72C4×C3⋊Dic3—64no—
73C4×C3⋊Dic3—64no—
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C2×C3×C3⋊C8—34no—
77C2×C3×C3⋊C8—34no—
78C2×C3×C3⋊C8C2×D314yes—
79C2×C3×C3⋊C8C2×D314yes—
80C22×C33—21no—
81C22×C33—21no—
82C22×C33—21no—
83C22×C33C22×C411yescharacteristic
84C33×C4—21no—
85C33×C4#4111yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—21no—
88C3⋊(C3⋊Dic3)—21no—
89C3⋊(C3⋊Dic3)—21no—
90C3⋊(C3⋊Dic3)—21no—
91#926—94no—
92#926—91no—
93#926—184no—
94C3×C22⋊C8—34no—
95C3×C22⋊C8—91no—
96#830—34no—
97#830C3⋊D311yescharacteristic
98#815—94no—
99#815—91no—
100Dic3⋊C8—94no—
101Dic3⋊C8—94no—
102Dic3⋊C8—91no—
103Dic3⋊C8—91no—
104C8×Dic3—94no—
105C8×Dic3—91no—
106C23×C32—24no—
107C23×C32C4×D314yes—
108C23×C32C4×D311yescharacteristic
109C2×C32×C4—28no—
110C2×C32×C4—28no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4C4×D314yes—
118C2×C32×C4C4×D311yescharacteristic
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—31no—
122C2×C3⋊Dic3—31no—
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—31no—
125C2×C3⋊Dic3—34no—
126C2×C3⋊Dic3—34no—
127C2×C3⋊Dic3—34no—
128C2×C3⋊Dic3—34no—
129C2×C3⋊Dic3—64no—
130C2×C3⋊Dic3—64no—
131C2×C3⋊Dic3—64no—
132C2×C3⋊Dic3—64no—
133C32×C8—61no—
134C32×C8—61no—
135C3⋊(C3⋊C8)—21no—
136C3⋊(C3⋊C8)—21no—
137C3×C3⋊C8—24no—
138C3×C3⋊C8—24no—
139C3×C3⋊C8—64no—
140C3×C3⋊C8—64no—
141#433—271noSylow
142C2×C33—41no—
143C2×C33#11711yescharacteristic
144C2×C33#13111yescharacteristic
145C2×C33#13111yescharacteristic
146C22×C3×C4—24no—
147C22×C3×C4D3214yes—
148C22×C3×C4C2×C3⋊D311yescharacteristic
149C2×C3×C8—34no—
150C2×C3×C8—34no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153#217—94no—
154#217—94no—
155#217—91no—
156#217—91no—
157#217—184no—
158#217—184no—
159C4⋊Dic3—94no—
160C4⋊Dic3—94no—
161C4⋊Dic3—91no—
162C4⋊Dic3—91no—
163C4⋊Dic3—184no—
164C4⋊Dic3—184no—
165C4×Dic3—94no—
166C4×Dic3—94no—
167C4×Dic3—91no—
168C4×Dic3—91no—
169C4×Dic3—184no—
170C4×Dic3—184no—
171C2×C3⋊C8—34no—
172C2×C3⋊C8—34no—
173C2×C3⋊C8C2×C3⋊D311yescharacteristic
174C2×C3⋊C8C2×C3⋊D311yescharacteristic
175C22×C32—28no—
176C22×C32—24no—
177C22×C32—21no—
178C22×C32—21no—
179C22×C32—21no—
180C22×C32—24no—
181C22×C32—24no—
182C22×C32—24no—
183C22×C32—44no—
184C22×C32—44no—
185C22×C32C2×C4×D314yes—
186C22×C32C2×C4×D311yescharacteristic
187C32×C4—28no—
188C32×C4—24no—
189C32×C4—21no—
190C32×C4—24no—
191C32×C4—24no—
192C32×C4#23714yes—
193C32×C4#23714yes—
194C32×C4#23711yescharacteristic
195C32×C4#23711yescharacteristic
196C3⋊Dic3—61no—
197C3⋊Dic3—61no—
198C3⋊Dic3—61no—
199C3⋊Dic3—61no—
200C3⋊Dic3—64no—
201C3⋊Dic3—64no—
202C3⋊Dic3—64no—
203C3⋊Dic3—64no—
204C3⋊Dic3—124no—
205C3⋊Dic3—124no—
206C3⋊Dic3—124no—
207C3⋊Dic3—124no—
208#117—271no—
209C4⋊C8—271no—
210C4⋊C8—271no—
211C22⋊C8—31no—
212C22⋊C8—91no—
213#97—271no—
214C4×C8—271no—
215C33#43311yescharacteristic, complemented, Sylow
216C23×C3—24no—
217C23×C3#62414yes—
218C23×C3C4×C3⋊D311yescharacteristic
219C2×C3×C4—28no—
220C2×C3×C4—28no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4#62414yes—
228C2×C3×C4C4×C3⋊D311yescharacteristic
229C2×C3×C4C2×D3214yes—
230C2×C3×C4C22×C3⋊D311yescharacteristic
231C2×Dic3—94no—
232C2×Dic3—94no—
233C2×Dic3—94no—
234C2×Dic3—94no—
235C2×Dic3—91no—
236C2×Dic3—91no—
237C2×Dic3—91no—
238C2×Dic3—91no—
239C2×Dic3—184no—
240C2×Dic3—184no—
241C2×Dic3—184no—
242C2×Dic3—184no—
243C3×C8—64no—
244C3×C8—64no—
245C3×C8—181no—
246C3×C8—181no—
247C3⋊C8—21no—
248C3⋊C8—21no—
249C3⋊C8—64no—
250C3⋊C8—64no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32—48no—
255C2×C32—44no—
256C2×C32—41no—
257C2×C32#87714yes—
258C2×C32#87711yescharacteristic
259C2×C32#90114yes—
260C2×C32#90111yescharacteristic
261C2×C32#90714yes—
262C2×C32#90711yescharacteristic
263C22×C4D3×C3⋊D311yescharacteristic
264C2×C8—31no—
265C2×C8—31no—
266C2×C8—91no—
267C2×C8—91no—
268C4⋊C4—271no—
269C4⋊C4—271no—
270C22⋊C4—271no—
271C22⋊C4—271no—
272C42—271no—
273C42—271no—
274C22×C3—28no—
275C22×C3—24no—
276C22×C3—24no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—24no—
281C22×C3—24no—
282C22×C3—44no—
283C22×C3—44no—
284C22×C3C2×#62414yes—
285C22×C3C2×C4×C3⋊D311yescharacteristic
286C3×C4—28no—
287C3×C4—21no—
288C3×C4—24no—
289C3×C4—24no—
290C3×C4—24no—
291C3×C4#380514yes—
292C3×C4#380514yes—
293C3×C4#383011yescharacteristic
294C3×C4#383011yescharacteristic
295Dic3—184no—
296Dic3—184no—
297Dic3—184no—
298Dic3—184no—
299Dic3—181no—
300Dic3—181no—
301Dic3—181no—
302Dic3—181no—
303Dic3—364no—
304Dic3—364no—
305Dic3—364no—
306Dic3—364no—
307C32—24no—
308C32#472014yescomplemented
309C32#472011yescharacteristic, complemented
310C23#628911yescharacteristic
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—271no—
314C2×C4—271no—
315C2×C4—271no—
316C2×C4—271no—
317C2×C4#628911yescharacteristic
318C2×C4C2×D3×C3⋊D311yescharacteristic
319C8—61no—
320C8—61no—
321C8—181no—
322C8—181no—
323C2×C3—24no—
324C2×C3—24no—
325C2×C3—24no—
326C2×C3—48no—
327C2×C3—44no—
328C2×C3—41no—
329C2×C3#6381714yes—
330C2×C3#6381714yes—
331C2×C3#6395714yes—
332C2×C3#6408611yescharacteristic
333C2×C3#6411011yescharacteristic
334C2×C3#6411611yescharacteristic
335C22—21no—
336C22—21no—
337C22—21no—
338C22C2×#628911yescharacteristic
339C4—21no—
340C4—541no—
341C4—541no—
342C4—541no—
343C4—541no—
344C4#8902911yescharacteristic
345C4#8902911yescharacteristic
346C3—24no—
347C3#1059106214yescomplemented
348C3#1059241911yescharacteristic, complemented
349C2—41no—
350C2#1172335611yescharacteristic
351C2#1172335711yescharacteristic
352C2#1172359811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC22×C33×C4C22
prime radicalC23×C33C2×C4
cosocleC2×C33×C4C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11722050C2semidirect
5. #11722050C2semidirect
6. #11722021C2semidirect
7. #11722018C2semidirect
215. C33#433semidirect
308. C32#4720semidirect
309. C32#4720semidirect
347. C3#10591062semidirect
348. C3#10592419semidirect

Maximal subgroups

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

subgroupindex
#117239672
C3×#636572
C32×#8302
#117220502
#117220212
#117220182
#105924193
#105910623