Finite Group Databasebeta

← #13138734

Subgroups — #13138734

Order 1728.

356 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 65 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#11721779C211yesmaximal, characteristic, complemented
2#64113⋊C3C211yesmaximal, characteristic, complemented
3C2×(C3⋊(C3⋊C16))⋊C3C211yesmaximal, characteristic, complemented
4#11720256C211yesmaximal, characteristic, complemented
5#11720255C211yesmaximal, characteristic, complemented
6#11720254C211yesmaximal, characteristic, complemented
7#11720253C211yesmaximal, characteristic, complemented
8#10590632—31nomaximal
9#10590629—31nomaximal
10#88740—21no—
11#3828⋊C3—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13#88530C2211yescharacteristic
14#88529C2211yescharacteristic
15#88528—21no—
16#3745⋊C3C2211yescharacteristic
17#3744⋊C3—21no—
18#3746⋊C3C2211yescharacteristic
19(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic
20(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic
21#64113D311yescharacteristic, complemented
22C3×#904—31no—
23C3×#904—31no—
24C3×#904—31no—
25C3×#904—61no—
26C2×C3⋊(C3⋊C16)D311yescharacteristic, complemented
27C2×C3×C3⋊C16—31no—
28#63551—31no—
29#63550—31no—
30#63549—31no—
31#63548—31no—
32#63547—31no—
33#63547—31no—
34#63546—31no—
35#63545—31no—
36C2×C4×H(3,𝔽3)D411yescharacteristic
37#6235—41no—
38C4⋊#272—21no—
39C4×#272—41no—
40#6229—21no—
41#638⋊C3—41no—
42C4⋊((C3⋊D3)⋊C3)—21no—
43C4×(C3⋊D3)⋊C3—41no—
44#634⋊C3—21no—
45C8×H(3,𝔽3)D411yescharacteristic
46C8×H(3,𝔽3)C2311yescharacteristic
47#5194—91no—
48#5194—91no—
49#4904—91no—
50#3828—21no—
51C3×#235—62no—
52C3×#235—61no—
53C3×#235—61no—
54C3×#235—61no—
55C2×C32×C8—21no—
56C2×C32×C8C2×D311yescharacteristic
57C2×C32×C8C2×D311yescharacteristic
58#3746C2×D311yescharacteristic
59#3745C2×D311yescharacteristic
60#3744—21no—
61C3×Dic12—31no—
62C3×Dic12—31no—
63C3×Dic12—31no—
64C3×Dic12—61no—
65C3×D24—31no—
66C3×D24—31no—
67C3×D24—31no—
68C3×D24—61no—
69C3×#204—62no—
70C3×#204—61no—
71C3×#204—61no—
72C3×#204—61no—
73C3⋊(C3⋊C16)C2×D311yescharacteristic
74C3⋊(C3⋊C16)C2×D311yescharacteristic
75C3×C3⋊C16—31no—
76C3×C3⋊C16—31no—
77C22×H(3,𝔽3)D811yescharacteristic
78C2×#272—41no—
79C2×(C3⋊D3)⋊C3—41no—
80C4×H(3,𝔽3)D811yescharacteristic
81C4×H(3,𝔽3)C2×D411yescharacteristic
82#1084—41no—
83(C3⋊Dic3)⋊C3—41no—
84C3×#135—91no—
85C3×#135—91no—
86#904—31no—
87#904—31no—
88#904—31no—
89#904—61no—
90#904—91no—
91C2×C3×C16—91no—
92#829—91no—
93#829—91no—
94#828—91no—
95#828—91no—
96#827—91no—
97#827—91no—
98#826—91no—
99#826—91no—
100C2×C3⋊C16—31no—
101C2×C3⋊C16—91no—
102Dic24—91no—
103#800—91no—
104#800—91no—
105D48—91no—
106C2×C32×C4—21no—
107C2×C32×C4D1211yescharacteristic
108C2×C32×C4#6711yescharacteristic
109#638—41no—
110C4⋊(C3⋊D3)—21no—
111C4×C3⋊D3—41no—
112#634—21no—
113C3×#67—122no—
114C3×#67—121no—
115C3×#67—121no—
116C3×#67—121no—
117C3×D12—62no—
118C3×D12—61no—
119C3×D12—61no—
120C3×D12—61no—
121C3×C4×D3—122no—
122C3×C4×D3—121no—
123C3×C4×D3—121no—
124C3×C4×D3—121no—
125C3×Dic6—62no—
126C3×Dic6—61no—
127C3×Dic6—61no—
128C3×Dic6—61no—
129C32×C8—21no—
130C32×C8—21no—
131C32×C8D1211yescharacteristic
132C32×C8#6711yescharacteristic
133C32×C8C22×D311yescharacteristic
134C32×C8C22×D311yescharacteristic
135#508—271noSylow
136C2×H(3,𝔽3)—21no—
137C2×H(3,𝔽3)C2×D811yescharacteristic
138#272—81no—
139(C3⋊D3)⋊C3—81no—
140C3×#41—181no—
141C3×#41—181no—
142#235—61no—
143#235—61no—
144#235—62no—
145#235—61no—
146#235—181no—
147C3×Q16—91no—
148C3×Q16—91no—
149C3×SD16—181no—
150C3×SD16—181no—
151C3×D8—91no—
152C3×D8—91no—
153C2×C3×C8—31no—
154C2×C3×C8—31no—
155C2×C3×C8—61no—
156C2×C3×C8D3211yescharacteristic
157Dic12—31no—
158Dic12—31no—
159Dic12—31no—
160Dic12—61no—
161Dic12—91no—
162D24—31no—
163D24—31no—
164D24—31no—
165D24—61no—
166D24—91no—
167#204—61no—
168#204—61no—
169#204—62no—
170#204—61no—
171#204—181no—
172C3×C16—91no—
173C3×C16—91no—
174C3⋊C16—31no—
175C3⋊C16—31no—
176C3⋊C16—91no—
177C3⋊C16—91no—
178C22×C32—21no—
179C22×C32D2411yescharacteristic
180C22×C32#21311yescharacteristic
181C2×C3⋊D3—41no—
182C2×C3×D3—122no—
183C2×C3×D3—121no—
184C2×C3×D3—121no—
185C2×C3×D3—121no—
186C32×C4—21no—
187C32×C4—21no—
188C32×C4D2411yescharacteristic
189C32×C4#21311yescharacteristic
190C32×C4C2×D1211yescharacteristic
191C32×C4C2×#6711yescharacteristic
192C3⋊Dic3—41no—
193C3×Dic3—122no—
194C3×Dic3—121no—
195C3×Dic3—121no—
196C3×Dic3—121no—
197#135—91no—
198#135—91no—
199Q32—271no—
200SD32—271no—
201SD32—271no—
202D16—271no—
203C2×C16—91no—
204H(3,𝔽3)#50811yescharacteristic, complemented, Sylow
205C3×Q8—181no—
206C3×Q8—181no—
207C3×D4—181no—
208C3×D4—181no—
209C3×D4—361no—
210C3×D4—361no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—61no—
214C2×C3×C4—361no—
215C2×C3×C4—361no—
216C2×C3×C4#62611yescharacteristic
217#67—121no—
218#67—121no—
219#67—122no—
220#67—121no—
221#67—361no—
222D12—61no—
223D12—61no—
224D12—62no—
225D12—61no—
226D12—181no—
227C4×D3—121no—
228C4×D3—121no—
229C4×D3—122no—
230C4×D3—121no—
231C4×D3—361no—
232Dic6—61no—
233Dic6—61no—
234Dic6—62no—
235Dic6—61no—
236Dic6—181no—
237C3×C8—31no—
238C3×C8—31no—
239C3×C8—31no—
240C3×C8—31no—
241C3×C8—61no—
242C3×C8—61no—
243C3×C8#62611yescharacteristic
244C3×C8C2×D3211yescharacteristic
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—41no—
249C2×C32C2×D2411yescharacteristic
250C2×C32C2×#21311yescharacteristic
251C3⋊D3—81no—
252C3×D3—242no—
253C3×D3—241no—
254C3×D3—241no—
255C3×D3—241no—
256#41—181no—
257#41—181no—
258Q16—91no—
259Q16—91no—
260SD16—181no—
261SD16—181no—
262D8—91no—
263D8—91no—
264C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
265C16—91no—
266C16—91no—
267C22×C3—31no—
268C22×C3—31no—
269C22×C3—61no—
270C22×C3—361no—
271C22×C3—361no—
272C22×C3#371411yescharacteristic
273C2×D3—121no—
274C2×D3—121no—
275C2×D3—122no—
276C2×D3—121no—
277C2×D3—361no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—31no—
281C3×C4—31no—
282C3×C4—61no—
283C3×C4—61no—
284C3×C4—361no—
285C3×C4—361no—
286C3×C4#371411yescharacteristic
287C3×C4C2×#62611yescharacteristic
288Dic3—121no—
289Dic3—121no—
290Dic3—122no—
291Dic3—121no—
292Dic3—361no—
293C32—21no—
294C32#490411yescharacteristic, complemented
295C32#519411yescharacteristic, complemented
296Q8—181no—
297Q8—181no—
298D4—181no—
299D4—181no—
300D4—361no—
301D4—361no—
302C2×C4—361no—
303C2×C4—361no—
304C2×C4#620011yescharacteristic
305C8#620011yescharacteristic
306C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
307C2×C3—21no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—61no—
311C2×C3—61no—
312C2×C3—61no—
313C2×C3—121no—
314C2×C3—721no—
315C2×C3—721no—
316C2×C3C2×#371411yescharacteristic
317D3—241no—
318D3—241no—
319D3—242no—
320D3—241no—
321D3—721no—
322C22—361no—
323C22—361no—
324C22#8843611yescharacteristic
325C4—361no—
326C4—361no—
327C4#8843611yescharacteristic
328C4C2×#620011yescharacteristic
329C3—31no—
330C3—31no—
331C3—61no—
332C3#1059063211yescharacteristic
333C2—21no—
334C2—721no—
335C2—721no—
336C2C2×#8843611yescharacteristic
subgroupG/H
centerC4#88436
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C2×#3714
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#508
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11721779C2semidirect
2. #64113⋊C3C2semidirect
3. C2×(C3⋊(C3⋊C16))⋊C3C2semidirect
4. #11720256C2semidirect
5. #11720255C2semidirect
6. #11720254C2semidirect
7. #11720253C2semidirect
21. #64113D3semidirect
26. C2×C3⋊(C3⋊C16)D3semidirect
204. H(3,𝔽3)#508semidirect
294. C32#4904semidirect
295. C32#5194semidirect

Maximal subgroups

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

subgroupindex
#117217792
#64113⋊C32
C2×(C3⋊(C3⋊C16))⋊C32
#117202562
#117202552
#117202542
#117202532
#105906323
#105906293