Finite Group Databasebeta

← #13167095

Subgroups — #13167095

Order 1728.

372 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 47 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#11723660C211yesmaximal, characteristic, complemented
2C3×#63822C211yesmaximal, characteristic, complemented
3C2×C3⋊#3708C211yesmaximal, characteristic, complemented
4#11722768C211yesmaximal, characteristic, complemented
5#11722767C211yesmaximal, characteristic, complemented
6#11722766C211yesmaximal, characteristic, complemented
7#11722765C211yesmaximal, characteristic, complemented
8#10593333—31nomaximal
9C3×#3828—21no—
10C32×#235—21no—
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12#88812C2211yescharacteristic
13#88811—21no—
14#88810C2211yescharacteristic
15C3×#3718C2211yescharacteristic
16C3×#3715—21no—
17C3×#3713C2211yescharacteristic
18C3⋊#3708C2211yescharacteristic
19C3⋊#3708C2211yescharacteristic
20C3×#950—61no—
21#63829—61no—
22#63822—31no—
23#63822D311yescharacteristic, complemented
24C2×#3708—31no—
25#63736—31no—
26#63735—31no—
27#63735—31no—
28#63734—31no—
29C2×C33×C4D411yescharacteristic
30C3×#638—41no—
31C3×C4⋊(C3⋊D3)—21no—
32C3×C4×C3⋊D3—41no—
33C3×#634—21no—
34C32×#67—41no—
35C32×D12—21no—
36C32×C4×D3—41no—
37C32×Dic6—21no—
38C3×C3⋊(C3⋊C8)D411yescharacteristic
39C3×C3⋊(C3⋊C8)C2311yescharacteristic
40#5194—91nomaximal
41C32×#41—61no—
42#3828—21no—
43C3×#235—21no—
44C3×#235—21no—
45C3×#235—41no—
46C3×#235—61no—
47C3×#235—61no—
48C3×#235—121no—
49C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
50C3×#216—61no—
51C3×#215—61no—
52C3×#214—61no—
53C3×#213—61no—
54C2×C3×C3⋊C8—61no—
55C2×C3×C3⋊C8—61no—
56#3719—61no—
57#3718—31no—
58#3718C2×D311yescharacteristic
59#3717—61no—
60#3716—61no—
61#3715—21no—
62#3715—61no—
63#3714—61no—
64#3713—31no—
65#3713C2×D311yescharacteristic
66#3708—31no—
67#3708—31no—
68C22×C33D811yescharacteristic
69C2×C3×C3⋊D3—41no—
70C2×C32×D3—41no—
71C33×C4D811yescharacteristic
72C33×C4C2×D411yescharacteristic
73C3×C3⋊Dic3—41no—
74C32×Dic3—41no—
75C3×#135—91no—
76#950—61no—
77#950—181no—
78#904—91no—
79#829—91no—
80#828—91no—
81#827—91no—
82#826—91no—
83C2×C3⋊C16—91no—
84C32×Q8—61no—
85C32×D4—61no—
86C32×D4—121no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4—41no—
90C2×C32×C4—41no—
91C2×C32×C4—121no—
92C2×C32×C4#6711yescharacteristic
93#638—41no—
94C4⋊(C3⋊D3)—21no—
95C4×C3⋊D3—41no—
96#634—21no—
97C3×#67—41no—
98C3×#67—42no—
99C3×#67—41no—
100C3×#67—121no—
101C3×#67—122no—
102C3×#67—121no—
103C3×D12—21no—
104C3×D12—21no—
105C3×D12—41no—
106C3×D12—61no—
107C3×D12—61no—
108C3×D12—121no—
109C3×C4×D3—41no—
110C3×C4×D3—42no—
111C3×C4×D3—41no—
112C3×C4×D3—121no—
113C3×C4×D3—122no—
114C3×C4×D3—121no—
115C3×Dic6—21no—
116C3×Dic6—21no—
117C3×Dic6—41no—
118C3×Dic6—61no—
119C3×Dic6—61no—
120C3×Dic6—121no—
121C3⋊(C3⋊C8)#6711yescharacteristic
122C3⋊(C3⋊C8)C22×D311yescharacteristic
123C3×C3⋊C8—61no—
124C3×C3⋊C8—61no—
125C3×C3⋊C8—61no—
126C3×C3⋊C8—61no—
127#508—271noSylow
128C2×C33—21no—
129C2×C33C2×D811yescharacteristic
130C3×C3⋊D3—81no—
131C32×D3—81no—
132C3×#41—61no—
133C3×#41—61no—
134C3×#41—121no—
135C3×#41—181no—
136#235—21no—
137#235—61no—
138#235—61no—
139#235—121no—
140C3×Q16—91no—
141C3×SD16—181no—
142C3×D8—91no—
143C2×C3×C8—91no—
144#216—61no—
145#216—181no—
146#215—61no—
147#215—181no—
148#214—61no—
149#214—181no—
150#213—61no—
151#213—181no—
152C2×C3⋊C8—61no—
153C2×C3⋊C8—61no—
154Dic12—91no—
155D24—91no—
156#204—181no—
157C3⋊C16—91no—
158C3⋊C16—91no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32—41no—
162C22×C32—41no—
163C22×C32—121no—
164C22×C32#21311yescharacteristic
165C2×C3⋊D3—41no—
166C2×C3×D3—41no—
167C2×C3×D3—42no—
168C2×C3×D3—41no—
169C2×C3×D3—121no—
170C2×C3×D3—122no—
171C2×C3×D3—121no—
172C32×C4—21no—
173C32×C4—21no—
174C32×C4—21no—
175C32×C4—21no—
176C32×C4—41no—
177C32×C4—41no—
178C32×C4—41no—
179C32×C4—41no—
180C32×C4—121no—
181C32×C4#21311yescharacteristic
182C32×C4C2×#6711yescharacteristic
183C3⋊Dic3—41no—
184C3×Dic3—41no—
185C3×Dic3—42no—
186C3×Dic3—41no—
187C3×Dic3—121no—
188C3×Dic3—122no—
189C3×Dic3—121no—
190#135—91no—
191#135—271no—
192Q32—271no—
193SD32—271no—
194SD32—271no—
195D16—271no—
196C2×C16—271no—
197C33#50811yescharacteristic, complemented, Sylow
198C3×Q8—61no—
199C3×Q8—61no—
200C3×Q8—121no—
201C3×Q8—181no—
202C3×D4—61no—
203C3×D4—61no—
204C3×D4—121no—
205C3×D4—121no—
206C3×D4—122no—
207C3×D4—121no—
208C3×D4—181no—
209C3×D4—361no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—121no—
215C2×C3×C4—122no—
216C2×C3×C4—121no—
217C2×C3×C4—361no—
218C2×C3×C4GO+(4,𝔽2)11yescharacteristic
219#67—41no—
220#67—121no—
221#67—122no—
222#67—121no—
223D12—21no—
224D12—61no—
225D12—61no—
226D12—121no—
227C4×D3—41no—
228C4×D3—121no—
229C4×D3—122no—
230C4×D3—121no—
231Dic6—21no—
232Dic6—61no—
233Dic6—61no—
234Dic6—121no—
235C3×C8—91no—
236C3×C8—91no—
237C3⋊C8—61no—
238C3⋊C8—61no—
239C3⋊C8—61no—
240C3⋊C8—61no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—41no—
248C2×C32—81no—
249C2×C32—81no—
250C2×C32—241no—
251C2×C32C2×#21311yescharacteristic
252C3⋊D3—81no—
253C3×D3—81no—
254C3×D3—82no—
255C3×D3—81no—
256C3×D3—241no—
257C3×D3—242no—
258C3×D3—241no—
259#41—61no—
260#41—181no—
261Q16—91no—
262Q16—271no—
263SD16—181no—
264SD16—541no—
265D8—91no—
266D8—271no—
267C2×C8—91no—
268C16—271no—
269C16—271no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—121no—
275C22×C3—122no—
276C22×C3—121no—
277C22×C3—361no—
278C22×C3#377411yescharacteristic
279C2×D3—41no—
280C2×D3—121no—
281C2×D3—122no—
282C2×D3—121no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—121no—
292C3×C4—122no—
293C3×C4—121no—
294C3×C4—361no—
295C3×C4#377411yescharacteristic
296C3×C4C2×GO+(4,𝔽2)11yescharacteristic
297Dic3—41no—
298Dic3—121no—
299Dic3—122no—
300Dic3—121no—
301C32—21no—
302C32—21no—
303C32—41no—
304C32—41no—
305C32#519411yescharacteristic, complemented
306Q8—61no—
307Q8—181no—
308D4—61no—
309D4—121no—
310D4—181no—
311D4—361no—
312C2×C4—121no—
313C2×C4—361no—
314C2×C4D32⋊D311yescharacteristic
315C8—91no—
316C8—91no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—81no—
325C2×C3—81no—
326C2×C3—241no—
327C2×C3—242no—
328C2×C3—241no—
329C2×C3—721no—
330C2×C3C2×#377411yescharacteristic
331D3—81no—
332D3—241no—
333D3—242no—
334D3—241no—
335C22—121no—
336C22—361no—
337C22#8893511yescharacteristic
338C4—121no—
339C4—361no—
340C4#8893511yescharacteristic
341C4C2×D32⋊D311yescharacteristic
342C3—21no—
343C3—21no—
344C3—41no—
345C3—41no—
346C3#1059333311yescharacteristic, complemented
347C2—21no—
348C2—241no—
349C2—721no—
350C2C2×#8893511yescharacteristic
subgroupG/H
centerC4#88935
commutator, cosocleC3×C3⋊(C3⋊C8)C23
FrattiniC4C2×D32⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×D8
hypercenterC2×C4D32⋊D3
hypocenterC33#508
LCSG ▷ C3×C3⋊(C3⋊C8) ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723660C2semidirect
2. C3×#63822C2semidirect
3. C2×C3⋊#3708C2semidirect
4. #11722768C2semidirect
5. #11722767C2semidirect
6. #11722766C2semidirect
7. #11722765C2semidirect
23. #63822D3semidirect
197. C33#508semidirect
305. C32#5194semidirect
346. C3#10593333semidirect

Maximal subgroups

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

subgroupindex
#117236602
C3×#638222
C2×C3⋊#37082
#117227682
#117227672
#117227662
#117227652
#105933333
#51949