Finite Group Databasebeta

← #13149410

Subgroups — #13149410

Order 1728.

892 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 149 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#11721761C211yesmaximal, characteristic
2#64099⋊C3C212yesmaximal, complemented
3#11721152C212yesmaximal, complemented
4#11721150C214yesmaximal, complemented
5#11721149C212yesmaximal
6#11721117C211yesmaximal, characteristic, complemented
7C4×#6196C211yesmaximal, characteristic, complemented
8#11721096C212yesmaximal, complemented
9#10594452—32nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×#272C2212yes—
12C2×#6199C2214yes—
13C2×#6196C2211yescharacteristic
14C2×#6196C2212yes—
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#88542C2212yes—
17C4×#1084C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2212yes—
19C4⋊((C3⋊Dic3)⋊C3)C2214yes—
20#3750⋊C3C2214yes—
21C4×(C3⋊Dic3)⋊C3C2212yes—
22#88451C2214yes—
23#88449C2212yes—
24#88449C2211yescharacteristic
25#88448C2212yes—
26#64099D312yescomplemented
27C3×#893—32no—
28C3×#893—61no—
29C3×#890—32no—
30#63918—32no—
31C4×#627—32no—
32#63899—34no—
33#63898—32no—
34#63893—34no—
35Dic3×D12—32no—
36D3×C4⋊Dic3—34no—
37#63851—34no—
38C22×#272C2312yes—
39C2×C4×H(3,𝔽3)C2312yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084C2311yescharacteristic
42C2×#1084C2311yescharacteristic
43C4⋊#272Q814yescomplemented
44C4×#272—24no—
45C2×(C3⋊Dic3)⋊C3C2314yes—
46C2×(C3⋊Dic3)⋊C3C2312yes—
47C2×(C3⋊Dic3)⋊C3C2312yes—
48#6199—44no—
49#6196—22no—
50#6196—42no—
51#5690—92no—
52#5543—91no—
53C2×C3×D12—32no—
54C2×C3×D12—61no—
55C2×C3×C4×D3—34no—
56C2×C3×C4×D3—62no—
57C2×#627—32no—
58C2×#627—34no—
59C2×D3×Dic3—34no—
60C2×D3×Dic3—34no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4C2×D312yes—
63C4⋊(C3⋊Dic3)C2×D314yes—
64C4⋊(C3⋊Dic3)C2×D312yes—
65#3750C2×D314yes—
66C4×C3⋊Dic3C2×D312yes—
67C3×(C2×D3)⋊C4—34no—
68C3×(C2×D3)⋊C4—62no—
69C3×C4⋊Dic3—34no—
70C3×C4⋊Dic3—32no—
71C3×Dic3⋊C4—34no—
72C3×C4×Dic3—32no—
73C3×C4×Dic3—32no—
74C3×C4×Dic3—61no—
75#3724—32no—
76Dic3⋊Dic3—34no—
77Dic3⋊Dic3—32no—
78Dic3⋊Dic3—32no—
79(C2×D3)⋊Dic3—34no—
80(C2×D3)⋊Dic3—34no—
81C22×H(3,𝔽3)C2411yescharacteristic
82C2×#272—22no—
83C2×#272—22no—
84C2×#272C2×Q814yes—
85C4×H(3,𝔽3)—22no—
86C4×H(3,𝔽3)C2×Q812yes—
87#1084—21no—
88#1084#4112yes—
89(C3⋊Dic3)⋊C3—22no—
90(C3⋊Dic3)⋊C3—24no—
91(C3⋊Dic3)⋊C3—22no—
92C3×#125—92no—
93C3×C4×D4—91no—
94D4×Dic3—92no—
95C2×C4⋊Dic3—92no—
96#924—92no—
97#897—94no—
98#896—94no—
99#896—94no—
100#893—32no—
101#893—61no—
102#893—92no—
103D3×C4⋊C4—94no—
104#890—32no—
105#890—92no—
106#888—92no—
107#878—94no—
108C4×D12—92no—
109#869—91no—
110C4×Dic6—91no—
111C4×Dic6—92no—
112C22×C3×D3—34no—
113C22×C3×D3—62no—
114C2×C32×C4—21no—
115C2×C32×C4—22no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D312yes—
118C2×C3⋊Dic3C22×D312yes—
119C2×C3⋊Dic3C22×D314yes—
120C2×C3⋊Dic3C22×D312yes—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—32no—
125C2×C3×Dic3—32no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C3×D12—38no—
129C3×D12—64no—
130C3×C4×D3—68no—
131C3×C4×D3—124no—
132#627—64no—
133#627—124no—
134D3×Dic3—124no—
135D3×Dic3—124no—
136#554—271noSylow
137C2×H(3,𝔽3)C22×Q811yescharacteristic
138C2×H(3,𝔽3)C2×#4111yescharacteristic
139C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
140#272—24no—
141C2×C3×D4—91no—
142C22×C3×C4—92no—
143C22×Dic3—94no—
144C2×D12—32no—
145C2×D12—61no—
146C2×C4×D3—34no—
147C2×C4×D3—62no—
148C2×C4×D3—94no—
149C2×C4×D3—94no—
150C2×Dic6—91no—
151C2×Dic6—92no—
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—94no—
159C3×C4⋊C4D3211yescharacteristic
160C3×C22⋊C4—92no—
161C3×C42—92no—
162C3×C42—91no—
163#217—94no—
164(C2×D3)⋊C4—34no—
165(C2×D3)⋊C4—62no—
166(C2×D3)⋊C4—94no—
167(C2×D3)⋊C4—94no—
168C4⋊Dic3—32no—
169C4⋊Dic3—34no—
170C4⋊Dic3—92no—
171C4⋊Dic3—94no—
172C4⋊Dic3—91no—
173C4⋊Dic3—92no—
174C4⋊Dic3—92no—
175Dic3⋊C4—34no—
176Dic3⋊C4—94no—
177Dic3⋊C4—92no—
178Dic3⋊C4—92no—
179Dic3⋊C4—92no—
180Dic3⋊C4—94no—
181C4×Dic3—32no—
182C4×Dic3—32no—
183C4×Dic3—61no—
184C4×Dic3—92no—
185C22×C32—21no—
186C22×C32C23×D312yes—
187C2×C3×D3—38no—
188C2×C3×D3—64no—
189C2×C3×D3—64no—
190C2×C3×D3—64no—
191C2×C3×D3—122no—
192C2×C3×D3—122no—
193C32×C4—24no—
194C32×C4—22no—
195C32×C4—42no—
196C32×C4D3×Q814yes—
197C3⋊Dic3—22no—
198C3⋊Dic3—22no—
199C3⋊Dic3—24no—
200C3×Dic3—34no—
201C3×Dic3—62no—
202C3×Dic3—62no—
203C3×Dic3—62no—
204C3×Dic3—64no—
205C3×Dic3—62no—
206C3×Dic3—121no—
207C4⋊Q8—271no—
208#125—92no—
209C22⋊Q8—274no—
210C22⋊Q8—272no—
211C4×Q8—271no—
212C4×D4—91no—
213C4×D4—272no—
214C2×C4⋊C4—272no—
215H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
216C23×C3—92no—
217C22×D3—34no—
218C22×D3—62no—
219C3×D4—94no—
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×C4C2×D3212yes—
231C2×C3×C4C2×D3211yescharacteristic
232C2×Dic3—32no—
233C2×Dic3—32no—
234C2×Dic3—32no—
235C2×Dic3—32no—
236C2×Dic3—34no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—92no—
240C2×Dic3—92no—
241C2×Dic3—94no—
242C2×Dic3—364no—
243D12—38no—
244D12—64no—
245C4×D3—68no—
246C4×D3—124no—
247C4×D3—364no—
248C4×D3—364no—
249Dic6—184no—
250Dic6—182no—
251Dic6—362no—
252Dic6—364no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32C2×D3×Q812yes—
257C2×C32D3×#4112yes—
258C2×C32#100912yes—
259C3×D3—68no—
260C3×D3—124no—
261C2×Q8—271no—
262C2×Q8—272no—
263C2×D4—91no—
264C22×C4—92no—
265C22×C4—274no—
266C4⋊C4—94no—
267C4⋊C4—92no—
268C4⋊C4—94no—
269C4⋊C4—271no—
270C4⋊C4—272no—
271C4⋊C4—272no—
272C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
273C22⋊C4—92no—
274C22⋊C4—274no—
275C42—91no—
276C42—92no—
277C22×C3—32no—
278C22×C3—61no—
279C22×C3—94no—
280C22×C3—182no—
281C22×C3—182no—
282C22×C3C22×D3211yescharacteristic
283C2×D3—38no—
284C2×D3—64no—
285C2×D3—64no—
286C2×D3—64no—
287C2×D3—122no—
288C2×D3—122no—
289C3×C4—22no—
290C3×C4—34no—
291C3×C4—64no—
292C3×C4—62no—
293C3×C4—92no—
294C3×C4—122no—
295C3×C4—182no—
296C3×C4—184no—
297C3×C4—182no—
298C3×C4—181no—
299C3×C4#379712yes—
300Dic3—34no—
301Dic3—62no—
302Dic3—62no—
303Dic3—62no—
304Dic3—64no—
305Dic3—62no—
306Dic3—121no—
307Dic3—182no—
308Dic3—184no—
309Dic3—182no—
310C32—21no—
311C32#569012yescomplemented
312C23—92no—
313Q8—542no—
314Q8—1082no—
315D4—94no—
316C2×C4—91no—
317C2×C4—91no—
318C2×C4—92no—
319C2×C4—92no—
320C2×C4—94no—
321C2×C4—184no—
322C2×C4—1084no—
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×#379711yescharacteristic
333C2×C3#6431211yescharacteristic
334C2×C3#6432011yescharacteristic
335D3—68no—
336D3—124no—
337C22—94no—
338C22—182no—
339C22—182no—
340C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
341C4—22no—
342C4—92no—
343C4—182no—
344C4—182no—
345C4—184no—
346C4—181no—
347C4#8865112yes—
348C3—32no—
349C3—61no—
350C3#1059448411yescharacteristic
351C2—184no—
352C2C2×#8865111yescharacteristic
353C2#1172250511yescharacteristic
354C2#1172252111yescharacteristic
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×C4⋊#272C22
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
2. #64099⋊C3C2semidirect
3. #11721152C2semidirect
4. #11721150C2semidirect
6. #11721117C2semidirect
7. C4×#6196C2semidirect
8. #11721096C2semidirect
26. #64099D3semidirect
43. C4⋊#272Q8semidirect
215. H(3,𝔽3)#554semidirect
311. C32#5690semidirect

Maximal subgroups

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

subgroupindex
#117217612
#64099⋊C32
#117211522
#117211502
#117211492
#117211172
C4×#61962
#117210962
#105944523