Finite Group Databasebeta

← #13149545

Subgroups — #13149545

Order 1728.

868 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 153 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#11721755C211yesmaximal, characteristic, complemented
2C4⋊C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
3C4×#6198C211yesmaximal, characteristic
4C4×#6198C212yesmaximal
5#11721117C211yesmaximal, characteristic, complemented
6#11721107C212yesmaximal, complemented
7#11721101C214yesmaximal, complemented
8#11721098C212yesmaximal, complemented
9#10594536—32nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2212yes—
12C2×C4×(C3⋊D3)⋊C3C2214yes—
13C2×#6198C2211yescharacteristic
14C2×#6196C2212yes—
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#88540C2212yes—
17C4×#1084C2212yes—
18C4×#1084C2211yescharacteristic
19C4⋊((C3⋊Dic3)⋊C3)C2212yes—
20#3750⋊C3C2214yes—
21#88450C2212yes—
22#88450C2214yes—
23#88449C2211yescharacteristic
24#88449C2212yes—
25#88448C2214yes—
26C4⋊C4×C3⋊D3D312yescomplemented
27C3×D3×C4⋊C4—32no—
28C3×#887—32no—
29C3×#887—61no—
30#63918—32no—
31C4×#626—32no—
32C4×#626—34no—
33#63871—32no—
34#63867—34no—
35#63865—34no—
36#63856—32no—
37#63851—34no—
38C22×(C3⋊D3)⋊C3C2312yes—
39C2×C4×H(3,𝔽3)C2312yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084C2311yescharacteristic
42C2×#1084C2312yes—
43C2×#1084C2311yescharacteristic
44#6229—22no—
45C2×(C3⋊Dic3)⋊C3C2314yes—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47C4×(C3⋊D3)⋊C3—24no—
48C4×(C3⋊D3)⋊C3—44no—
49#6198Q814yescomplemented
50#6196—42no—
51#5579—91no—
52#5511—92no—
53C2×C4×C3⋊D3C2×D312yes—
54C2×C4×C3⋊D3C2×D314yes—
55C2×C3×C4×D3—32no—
56C2×C3×C4×D3—34no—
57C2×C3×Dic6—32no—
58C2×C3×Dic6—61no—
59C2×#627—34no—
60C2×#626—32no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4C2×D312yes—
63C4⋊(C3⋊Dic3)C2×D312yes—
64#3750C2×D314yes—
65C3×C4⋊Dic3—32no—
66C3×Dic3⋊C4—34no—
67C3×Dic3⋊C4—34no—
68C3×Dic3⋊C4—62no—
69C3×C4×Dic3—34no—
70C3×C4×Dic3—32no—
71C3×C4×Dic3—62no—
72C3×C4×Dic3—61no—
73#3724—34no—
74Dic3⋊Dic3—34no—
75Dic3⋊Dic3—32no—
76Dic3⋊Dic3—34no—
77#3722—32no—
78#3722—34no—
79(C2×D3)⋊Dic3—32no—
80(C2×D3)⋊Dic3—34no—
81C22×H(3,𝔽3)C2411yescharacteristic
82C2×(C3⋊D3)⋊C3—22no—
83C2×(C3⋊D3)⋊C3—22no—
84C2×(C3⋊D3)⋊C3C2×Q814yes—
85C4×H(3,𝔽3)—22no—
86C4×H(3,𝔽3)#4112yes—
87#1084—22no—
88#1084—21no—
89#1084C2×Q812yes—
90(C3⋊Dic3)⋊C3—22no—
91(C3⋊Dic3)⋊C3—24no—
92C3×C4×Q8—91no—
93C2×C3×C4⋊C4—92no—
94#946—92no—
95C4×#67—92no—
96C4×#67—94no—
97#924—94no—
98#924—92no—
99#897—92no—
100#896—94no—
101#896—94no—
102D3×C4⋊C4—32no—
103D3×C4⋊C4—92no—
104#889—94no—
105#888—92no—
106#887—32no—
107#887—61no—
108C4×D12—92no—
109C4×D12—91no—
110#870—92no—
111#869—91no—
112C22×C3⋊D3C22×D312yes—
113C22×C3×D3—32no—
114C2×C32×C4—21no—
115C2×C32×C4—22no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D312yes—
118C2×C3⋊Dic3C22×D312yes—
119C2×C3⋊Dic3C22×D314yes—
120C4×C3⋊D3—24no—
121C4×C3⋊D3—44no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—32no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—61no—
130C3×C4×D3—64no—
131C3×C4×D3—124no—
132C3×Dic6—64no—
133C3×Dic6—64no—
134#627—124no—
135#626—38no—
136#554—271noSylow
137C2×H(3,𝔽3)C22×Q811yescharacteristic
138C2×H(3,𝔽3)C2×#4111yescharacteristic
139C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
140(C3⋊D3)⋊C3—24no—
141C2×C3×Q8—91no—
142C22×C3×C4—92no—
143C22×C3×C4—94no—
144C2×#67—92no—
145C2×D12—91no—
146C2×C4×D3—32no—
147C2×C4×D3—34no—
148C2×C4×D3—92no—
149C2×C4×D3—94no—
150C2×Dic6—32no—
151C2×Dic6—61no—
152C2×Dic6—92no—
153C2×Dic6—94no—
154C3×C4⋊C4—32no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4—92no—
157C3×C4⋊C4—94no—
158C3×C4⋊C4—92no—
159C3×C4⋊C4D3211yescharacteristic
160C3×C42—91no—
161C3×C42—92no—
162#217—92no—
163#217—94no—
164(C2×D3)⋊C4—92no—
165(C2×D3)⋊C4—94no—
166(C2×D3)⋊C4—92no—
167(C2×D3)⋊C4—94no—
168C4⋊Dic3—32no—
169C4⋊Dic3—92no—
170C4⋊Dic3—92no—
171C4⋊Dic3—91no—
172C4⋊Dic3—94no—
173Dic3⋊C4—34no—
174Dic3⋊C4—34no—
175Dic3⋊C4—62no—
176Dic3⋊C4—94no—
177Dic3⋊C4—94no—
178Dic3⋊C4—94no—
179Dic3⋊C4—92no—
180Dic3⋊C4—94no—
181C4×Dic3—32no—
182C4×Dic3—34no—
183C4×Dic3—62no—
184C4×Dic3—61no—
185C22×C32—21no—
186C22×C32C23×D312yes—
187C2×C3⋊D3—22no—
188C2×C3⋊D3—22no—
189C2×C3⋊D3D3×Q814yes—
190C2×C3×D3—34no—
191C2×C3×D3—62no—
192C2×C3×D3—62no—
193C32×C4—24no—
194C32×C4—22no—
195C32×C4—42no—
196C32×C4#23514yes—
197C3⋊Dic3—22no—
198C3⋊Dic3—24no—
199C3×Dic3—34no—
200C3×Dic3—62no—
201C3×Dic3—64no—
202C3×Dic3—62no—
203C3×Dic3—64no—
204C3×Dic3—64no—
205C3×Dic3—62no—
206C3×Dic3—121no—
207C4⋊Q8—271no—
208#125—272no—
209C22⋊Q8—274no—
210C22⋊Q8—272no—
211C4×Q8—91no—
212C4×D4—272no—
213C4×D4—271no—
214C2×C4⋊C4—92no—
215H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
216C23×C3—92no—
217C22×D3—32no—
218C22×D3—92no—
219C3×Q8—182no—
220C2×C3×C4—32no—
221C2×C3×C4—34no—
222C2×C3×C4—61no—
223C2×C3×C4—62no—
224C2×C3×C4—91no—
225C2×C3×C4—92no—
226C2×C3×C4—94no—
227C2×C3×C4—92no—
228C2×C3×C4—91no—
229C2×C3×C4—184no—
230C2×C3×C4—364no—
231C2×C3×C4C2×D3212yes—
232C2×C3×C4C2×D3211yescharacteristic
233#67—98no—
234C2×Dic3—32no—
235C2×Dic3—32no—
236C2×Dic3—34no—
237C2×Dic3—32no—
238C2×Dic3—34no—
239C2×Dic3—61no—
240C2×Dic3—62no—
241C2×Dic3—61no—
242C2×Dic3—92no—
243C2×Dic3—94no—
244D12—94no—
245C4×D3—64no—
246C4×D3—124no—
247C4×D3—184no—
248C4×D3—364no—
249Dic6—64no—
250Dic6—64no—
251Dic6—362no—
252Dic6—364no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32C2×#23512yes—
257C2×C32#100412yes—
258C2×C32C2×D3×Q812yes—
259C3⋊D3—24no—
260C3×D3—64no—
261C2×Q8—91no—
262C2×Q8—272no—
263C2×D4—271no—
264C22×C4—92no—
265C22×C4—94no—
266C4⋊C4—94no—
267C4⋊C4—92no—
268C4⋊C4—92no—
269C4⋊C4—271no—
270C4⋊C4—272no—
271C4⋊C4—274no—
272C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
273C22⋊C4—272no—
274C22⋊C4—274no—
275C42—92no—
276C42—91no—
277C22×C3—32no—
278C22×C3—61no—
279C22×C3—94no—
280C22×C3—182no—
281C22×C3—182no—
282C22×C3C22×D3211yescharacteristic
283C2×D3—34no—
284C2×D3—62no—
285C2×D3—62no—
286C2×D3—94no—
287C2×D3—182no—
288C2×D3—182no—
289C3×C4—22no—
290C3×C4—34no—
291C3×C4—64no—
292C3×C4—62no—
293C3×C4—92no—
294C3×C4—122no—
295C3×C4—184no—
296C3×C4—182no—
297C3×C4—182no—
298C3×C4—181no—
299C3×C4#379812yes—
300Dic3—34no—
301Dic3—62no—
302Dic3—64no—
303Dic3—62no—
304Dic3—64no—
305Dic3—62no—
306Dic3—64no—
307Dic3—121no—
308Dic3—184no—
309Dic3—182no—
310C32—21no—
311C32#551112yescomplemented
312C23—92no—
313Q8—182no—
314Q8—1082no—
315D4—274no—
316C2×C4—91no—
317C2×C4—91no—
318C2×C4—92no—
319C2×C4—92no—
320C2×C4—94no—
321C2×C4—184no—
322C2×C4—364no—
323C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
324C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
325C2×C3—32no—
326C2×C3—32no—
327C2×C3—32no—
328C2×C3—61no—
329C2×C3—61no—
330C2×C3—61no—
331C2×C3—184no—
332C2×C3C2×#379811yescharacteristic
333C2×C3#6431311yescharacteristic
334C2×C3D32×Q811yescharacteristic
335D3—64no—
336D3—184no—
337C22—94no—
338C22—182no—
339C22—182no—
340C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
341C4—22no—
342C4—92no—
343C4—184no—
344C4—181no—
345C4—182no—
346C4—182no—
347C4#8865012yes—
348C3—32no—
349C3—61no—
350C3#1059453111yescharacteristic
351C2—184no—
352C2C2×#8865011yescharacteristic
353C2#1172250611yescharacteristic
354C2Q8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC2×#6198C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#554
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
1. #11721755C2semidirect
2. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
5. #11721117C2semidirect
6. #11721107C2semidirect
7. #11721101C2semidirect
8. #11721098C2semidirect
26. C4⋊C4×C3⋊D3D3semidirect
49. #6198Q8semidirect
215. H(3,𝔽3)#554semidirect
311. C32#5511semidirect

Maximal subgroups

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

subgroupindex
#117217552
C4⋊C4×(C3⋊D3)⋊C32
C4×#61982
#117211172
#117211072
#117211012
#117210982
#105945363