Finite Group Databasebeta

← #13151319

Subgroups — #13151319

Order 1728.

836 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 141 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
1D4×#1084C211yesmaximal, characteristic
2#64145⋊C3C212yesmaximal
3#11721309C212yesmaximal, complemented
4#11721298C212yesmaximal, complemented
5#11721297C214yesmaximal, complemented
6C4×#6196C211yesmaximal, characteristic, complemented
7#11721099C211yesmaximal, characteristic, complemented
8#11721082C212yesmaximal, complemented
9#10595273—32nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2212yes—
12C2×#634⋊C3C2212yes—
13C2×#6196C2211yescharacteristic
14#88552C2212yes—
15C4⋊#1084C2211yescharacteristic
16C4×#1084C2211yescharacteristic
17#3757⋊C3C2214yes—
18#3757⋊C3C2214yes—
19C4×(C3⋊Dic3)⋊C3C2212yes—
20#88449C2212yes—
21#88449C2211yescharacteristic
22#88449C2212yes—
23#88448C2212yes—
24#88448C2214yes—
25#88447C2214yes—
26#64145D312yes—
27C3×#936—32no—
28C3×D4×Dic3—32no—
29C3×D4×Dic3—61no—
30#63964—34no—
31#63956—34no—
32#63956—34no—
33#63955—34no—
34C4×#627—32no—
35#63866—32no—
36#63844—32no—
37#63843—32no—
38C23×H(3,𝔽3)C2312yes—
39D4×H(3,𝔽3)—22no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—24no—
42C2×#1084C2312yes—
43C2×#1084C2311yescharacteristic
44C2×#1084C2311yescharacteristic
45C2×(C3⋊Dic3)⋊C3C2312yes—
46C2×(C3⋊Dic3)⋊C3C2314yes—
47C2×(C3⋊Dic3)⋊C3C2312yes—
48#634⋊C3—24no—
49#6196—22no—
50#5671—92no—
51#5542—91no—
52C2×C32×D4—21no—
53C2×C32×D4C2×D312yes—
54C2×#634C2×D312yes—
55C22×C3×Dic3—34no—
56C22×C3×Dic3—62no—
57C2×C3×Dic6—32no—
58C2×#627—32no—
59#3757C2×D314yes—
60#3757C2×D314yes—
61C4×C3⋊Dic3C2×D312yes—
62C3×#217—34no—
63C3×#217—34no—
64C3×#217—34no—
65C3×#217—62no—
66C3×C4⋊Dic3—32no—
67C3×C4⋊Dic3—61no—
68C3×C4×Dic3—32no—
69C3×C4×Dic3—32no—
70C3×C4×Dic3—61no—
71#3724—34no—
72#3724—32no—
73Dic3⋊Dic3—34no—
74Dic3⋊Dic3—34no—
75Dic3⋊Dic3—34no—
76Dic3⋊Dic3—32no—
77Dic3⋊Dic3—32no—
78Dic32—34no—
79Dic32—34no—
80C22×H(3,𝔽3)—22no—
81C22×H(3,𝔽3)—22no—
82C22×H(3,𝔽3)—22no—
83C22×H(3,𝔽3)C2411yescharacteristic
84C4×H(3,𝔽3)#4112yes—
85#1084—22no—
86#1084—21no—
87#1084#4112yes—
88(C3⋊Dic3)⋊C3—22no—
89(C3⋊Dic3)⋊C3—24no—
90(C3⋊Dic3)⋊C3—22no—
91C3×#124—92no—
92C3×C4×D4—91no—
93Q8×Dic3—92no—
94#944—92no—
95#936—32no—
96#936—92no—
97D4×Dic3—32no—
98D4×Dic3—61no—
99#926—92no—
100#924—92no—
101#887—92no—
102#879—94no—
103#879—94no—
104#879—94no—
105#878—94no—
106#877—94no—
107#869—91no—
108C4×Dic6—91no—
109C4×Dic6—92no—
110C4×Dic6—92no—
111C23×C32—22no—
112C23×C32C22×D314yes—
113C32×D4—24no—
114C32×D4—24no—
115C2×C32×C4—21no—
116C2×C32×C4C22×D312yes—
117C2×C3⋊Dic3C22×D312yes—
118C2×C3⋊Dic3C22×D314yes—
119C2×C3⋊Dic3C22×D312yes—
120#634—24no—
121C2×C3×Dic3—32no—
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—61no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—68no—
131C2×C3×Dic3—124no—
132C3×Dic6—64no—
133#627—64no—
134#556—271noSylow
135C2×H(3,𝔽3)—42no—
136C2×H(3,𝔽3)C2×#4111yescharacteristic
137C2×H(3,𝔽3)C2×#4111yescharacteristic
138C2×H(3,𝔽3)2−1+411yescharacteristic
139C2×C3×Q8—92no—
140C2×C3×D4—32no—
141C2×C3×D4—61no—
142C2×C3×D4D3211yescharacteristic
143C22×C3×C4—92no—
144C22×Dic3—34no—
145C22×Dic3—62no—
146C2×Dic6—32no—
147C2×Dic6—92no—
148C2×Dic6—91no—
149C2×Dic6—92no—
150C3×C4⋊C4—91no—
151C3×C22⋊C4—94no—
152C3×C22⋊C4—94no—
153C3×C22⋊C4—92no—
154C3×C42—92no—
155C3×C42—91no—
156#217—34no—
157#217—34no—
158#217—34no—
159#217—62no—
160#217—94no—
161#217—94no—
162C4⋊Dic3—32no—
163C4⋊Dic3—61no—
164C4⋊Dic3—91no—
165C4⋊Dic3—92no—
166C4⋊Dic3—92no—
167C4⋊Dic3—92no—
168C4⋊Dic3—94no—
169Dic3⋊C4—92no—
170Dic3⋊C4—94no—
171Dic3⋊C4—92no—
172Dic3⋊C4—94no—
173Dic3⋊C4—92no—
174Dic3⋊C4—94no—
175Dic3⋊C4—94no—
176C4×Dic3—32no—
177C4×Dic3—32no—
178C4×Dic3—61no—
179C4×Dic3—92no—
180C4×Dic3—94no—
181C4×Dic3—94no—
182C4×Dic3—94no—
183C22×C32—24no—
184C22×C32—24no—
185C22×C32—24no—
186C22×C32—24no—
187C22×C32—21no—
188C22×C32—42no—
189C22×C32—42no—
190C22×C32C23×D312yes—
191C32×C4—22no—
192C32×C4#23714yes—
193C3⋊Dic3—24no—
194C3⋊Dic3—22no—
195C3⋊Dic3—22no—
196C3×Dic3—34no—
197C3×Dic3—62no—
198C3×Dic3—64no—
199C3×Dic3—62no—
200C3×Dic3—64no—
201C3×Dic3—62no—
202C3×Dic3—62no—
203C3×Dic3—121no—
204C3×Dic3—122no—
205C4⋊Q8—271no—
206#126—274no—
207#124—92no—
208C22⋊Q8—272no—
209C4×Q8—271no—
210C4×Q8—272no—
211C4×D4—91no—
212#117—272no—
213H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
214C23×C3—34no—
215C23×C3—62no—
216C23×C3C2×D3212yes—
217C3×Q8—184no—
218C3×D4—22no—
219C3×D4—64no—
220C3×D4—64no—
221C2×C3×C4—32no—
222C2×C3×C4—61no—
223C2×C3×C4—91no—
224C2×C3×C4—92no—
225C2×C3×C4—94no—
226C2×C3×C4—92no—
227C2×C3×C4—92no—
228C2×C3×C4—91no—
229C2×C3×C4—184no—
230C2×C3×C4C2×D3211yescharacteristic
231C2×Dic3—32no—
232C2×Dic3—32no—
233C2×Dic3—34no—
234C2×Dic3—32no—
235C2×Dic3—32no—
236C2×Dic3—34no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—62no—
240C2×Dic3—68no—
241C2×Dic3—92no—
242C2×Dic3—94no—
243C2×Dic3—92no—
244C2×Dic3—124no—
245Dic6—64no—
246Dic6—184no—
247Dic6—184no—
248Dic6—182no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—44no—
253C2×C32—44no—
254C2×C32C2×#23712yes—
255C2×C32D3×#4112yes—
256C2×C32#101012yes—
257C2×Q8—92no—
258C2×Q8—271no—
259C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
260C22×C4—92no—
261C4⋊C4—91no—
262C4⋊C4—271no—
263C4⋊C4—272no—
264C4⋊C4—274no—
265C4⋊C4—272no—
266C4⋊C4—272no—
267C22⋊C4—94no—
268C22⋊C4—92no—
269C22⋊C4—94no—
270C42—91no—
271C42—92no—
272C42—274no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—32no—
277C22×C3—64no—
278C22×C3—64no—
279C22×C3—64no—
280C22×C3—64no—
281C22×C3—61no—
282C22×C3—122no—
283C22×C3—122no—
284C22×C3C22×D3211yescharacteristic
285C3×C4—34no—
286C3×C4—62no—
287C3×C4—92no—
288C3×C4—182no—
289C3×C4—184no—
290C3×C4—182no—
291C3×C4—181no—
292C3×C4—182no—
293C3×C4#380512yes—
294Dic3—34no—
295Dic3—62no—
296Dic3—64no—
297Dic3—62no—
298Dic3—62no—
299Dic3—62no—
300Dic3—64no—
301Dic3—122no—
302Dic3—121no—
303Dic3—182no—
304Dic3—184no—
305Dic3—182no—
306C32—21no—
307C32#567112yescomplemented
308C23C2×((C3⋊D3)⋊C3)⋊C212yes—
309Q8—184no—
310Q8—542no—
311D4—22no—
312C2×C4—91no—
313C2×C4—92no—
314C2×C4—91no—
315C2×C4—92no—
316C2×C4—92no—
317C2×C4—94no—
318C2×C4—184no—
319C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
320C2×C3—32no—
321C2×C3—32no—
322C2×C3—32no—
323C2×C3—42no—
324C2×C3—61no—
325C2×C3—61no—
326C2×C3—61no—
327C2×C3—124no—
328C2×C3—124no—
329C2×C3#6430911yescharacteristic
330C2×C3#6431211yescharacteristic
331C2×C3C2×#380511yescharacteristic
332C22—22no—
333C22—22no—
334C22—22no—
335C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
336C4—92no—
337C4—181no—
338C4—182no—
339C4—182no—
340C4—182no—
341C4—184no—
342C4#8867212yes—
343C3—32no—
344C3—61no—
345C3#1059511911yescharacteristic
346C2—42no—
347C2#1172250211yescharacteristic
348C2#1172250511yescharacteristic
349C2C2×#8867211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#556
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. #11721309C2semidirect
4. #11721298C2semidirect
5. #11721297C2semidirect
6. C4×#6196C2semidirect
7. #11721099C2semidirect
8. #11721082C2semidirect
213. H(3,𝔽3)#556semidirect
307. C32#5671semidirect

Maximal subgroups

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

subgroupindex
D4×#10842
#64145⋊C32
#117213092
#117212982
#117212972
C4×#61962
#117210992
#117210822
#105952733