Finite Group Databasebeta

← #13137430

Subgroups — #13137430

Order 1728.

488 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 89 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#11721715C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3C2×#88438C212yesmaximal
4C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
5#11720326C212yesmaximal, complemented
6#10590379—31nomaximal
7#10590363—31nomaximal
8C2×#6229C2212yes—
9C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
10C2×C4×(C3⋊D3)⋊C3—21no—
11C42×H(3,𝔽3)C2211yescharacteristic
12C4⋊#1084—21no—
13C4⋊#1084—21no—
14#3752⋊C3—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17#88438—24no—
18C4×C4⋊(C3⋊D3)D311yescharacteristic
19C3×C4×D12—31no—
20C3×#869—31no—
21C3×#869—31no—
22C3×#869—61no—
23C2×#3717—32no—
24C2×#3715—32no—
25C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
26C3×C4⋊(C3⋊C8)—31no—
27#63566—32no—
28#63565—32no—
29C22×(C3⋊D3)⋊C3—21no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084—22no—
34#6229—22no—
35#6229—22no—
36C2×(C3⋊Dic3)⋊C3—21no—
37C4⋊((C3⋊D3)⋊C3)—21no—
38C4⋊((C3⋊D3)⋊C3)D412yes—
39C4×(C3⋊D3)⋊C3—42no—
40(C3⋊(C3⋊C8))⋊C3—22no—
41#5083—91no—
42#5019—91no—
43#4841—91no—
44C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
45C2×C4×C3⋊D3—21no—
46C2×C3×D12—31no—
47C2×C3×C4×D3—61no—
48C2×C3×Dic6—32no—
49C2×C3×Dic6—32no—
50C2×C3×Dic6—62no—
51C32×C42—21no—
52C32×C42C2×D311yescharacteristic
53C32×C42C2×D311yescharacteristic
54#3752—21no—
55C4⋊(C3⋊Dic3)C2×D311yescharacteristic
56C2×C3⋊(C3⋊C8)C2×D312yes—
57C3×(C2×D3)⋊C4—61no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3—62no—
60C3×C4⋊Dic3—62no—
61C3×C4⋊Dic3—61no—
62C3×C4⋊Dic3—61no—
63C3×C4⋊Dic3—61no—
64C3×C4⋊Dic3—61no—
65C2×C3×C3⋊C8—32no—
66#3717—64no—
67#3715—64no—
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C2×(C3⋊D3)⋊C3—22no—
70C2×(C3⋊D3)⋊C3—41no—
71C2×(C3⋊D3)⋊C3—41no—
72C4×H(3,𝔽3)—21no—
73C4×H(3,𝔽3)SD1612yescomplemented
74C4×H(3,𝔽3)C2×D411yescharacteristic
75C4×H(3,𝔽3)#4111yescharacteristic
76#1084—42no—
77(C3⋊Dic3)⋊C3—41no—
78C3×C4⋊Q8—91no—
79C3×C4×D4—91no—
80C2×#215—92no—
81C2×#214—92no—
82C2×#204—92no—
83C4×D12—31no—
84C4×D12—91no—
85#869—31no—
86#869—31no—
87#869—61no—
88C3×C4⋊C8—91no—
89#835—92no—
90#816—92no—
91#810—92no—
92C4⋊(C3⋊C8)—31no—
93C4⋊(C3⋊C8)—91no—
94C22×C3⋊D3—21no—
95C22×C3×D3—61no—
96C2×C32×C4—21no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4D1211yescharacteristic
100C2×C32×C4D1211yescharacteristic
101C2×C32×C4#6711yescharacteristic
102C2×C32×C4#6711yescharacteristic
103C2×C32×C4C22×D311yescharacteristic
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—21no—
106C4⋊(C3⋊D3)—21no—
107C4⋊(C3⋊D3)D1212yes—
108C4×C3⋊D3—42no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C3×D12—32no—
114C3×D12—61no—
115C3×C4×D3—122no—
116C3×Dic6—62no—
117C3×Dic6—62no—
118C3×Dic6—62no—
119C3×Dic6—62no—
120C3×Dic6—122no—
121C3×Dic6—122no—
122C3⋊(C3⋊C8)—22no—
123C3×C3⋊C8—62no—
124#461—271noSylow
125C2×H(3,𝔽3)#12111yescharacteristic
126C2×H(3,𝔽3)C2×SD1611yescharacteristic
127C2×H(3,𝔽3)#13711yescharacteristic
128(C3⋊D3)⋊C3—42no—
129C2×C3×Q8—92no—
130C2×C3×D4—91no—
131C22×C3×C4—181no—
132C2×D12—31no—
133C2×D12—91no—
134C2×C4×D3—61no—
135C2×C4×D3—181no—
136C2×Dic6—32no—
137C2×Dic6—32no—
138C2×Dic6—62no—
139C2×C3×C8—92no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—181no—
142C3×C4⋊C4—181no—
143C3×C22⋊C4—181no—
144C3×C42—31no—
145C3×C42—31no—
146C3×C42—61no—
147C3×C42D3211yescharacteristic
148#215—184no—
149#214—184no—
150(C2×D3)⋊C4—61no—
151(C2×D3)⋊C4—181no—
152C4⋊Dic3—31no—
153C4⋊Dic3—61no—
154C4⋊Dic3—61no—
155C4⋊Dic3—61no—
156C4⋊Dic3—61no—
157C4⋊Dic3—62no—
158C4⋊Dic3—62no—
159C4⋊Dic3—91no—
160C2×C3⋊C8—32no—
161C2×C3⋊C8—92no—
162#204—184no—
163C22×C32—21no—
164C22×C32C2×D1211yescharacteristic
165C22×C32C2×#6711yescharacteristic
166C2×C3⋊D3—22no—
167C2×C3⋊D3—41no—
168C2×C3⋊D3—41no—
169C2×C3×D3—62no—
170C2×C3×D3—121no—
171C2×C3×D3—121no—
172C32×C4—22no—
173C32×C4—21no—
174C32×C4—21no—
175C32×C4—22no—
176C32×C4—21no—
177C32×C4—21no—
178C32×C4#20412yescomplemented
179C32×C4#21412yescomplemented
180C32×C4C2×D1211yescharacteristic
181C32×C4#23511yescharacteristic
182C32×C4D3×D411yescharacteristic
183C32×C4#23911yescharacteristic
184C3⋊Dic3—41no—
185C3×Dic3—124no—
186C3×Dic3—122no—
187C3×Dic3—121no—
188C3×Dic3—122no—
189C2×SD16—272no—
190C4⋊Q8—91no—
191C4×D4—91no—
192C4⋊C8—91no—
193#103—272no—
194H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
195C23×C3—181no—
196C22×D3—61no—
197C22×D3—181no—
198C3×Q8—182no—
199C3×Q8—182no—
200C3×D4—92no—
201C3×D4—181no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—61no—
211C2×C3×C4—181no—
212C2×C3×C4—182no—
213C2×C3×C4—362no—
214C2×C3×C4#62611yescharacteristic
215C2×C3×C4#62611yescharacteristic
216C2×C3×C4C2×D3211yescharacteristic
217C2×Dic3—62no—
218C2×Dic3—62no—
219C2×Dic3—64no—
220C2×Dic3—61no—
221C2×Dic3—181no—
222D12—32no—
223D12—61no—
224D12—92no—
225D12—181no—
226C4×D3—122no—
227C4×D3—362no—
228Dic6—62no—
229Dic6—62no—
230Dic6—62no—
231Dic6—62no—
232Dic6—122no—
233Dic6—122no—
234C3×C8—182no—
235C3⋊C8—62no—
236C3⋊C8—182no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32—21no—
240C2×C32#89511yescharacteristic
241C2×C32C2×#20411yescharacteristic
242C2×C32#90911yescharacteristic
243C2×C32#93011yescharacteristic
244C2×C32C2×#21411yescharacteristic
245C2×C32#95111yescharacteristic
246C3⋊D3—42no—
247C3×D3—122no—
248C2×Q8—92no—
249C2×D4—91no—
250C22×C4—181no—
251SD16—544no—
252C2×C8—92no—
253C4⋊C4—91no—
254C4⋊C4—181no—
255C4⋊C4—181no—
256C22⋊C4—181no—
257C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
258C22×C3—31no—
259C22×C3—31no—
260C22×C3—61no—
261C22×C3—182no—
262C22×C3—361no—
263C22×C3—361no—
264C22×C3C2×#62611yescharacteristic
265C2×D3—62no—
266C2×D3—121no—
267C2×D3—121no—
268C2×D3—182no—
269C2×D3—361no—
270C2×D3—361no—
271C3×C4—21no—
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—31no—
275C3×C4—31no—
276C3×C4—32no—
277C3×C4—32no—
278C3×C4—61no—
279C3×C4—61no—
280C3×C4—62no—
281C3×C4—61no—
282C3×C4—61no—
283C3×C4—62no—
284C3×C4—361no—
285C3×C4—362no—
286C3×C4#371612yes—
287C3×C4#379911yescharacteristic
288C3×C4D3×D1211yescharacteristic
289Dic3—122no—
290Dic3—122no—
291Dic3—124no—
292Dic3—121no—
293Dic3—361no—
294C32—21no—
295C32#484111yescharacteristic, complemented
296C32#501911yescharacteristic, complemented
297C23—181no—
298Q8—182no—
299Q8—182no—
300D4—92no—
301D4—181no—
302C2×C4—181no—
303C2×C4—182no—
304C2×C4—362no—
305C2×C4#620011yescharacteristic
306C2×C4#620011yescharacteristic
307C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
308C8—182no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—61no—
316C2×C3—61no—
317C2×C3—61no—
318C2×C3—362no—
319C2×C3C2×#371611yescharacteristic
320C2×C3#6383111yescharacteristic
321C2×C3#6390911yescharacteristic
322D3—122no—
323D3—362no—
324C22—182no—
325C22—361no—
326C22—361no—
327C22C2×#620011yescharacteristic
328C4—21no—
329C4—361no—
330C4—362no—
331C4#8843812yescomplemented
332C4#8865211yescharacteristic
333C4#8865411yescharacteristic
334C3—31no—
335C3—31no—
336C3—61no—
337C3#1059036511yescharacteristic
338C2—362no—
339C2C2×#8843811yescharacteristic
340C2#1172107411yescharacteristic
341C2#1172115611yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#461
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721715C2semidirect
4. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
5. #11720326C2semidirect
25. C4⋊(C3⋊(C3⋊C8))D3semidirect
73. C4×H(3,𝔽3)SD16semidirect
178. C32×C4#204semidirect
179. C32×C4#214semidirect
194. H(3,𝔽3)#461semidirect
295. C32#4841semidirect
296. C32#5019semidirect
331. C4#88438semidirect

Maximal subgroups

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

subgroupindex
#117217152
C4×C4⋊((C3⋊D3)⋊C3)2
C2×#884382
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#117203262
#105903793
#105903633