Finite Group Databasebeta

← #13149384

Subgroups — #13149384

Order 1728.

658 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 129 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#11721758C211yesmaximal, characteristic
2#64098⋊C3C212yesmaximal
3#11721098C211yesmaximal, characteristic
4#11721098C211yesmaximal, characteristic
5#11721093C212yesmaximal
6#11721091C212yesmaximal
7#11721086C212yesmaximal
8#11721085C212yesmaximal
9#11721085C212yesmaximal
10#10594326—32nomaximal
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C4⋊#1084C2211yescharacteristic
13C4⋊#1084C2212yes—
14#88540C2212yes—
15C4×#1084C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2212yes—
17#3750⋊C3C2212yes—
18#3750⋊C3C2212yes—
19#3750⋊C3C2212yes—
20#3750⋊C3C2212yes—
21C4×(C3⋊Dic3)⋊C3C2212yes—
22#88449C2211yescharacteristic
23#88449C2212yes—
24#88449C2211yescharacteristic
25#88449C2211yescharacteristic
26#88449C2211yescharacteristic
27#88448C2212yes—
28#88448C2212yes—
29#88448C2212yes—
30#88448C2212yes—
31#88447C2212yes—
32#64098D312yes—
33C3×#890—32no—
34C3×#890—61no—
35C3×#889—32no—
36#63872—32no—
37#63870—32no—
38#63869—32no—
39#63867—32no—
40#63867—32no—
41#63867—32no—
42#63847—32no—
43#63847—32no—
44#63846—32no—
45#63846—32no—
46#63846—32no—
47#63845—32no—
48C2×C4×H(3,𝔽3)C2312yes—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×#1084C2312yes—
51C2×#1084C2311yescharacteristic
52C2×#1084C2311yescharacteristic
53C2×(C3⋊Dic3)⋊C3C2312yes—
54C2×(C3⋊Dic3)⋊C3C2312yes—
55C2×(C3⋊Dic3)⋊C3C2312yes—
56C2×(C3⋊Dic3)⋊C3C2312yes—
57C3⋊#564—91no—
58C3⋊#564—92no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4C2×D312yes—
61C4⋊(C3⋊Dic3)C2×D312yes—
62#3750C2×D312yes—
63#3750C2×D312yes—
64#3750C2×D312yes—
65#3750C2×D312yes—
66C4×C3⋊Dic3C2×D312yes—
67C3×C4⋊Dic3—32no—
68C3×C4⋊Dic3—32no—
69C3×C4⋊Dic3—32no—
70C3×C4⋊Dic3—32no—
71C3×C4⋊Dic3—62no—
72C3×C4⋊Dic3—61no—
73C3×Dic3⋊C4—32no—
74C3×Dic3⋊C4—32no—
75C3×Dic3⋊C4—32no—
76C3×Dic3⋊C4—32no—
77C3×Dic3⋊C4—32no—
78C3×Dic3⋊C4—32no—
79C3×Dic3⋊C4—62no—
80C3×C4×Dic3—32no—
81C3×C4×Dic3—32no—
82C3×C4×Dic3—61no—
83#3724—32no—
84#3724—32no—
85#3724—32no—
86#3724—32no—
87Dic3⋊Dic3—32no—
88Dic3⋊Dic3—32no—
89Dic3⋊Dic3—32no—
90Dic3⋊Dic3—32no—
91Dic3⋊Dic3—32no—
92Dic3⋊Dic3—32no—
93Dic3⋊Dic3—32no—
94Dic3⋊Dic3—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97Dic32—32no—
98Dic32—32no—
99C22×H(3,𝔽3)C2411yescharacteristic
100C4×H(3,𝔽3)—21no—
101C4×H(3,𝔽3)—22no—
102#1084—22no—
103#1084—21no—
104#1084—21no—
105(C3⋊Dic3)⋊C3—22no—
106(C3⋊Dic3)⋊C3—22no—
107(C3⋊Dic3)⋊C3—22no—
108(C3⋊Dic3)⋊C3—22no—
109C3×#125—92no—
110C3×#125—91no—
111#890—32no—
112#890—61no—
113#890—92no—
114#890—92no—
115#890—92no—
116#890—92no—
117#890—92no—
118#889—32no—
119#889—92no—
120#889—92no—
121#889—92no—
122#889—92no—
123#889—92no—
124#889—92no—
125#889—92no—
126#889—92no—
127#889—92no—
128#889—92no—
129#889—92no—
130#870—91no—
131#870—91no—
132#870—92no—
133#870—92no—
134C2×C32×C4—21no—
135C2×C32×C4—22no—
136C2×C32×C4C22×D312yes—
137C2×C32×C4C22×D312yes—
138C2×C32×C4C22×D312yes—
139C2×C3⋊Dic3C22×D312yes—
140C2×C3⋊Dic3C22×D312yes—
141C2×C3⋊Dic3C22×D312yes—
142C2×C3⋊Dic3C22×D312yes—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—32no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—62no—
154#564—271noSylow
155C2×H(3,𝔽3)2−1+411yescharacteristic
156C2×H(3,𝔽3)2−1+411yescharacteristic
157C2×H(3,𝔽3)2−1+411yescharacteristic
158C3×C4⋊C4—32no—
159C3×C4⋊C4—61no—
160C3×C4⋊C4—92no—
161C3×C4⋊C4—92no—
162C3×C4⋊C4—92no—
163C3×C4⋊C4—92no—
164C3×C4⋊C4—92no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4—92no—
167C3×C4⋊C4—92no—
168C3×C4⋊C4D3211yescharacteristic
169C3×C42—92no—
170C3×C42—91no—
171C4⋊Dic3—32no—
172C4⋊Dic3—32no—
173C4⋊Dic3—32no—
174C4⋊Dic3—32no—
175C4⋊Dic3—62no—
176C4⋊Dic3—61no—
177C4⋊Dic3—92no—
178C4⋊Dic3—91no—
179C4⋊Dic3—91no—
180C4⋊Dic3—91no—
181C4⋊Dic3—91no—
182C4⋊Dic3—92no—
183C4⋊Dic3—92no—
184C4⋊Dic3—92no—
185C4⋊Dic3—92no—
186C4⋊Dic3—92no—
187Dic3⋊C4—32no—
188Dic3⋊C4—32no—
189Dic3⋊C4—32no—
190Dic3⋊C4—32no—
191Dic3⋊C4—32no—
192Dic3⋊C4—32no—
193Dic3⋊C4—62no—
194Dic3⋊C4—92no—
195Dic3⋊C4—92no—
196Dic3⋊C4—92no—
197Dic3⋊C4—92no—
198Dic3⋊C4—92no—
199Dic3⋊C4—92no—
200Dic3⋊C4—92no—
201Dic3⋊C4—92no—
202Dic3⋊C4—92no—
203Dic3⋊C4—92no—
204Dic3⋊C4—92no—
205Dic3⋊C4—92no—
206Dic3⋊C4—92no—
207Dic3⋊C4—92no—
208Dic3⋊C4—92no—
209Dic3⋊C4—92no—
210Dic3⋊C4—92no—
211Dic3⋊C4—92no—
212C4×Dic3—32no—
213C4×Dic3—32no—
214C4×Dic3—61no—
215C4×Dic3—92no—
216C4×Dic3—92no—
217C4×Dic3—92no—
218C4×Dic3—92no—
219C22×C32—21no—
220C22×C32C23×D312yes—
221C32×C4—22no—
222C32×C4—22no—
223C32×C4—22no—
224C32×C4—22no—
225C32×C4—42no—
226C3⋊Dic3—22no—
227C3⋊Dic3—22no—
228C3⋊Dic3—22no—
229C3⋊Dic3—22no—
230C3×Dic3—62no—
231C3×Dic3—62no—
232C3×Dic3—62no—
233C3×Dic3—62no—
234C3×Dic3—62no—
235C3×Dic3—62no—
236C3×Dic3—62no—
237C3×Dic3—62no—
238C3×Dic3—121no—
239C3×Dic3—121no—
240C3×Dic3—122no—
241#125—92no—
242#125—91no—
243#125—271no—
244#125—271no—
245#125—272no—
246#125—272no—
247#125—272no—
248#125—272no—
249#125—272no—
250H(3,𝔽3)#56411yescharacteristic, complemented, Sylow
251C2×C3×C4—32no—
252C2×C3×C4—32no—
253C2×C3×C4—32no—
254C2×C3×C4—61no—
255C2×C3×C4—62no—
256C2×C3×C4—92no—
257C2×C3×C4—92no—
258C2×C3×C4—92no—
259C2×C3×C4—92no—
260C2×C3×C4—92no—
261C2×C3×C4—91no—
262C2×C3×C4—91no—
263C2×C3×C4C2×D3212yes—
264C2×C3×C4C2×D3211yescharacteristic
265C2×Dic3—32no—
266C2×Dic3—32no—
267C2×Dic3—32no—
268C2×Dic3—32no—
269C2×Dic3—32no—
270C2×Dic3—32no—
271C2×Dic3—32no—
272C2×Dic3—32no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—62no—
276C2×Dic3—92no—
277C2×Dic3—92no—
278C2×Dic3—92no—
279C2×Dic3—92no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32#100712yes—
284C2×C32#101012yes—
285C2×C32#101012yes—
286C4⋊C4—92no—
287C4⋊C4—92no—
288C4⋊C4—92no—
289C4⋊C4—92no—
290C4⋊C4—92no—
291C4⋊C4—92no—
292C4⋊C4—91no—
293C4⋊C4—92no—
294C4⋊C4—271no—
295C4⋊C4—271no—
296C4⋊C4—272no—
297C4⋊C4—272no—
298C4⋊C4—271no—
299C4⋊C4—271no—
300C4⋊C4—272no—
301C4⋊C4—272no—
302C4⋊C4—272no—
303C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
304C42—91no—
305C42—92no—
306C42—272no—
307C22×C3—32no—
308C22×C3—61no—
309C22×C3C22×D3211yescharacteristic
310C3×C4—21no—
311C3×C4—22no—
312C3×C4—62no—
313C3×C4—62no—
314C3×C4—62no—
315C3×C4—62no—
316C3×C4—122no—
317C3×C4—182no—
318C3×C4—182no—
319C3×C4—182no—
320C3×C4—182no—
321C3×C4—181no—
322C3×C4—182no—
323C3×C4—181no—
324Dic3—62no—
325Dic3—62no—
326Dic3—62no—
327Dic3—62no—
328Dic3—62no—
329Dic3—62no—
330Dic3—62no—
331Dic3—62no—
332Dic3—121no—
333Dic3—121no—
334Dic3—122no—
335Dic3—182no—
336Dic3—182no—
337Dic3—182no—
338Dic3—182no—
339C32—21no—
340C32C3⋊#56412yescomplemented
341C2×C4—91no—
342C2×C4—91no—
343C2×C4—92no—
344C2×C4—92no—
345C2×C4—92no—
346C2×C4—92no—
347C2×C4—92no—
348C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
349C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
350C2×C3—32no—
351C2×C3—32no—
352C2×C3—32no—
353C2×C3—61no—
354C2×C3—61no—
355C2×C3—61no—
356C2×C3#6429811yescharacteristic
357C2×C3#6430911yescharacteristic
358C2×C3#6431611yescharacteristic
359C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
360C4—22no—
361C4—21no—
362C4—181no—
363C4—182no—
364C4—181no—
365C4—182no—
366C4—182no—
367C4—182no—
368C4—182no—
369C3—32no—
370C3—61no—
371C3#1059435411yescharacteristic
372C2#1172248311yescharacteristic
373C2#1172250211yescharacteristic
374C2#1172251711yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#564
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
250. H(3,𝔽3)#564semidirect
340. C32C3⋊#564semidirect

Maximal subgroups

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

subgroupindex
#117217582
#64098⋊C32
#117210982
#117210932
#117210912
#117210862
#117210852
#105943263