Finite Group Databasebeta

← #13150666

Subgroups — #13150666

Order 1728.

885 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 885 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C2×C4×#1084C211yesmaximal, characteristic
2#64133⋊C3C212yesmaximal
3#11721309C212yesmaximal
4#11721298C212yesmaximal
5#11721117C211yesmaximal, characteristic, complemented
6#11721099C211yesmaximal, characteristic, complemented
7#11721098C212yesmaximal, complemented
8#11721084C212yesmaximal, complemented
9#11721078C212yesmaximal, complemented
10#10594808—32nomaximal
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C2×#634⋊C3C2212yes—
15C2×#6196C2212yes—
16C4×#1084C2211yescharacteristic
17C4×#1084C2211yescharacteristic
18C4×#1084C2211yescharacteristic
19C4×#1084C2211yescharacteristic
20#3757⋊C3C2214yes—
21C4⋊((C3⋊Dic3)⋊C3)C2212yes—
22#3750⋊C3C2214yes—
23#88449C2212yes—
24#88449C2212yes—
25#88449C2212yes—
26#88448C2214yes—
27#88447C2214yes—
28#64133D312yes—
29C3×#924—32no—
30C2×C3×C4×Dic3—32no—
31C2×C3×C4×Dic3—61no—
32#63964—34no—
33#63955—34no—
34#63918—32no—
35#63867—34no—
36#63866—32no—
37#63844—32no—
38Dic3×Dic6—32no—
39#63837—34no—
40C23×H(3,𝔽3)C2311yescharacteristic
41C2×C4×H(3,𝔽3)—22no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×#1084—22no—
45C2×#1084Q814yescomplemented
46C2×#1084C2311yescharacteristic
47C2×#1084C2311yescharacteristic
48C2×#1084C2311yescharacteristic
49C2×#1084C2311yescharacteristic
50C2×(C3⋊Dic3)⋊C3C2314yes—
51C2×(C3⋊Dic3)⋊C3C2314yes—
52#634⋊C3—24no—
53#6196—24no—
54#5622—92no—
55#5470—91no—
56C22×C32×C4—21no—
57C22×C32×C4C2×D312yes—
58C2×#634C2×D312yes—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—61no—
62C22×C3×Dic3—61no—
63C2×C3×Dic6—32no—
64C2×#627—34no—
65#3757C2×D314yes—
66C4⋊(C3⋊Dic3)C2×D312yes—
67#3750C2×D314yes—
68C3×#217—34no—
69C3×C4⋊Dic3—32no—
70C3×Dic3⋊C4—34no—
71C3×C4×Dic3—32no—
72C3×C4×Dic3—32no—
73C3×C4×Dic3—32no—
74C3×C4×Dic3—32no—
75C3×C4×Dic3—61no—
76C3×C4×Dic3—61no—
77C3×C4×Dic3—61no—
78C3×C4×Dic3—61no—
79#3724—34no—
80Dic3⋊Dic3—34no—
81Dic3⋊Dic3—34no—
82Dic3⋊Dic3—34no—
83Dic3⋊Dic3—34no—
84Dic32—34no—
85Dic32—34no—
86C22×H(3,𝔽3)—21no—
87C22×H(3,𝔽3)—21no—
88C22×H(3,𝔽3)C2×Q812yes—
89C22×H(3,𝔽3)C2411yescharacteristic
90C4×H(3,𝔽3)—21no—
91C4×H(3,𝔽3)#4112yes—
92#1084—21no—
93#1084C2×Q812yes—
94#1084C2×Q812yes—
95#1084#4112yes—
96(C3⋊Dic3)⋊C3—24no—
97(C3⋊Dic3)⋊C3—24no—
98C3×C22⋊Q8—92no—
99C2×C3×C42—91no—
100Q8×Dic3—92no—
101#944—92no—
102#926—92no—
103#924—32no—
104#924—92no—
105#924—92no—
106C2×C4×Dic3—32no—
107C2×C4×Dic3—61no—
108#889—94no—
109#888—92no—
110#887—94no—
111#887—92no—
112#878—94no—
113#877—94no—
114#870—92no—
115#869—91no—
116#869—91no—
117C4×Dic6—92no—
118C4×Dic6—92no—
119C23×C32—21no—
120C23×C32C22×D312yes—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4C22×D312yes—
126C2×C32×C4C22×D312yes—
127C2×C3⋊Dic3C22×D314yes—
128C2×C3⋊Dic3C22×D314yes—
129#634—24no—
130C2×C3×Dic3—38no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—32no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—64no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3—64no—
144C3×Dic6—64no—
145#627—68no—
146#533—271noSylow
147C2×H(3,𝔽3)—22no—
148C2×H(3,𝔽3)C22×Q811yescharacteristic
149C2×H(3,𝔽3)C2×#4111yescharacteristic
150C2×H(3,𝔽3)C2×#4111yescharacteristic
151C2×C3×Q8—92no—
152C22×C3×C4—32no—
153C22×C3×C4—61no—
154C22×C3×C4—91no—
155C22×C3×C4—91no—
156C22×C3×C4D3211yescharacteristic
157C22×Dic3—32no—
158C22×Dic3—32no—
159C22×Dic3—61no—
160C22×Dic3—61no—
161C2×Dic6—32no—
162C2×Dic6—92no—
163C2×Dic6—92no—
164C2×Dic6—94no—
165C3×C4⋊C4—92no—
166C3×C4⋊C4—94no—
167C3×C22⋊C4—94no—
168C3×C42—91no—
169C3×C42—91no—
170C3×C42—91no—
171C3×C42—91no—
172#217—34no—
173#217—94no—
174C4⋊Dic3—32no—
175C4⋊Dic3—92no—
176C4⋊Dic3—92no—
177C4⋊Dic3—92no—
178C4⋊Dic3—92no—
179C4⋊Dic3—94no—
180Dic3⋊C4—34no—
181Dic3⋊C4—94no—
182Dic3⋊C4—94no—
183Dic3⋊C4—94no—
184Dic3⋊C4—94no—
185Dic3⋊C4—94no—
186Dic3⋊C4—94no—
187C4×Dic3—32no—
188C4×Dic3—32no—
189C4×Dic3—32no—
190C4×Dic3—32no—
191C4×Dic3—61no—
192C4×Dic3—61no—
193C4×Dic3—61no—
194C4×Dic3—61no—
195C4×Dic3—94no—
196C4×Dic3—94no—
197C4×Dic3—94no—
198C22×C32—22no—
199C22×C32—22no—
200C22×C32—22no—
201C22×C32—22no—
202C22×C32—21no—
203C22×C32—22no—
204C22×C32D3×Q814yes—
205C22×C32C23×D312yes—
206C32×C4—22no—
207C32×C4—22no—
208C32×C4—22no—
209C32×C4#23714yes—
210C3⋊Dic3—24no—
211C3⋊Dic3—24no—
212C3×Dic3—34no—
213C3×Dic3—34no—
214C3×Dic3—34no—
215C3×Dic3—62no—
216C3×Dic3—62no—
217C3×Dic3—62no—
218C3×Dic3—62no—
219C3×Dic3—64no—
220C3×Dic3—64no—
221C3×Dic3—62no—
222C4⋊Q8—271no—
223C4⋊Q8—271no—
224#125—272no—
225C22⋊Q8—92no—
226C22⋊Q8—272no—
227C4×Q8—272no—
228C4×Q8—272no—
229#117—272no—
230C2×C42—91no—
231H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
232C23×C3—32no—
233C23×C3—61no—
234C23×C3C2×D3211yescharacteristic
235C3×Q8—184no—
236C2×C3×C4—22no—
237C2×C3×C4—32no—
238C2×C3×C4—32no—
239C2×C3×C4—64no—
240C2×C3×C4—64no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—91no—
244C2×C3×C4—91no—
245C2×C3×C4—91no—
246C2×C3×C4—94no—
247C2×C3×C4—94no—
248C2×C3×C4—94no—
249C2×C3×C4—91no—
250C2×C3×C4—182no—
251C2×C3×C4C2×D3211yescharacteristic
252C2×C3×C4C2×D3211yescharacteristic
253C2×Dic3—32no—
254C2×Dic3—32no—
255C2×Dic3—32no—
256C2×Dic3—38no—
257C2×Dic3—32no—
258C2×Dic3—34no—
259C2×Dic3—34no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—64no—
265C2×Dic3—64no—
266C2×Dic3—64no—
267C2×Dic3—94no—
268C2×Dic3—94no—
269Dic6—64no—
270Dic6—184no—
271Dic6—188no—
272Dic6—184no—
273C2×C32—24no—
274C2×C32—24no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32C2×#23712yes—
279C2×C32C2×D3×Q812yes—
280C2×C32D3×#4112yes—
281C2×Q8—92no—
282C2×Q8—272no—
283C22×C4—91no—
284C22×C4—91no—
285C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
286C4⋊C4—94no—
287C4⋊C4—92no—
288C4⋊C4—272no—
289C4⋊C4—272no—
290C4⋊C4—274no—
291C4⋊C4—272no—
292C22⋊C4—94no—
293C42—91no—
294C42—91no—
295C42—91no—
296C42—91no—
297C42—274no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—32no—
301C22×C3—34no—
302C22×C3—62no—
303C22×C3—62no—
304C22×C3—62no—
305C22×C3—62no—
306C22×C3—61no—
307C22×C3—62no—
308C22×C3#379712yes—
309C22×C3C22×D3211yescharacteristic
310C3×C4—21no—
311C3×C4—34no—
312C3×C4—62no—
313C3×C4—62no—
314C3×C4—62no—
315C3×C4—92no—
316C3×C4—92no—
317C3×C4—92no—
318C3×C4—184no—
319C3×C4—184no—
320C3×C4—181no—
321C3×C4#380512yes—
322Dic3—34no—
323Dic3—34no—
324Dic3—34no—
325Dic3—62no—
326Dic3—62no—
327Dic3—62no—
328Dic3—62no—
329Dic3—62no—
330Dic3—64no—
331Dic3—64no—
332Dic3—184no—
333Dic3—184no—
334C32—21no—
335C32#562212yescomplemented
336C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
337Q8—184no—
338Q8—544no—
339C2×C4—22no—
340C2×C4—91no—
341C2×C4—91no—
342C2×C4—91no—
343C2×C4—94no—
344C2×C4—91no—
345C2×C4—94no—
346C2×C4—94no—
347C2×C4—182no—
348C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
349C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
350C2×C3—22no—
351C2×C3—32no—
352C2×C3—32no—
353C2×C3—32no—
354C2×C3—64no—
355C2×C3—64no—
356C2×C3—61no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3C2×#379711yescharacteristic
360C2×C3#6430611yescharacteristic
361C2×C3C2×#380511yescharacteristic
362C22—21no—
363C22—21no—
364C22#8865112yes—
365C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
366C4—21no—
367C4—92no—
368C4—92no—
369C4—92no—
370C4—184no—
371C4—181no—
372C4—184no—
373C4#8867212yes—
374C3—32no—
375C3—61no—
376C3#1059481011yescharacteristic
377C2—22no—
378C2C2×#8865111yescharacteristic
379C2#1172249111yescharacteristic
380C2C2×#8867211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#533
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
5. #11721117C2semidirect
6. #11721099C2semidirect
7. #11721098C2semidirect
8. #11721084C2semidirect
9. #11721078C2semidirect
45. C2×#1084Q8semidirect
231. H(3,𝔽3)#533semidirect
335. C32#5622semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10842
#64133⋊C32
#117213092
#117212982
#117211172
#117210992
#117210982
#117210842
#117210782
#105948083