Finite Group Databasebeta

← #13149604

Subgroups — #13149604

Order 1728.

883 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 883 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C4⋊C4×#272C211yesmaximal, characteristic
2#64099⋊C3C212yesmaximal, complemented
3#11721152C212yesmaximal
4#11721152C211yesmaximal, characteristic
5#11721147C212yesmaximal
6C4×#6199C211yesmaximal, characteristic
7#11721093C212yesmaximal, complemented
8#11721086C214yesmaximal, complemented
9D3×#890—32nomaximal
10C2×C4×#272C2212yes—
11C2×C4×#272C2211yescharacteristic
12C2×#6199C2211yescharacteristic
13C2×#6199C2211yescharacteristic
14C2×#6199C2212yes—
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16C4⋊#1084C2211yescharacteristic
17#88540C2212yes—
18C4⋊((C3⋊Dic3)⋊C3)C2214yes—
19C4⋊((C3⋊Dic3)⋊C3)C2212yes—
20#3750⋊C3C2214yes—
21C4×(C3⋊Dic3)⋊C3C2212yes—
22#88449C2212yes—
23#88449C2214yes—
24#88448C2214yes—
25#88447C2212yes—
26#64099D312yescomplemented
27C3×D3×C4⋊C4—32no—
28C3×D3×C4⋊C4—61no—
29C3×#890—32no—
30D3×C4⋊Dic3—32no—
31D3×C4⋊Dic3—34no—
32D3×Dic3⋊C4—34no—
33C4×D3×Dic3—32no—
34#63872—32no—
35#63869—32no—
36#63846—34no—
37#63845—34no—
38C22×#272C2311yescharacteristic
39C2×C4×H(3,𝔽3)C2312yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084C2312yes—
42C2×#1084C2311yescharacteristic
43C4×#272—24no—
44C4×#272Q814yes—
45C2×(C3⋊Dic3)⋊C3C2312yes—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47C2×(C3⋊Dic3)⋊C3C2314yes—
48#6199—22no—
49#6199—24no—
50#6199—22no—
51D3×#125—92no—
52C2×#890—91no—
53C2×C3×C4×D3—32no—
54C2×C3×C4×D3—34no—
55C2×C3×C4×D3—61no—
56C2×C3×C4×D3—62no—
57C2×D3×Dic3—32no—
58C2×D3×Dic3—32no—
59C2×D3×Dic3—34no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D312yes—
62C4⋊(C3⋊Dic3)C2×D314yes—
63C4⋊(C3⋊Dic3)C2×D312yes—
64#3750C2×D314yes—
65C4×C3⋊Dic3C2×D312yes—
66C3×C4⋊Dic3—34no—
67C3×C4⋊Dic3—32no—
68C3×C4⋊Dic3—32no—
69C3×C4⋊Dic3—61no—
70C3×Dic3⋊C4—34no—
71C3×Dic3⋊C4—34no—
72C3×Dic3⋊C4—62no—
73C3×C4×Dic3—32no—
74#3724—34no—
75Dic3⋊Dic3—34no—
76Dic3⋊Dic3—34no—
77Dic3⋊Dic3—34no—
78Dic3⋊Dic3—34no—
79Dic32—32no—
80Dic32—32no—
81C22×H(3,𝔽3)C2411yescharacteristic
82C2×#272C2×Q812yes—
83C2×#272#4112yes—
84C2×#272#4112yes—
85C4×H(3,𝔽3)—22no—
86C4×H(3,𝔽3)C2×Q812yes—
87#1084—22no—
88#1084C2×Q812yes—
89(C3⋊Dic3)⋊C3—22no—
90(C3⋊Dic3)⋊C3—24no—
91(C3⋊Dic3)⋊C3—22no—
92C3×#125—92no—
93C2×C3×C4⋊C4—91no—
94C2×C4⋊Dic3—92no—
95C2×C4⋊Dic3—91no—
96C2×Dic3⋊C4—92no—
97C2×C4×Dic3—91no—
98D3×C4⋊C4—32no—
99D3×C4⋊C4—61no—
100D3×C4⋊C4—94no—
101D3×C4⋊C4—94no—
102D3×C4⋊C4—92no—
103#890—32no—
104#890—92no—
105#890—92no—
106#890—94no—
107#890—92no—
108#889—94no—
109#889—94no—
110C42×D3—92no—
111#870—92no—
112C22×C3×D3—32no—
113C22×C3×D3—61no—
114C2×C32×C4—21no—
115C2×C32×C4—22no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D312yes—
118C2×C3⋊Dic3C22×D312yes—
119C2×C3⋊Dic3C22×D312yes—
120C2×C3⋊Dic3C22×D314yes—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—32no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—62no—
128C3×C4×D3—38no—
129C3×C4×D3—64no—
130C3×C4×D3—68no—
131C3×C4×D3—124no—
132D3×Dic3—64no—
133D3×Dic3—68no—
134D3×Dic3—64no—
135C2×#125—271noSylow
136C2×H(3,𝔽3)C22×Q811yescharacteristic
137C2×H(3,𝔽3)C2×#4111yescharacteristic
138C2×H(3,𝔽3)C2×#4111yescharacteristic
139#272#12514yescomplemented
140C22×C3×C4—91no—
141C22×C3×C4—92no—
142C22×Dic3—91no—
143C22×Dic3—91no—
144C22×Dic3—92no—
145C2×C4×D3—32no—
146C2×C4×D3—34no—
147C2×C4×D3—61no—
148C2×C4×D3—62no—
149C2×C4×D3—92no—
150C2×C4×D3—92no—
151C2×C4×D3—94no—
152C3×C4⋊C4—32no—
153C3×C4⋊C4—61no—
154C3×C4⋊C4—94no—
155C3×C4⋊C4—94no—
156C3×C4⋊C4—92no—
157C3×C4⋊C4—92no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4D3211yescharacteristic
160C3×C42—92no—
161C4⋊Dic3—32no—
162C4⋊Dic3—34no—
163C4⋊Dic3—32no—
164C4⋊Dic3—61no—
165C4⋊Dic3—92no—
166C4⋊Dic3—94no—
167C4⋊Dic3—94no—
168C4⋊Dic3—92no—
169C4⋊Dic3—94no—
170Dic3⋊C4—34no—
171Dic3⋊C4—34no—
172Dic3⋊C4—62no—
173Dic3⋊C4—94no—
174Dic3⋊C4—94no—
175Dic3⋊C4—94no—
176Dic3⋊C4—94no—
177Dic3⋊C4—94no—
178Dic3⋊C4—94no—
179C4×Dic3—32no—
180C4×Dic3—92no—
181C4×Dic3—92no—
182C4×Dic3—92no—
183C4×Dic3—92no—
184C22×C32—21no—
185C22×C32C23×D312yes—
186C2×C3×D3—34no—
187C2×C3×D3—34no—
188C2×C3×D3—34no—
189C2×C3×D3—62no—
190C2×C3×D3—62no—
191C2×C3×D3—62no—
192C32×C4—24no—
193C32×C4—22no—
194C32×C4—42no—
195C32×C4D3×Q814yes—
196C3⋊Dic3—22no—
197C3⋊Dic3—22no—
198C3⋊Dic3—24no—
199C3×Dic3—34no—
200C3×Dic3—62no—
201C3×Dic3—62no—
202C3×Dic3—62no—
203C3×Dic3—64no—
204C3×Dic3—64no—
205C3×Dic3—122no—
206#125—92no—
207#125—272no—
208#125—274no—
209C2×C4⋊C4—91no—
210C2×C4⋊C4—272no—
211C2×C4⋊C4—271no—
212C2×C4⋊C4—272no—
213C2×C42—271no—
214H(3,𝔽3)C2×#12511yescharacteristic, complemented, Sylow
215C23×C3—91no—
216C22×D3—32no—
217C22×D3—61no—
218C2×C3×C4—32no—
219C2×C3×C4—34no—
220C2×C3×C4—61no—
221C2×C3×C4—62no—
222C2×C3×C4—91no—
223C2×C3×C4—94no—
224C2×C3×C4—92no—
225C2×C3×C4—92no—
226C2×C3×C4—94no—
227C2×C3×C4—92no—
228C2×C3×C4—184no—
229C2×C3×C4C2×D3212yes—
230C2×C3×C4C2×D3211yescharacteristic
231C2×Dic3—32no—
232C2×Dic3—34no—
233C2×Dic3—32no—
234C2×Dic3—32no—
235C2×Dic3—34no—
236C2×Dic3—61no—
237C2×Dic3—62no—
238C2×Dic3—92no—
239C2×Dic3—92no—
240C2×Dic3—94no—
241C2×Dic3—182no—
242C2×Dic3—182no—
243C2×Dic3—184no—
244C4×D3—38no—
245C4×D3—64no—
246C4×D3—68no—
247C4×D3—124no—
248C4×D3—184no—
249C4×D3—184no—
250C4×D3—188no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32C2×D3×Q812yes—
255C2×C32D3×#4112yes—
256C2×C32D3×#4112yes—
257C3×D3—38no—
258C3×D3—64no—
259C22×C4—91no—
260C22×C4—92no—
261C22×C4—271no—
262C22×C4—271no—
263C22×C4—272no—
264C4⋊C4—92no—
265C4⋊C4—91no—
266C4⋊C4—94no—
267C4⋊C4—94no—
268C4⋊C4—92no—
269C4⋊C4—272no—
270C4⋊C4—274no—
271C4⋊C4—274no—
272C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
273C42—92no—
274C42—272no—
275C22×C3—32no—
276C22×C3—61no—
277C22×C3—92no—
278C22×C3—92no—
279C22×C3—92no—
280C22×C3C22×D3211yescharacteristic
281C2×D3—34no—
282C2×D3—34no—
283C2×D3—34no—
284C2×D3—62no—
285C2×D3—62no—
286C2×D3—62no—
287C3×C4—22no—
288C3×C4—34no—
289C3×C4—64no—
290C3×C4—62no—
291C3×C4—92no—
292C3×C4—122no—
293C3×C4—182no—
294C3×C4—182no—
295C3×C4—184no—
296C3×C4—182no—
297C3×C4#379712yes—
298Dic3—34no—
299Dic3—62no—
300Dic3—64no—
301Dic3—62no—
302Dic3—62no—
303Dic3—64no—
304Dic3—122no—
305Dic3—182no—
306Dic3—182no—
307Dic3—184no—
308C32—21no—
309C32D3×#12512yescomplemented
310C23—91no—
311C2×C4—91no—
312C2×C4—94no—
313C2×C4—92no—
314C2×C4—92no—
315C2×C4—94no—
316C2×C4—92no—
317C2×C4—184no—
318C2×C4—542no—
319C2×C4—542no—
320C2×C4—544no—
321C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
322C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
323C2×C3—32no—
324C2×C3—32no—
325C2×C3—32no—
326C2×C3—61no—
327C2×C3—61no—
328C2×C3—61no—
329C2×C3—94no—
330C2×C3C2×#379711yescharacteristic
331C2×C3#6431211yescharacteristic
332C2×C3#6431911yescharacteristic
333D3—38no—
334D3—64no—
335C22—92no—
336C22—92no—
337C22—92no—
338C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
339C4—22no—
340C4—92no—
341C4—182no—
342C4—182no—
343C4—182no—
344C4—184no—
345C4#8865112yes—
346C3—32no—
347C3—61no—
348C3#1059448211yescharacteristic
349C2—94no—
350C2C2×#8865111yescharacteristic
351C2#1172250511yescharacteristic
352C2#1172252011yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC22×#272C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#125
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
2. #64099⋊C3C2semidirect
7. #11721093C2semidirect
8. #11721086C2semidirect
26. #64099D3semidirect
139. #272#125semidirect
214. H(3,𝔽3)C2×#125semidirect
309. C32D3×#125semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#2722
#64099⋊C32
#117211522
#117211472
C4×#61992
#117210932
#117210862
D3×#8903