Finite Group Databasebeta

← #13140225

Subgroups — #13140225

Order 1728.

759 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 759 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C22×#6196C211yesmaximal, characteristic
2C2×C4×#1084C211yesmaximal, characteristic
3C2×#88449C211yesmaximal, characteristic
4#63658⋊C3C212yesmaximal
5#11720334C212yesmaximal
6#10591191—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C2×#6196—22no—
13C2×#6196—22no—
14C2×#6196C414yescomplemented
15C4×#1084—22no—
16#88449—22no—
17C22×#627—32no—
18C2×C3×C4×Dic3—32no—
19C2×C3×C4×Dic3—61no—
20C2×Dic3⋊Dic3—32no—
21#63658D312yes—
22C3×C3⋊#95—32no—
23#63579—32no—
24#63579—32no—
25C23×H(3,𝔽3)C2311yescharacteristic
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)—21no—
28C2×C4×H(3,𝔽3)Q812yescomplemented
29C2×#1084—21no—
30C2×#1084—21no—
31C2×#1084C2×C412yes—
32C2×#1084D412yes—
33C2×#1084D412yes—
34C2×#1084Q812yescomplemented
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3C2×C414yes—
41#6196—28no—
42#4929—91no—
43#4646—92no—
44C22×C32×C4—21no—
45C22×C32×C4C2×D312yes—
46C22×C3⋊Dic3C2×D312yes—
47C22×C3⋊Dic3C2×D312yes—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—32no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—61no—
53C22×C3×Dic3—61no—
54C2×#627—38no—
55C2×#627—64no—
56C2×#627—64no—
57C3×C4×Dic3—64no—
58C3×C4×Dic3—64no—
59Dic3⋊Dic3—64no—
60C22×H(3,𝔽3)C22×C411yescharacteristic
61C22×H(3,𝔽3)C2×D411yescharacteristic
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C22×H(3,𝔽3)C2×Q811yescharacteristic
64C22×H(3,𝔽3)C2×Q811yescharacteristic
65C22×H(3,𝔽3)#4111yescharacteristic
66C22×H(3,𝔽3)#4111yescharacteristic
67C4×H(3,𝔽3)—22no—
68#1084—22no—
69#1084C22⋊C414yes—
70(C3⋊Dic3)⋊C3—24no—
71(C3⋊Dic3)⋊C3—42no—
72C22×Dic6—91no—
73C22×Dic6—92no—
74C2×C3×C42—91no—
75C2×C4⋊Dic3—91no—
76C2×Dic3⋊C4—92no—
77C2×C4×Dic3—32no—
78C2×C4×Dic3—61no—
79C3×#95—92no—
80C3⋊#95—32no—
81C3⋊#95—92no—
82C3⋊#95—92no—
83C3⋊#95—92no—
84C3⋊#95—92no—
85C23×C32—21no—
86C23×C32C22×D312yes—
87C2×C32×C4—22no—
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—22no—
92C2×C32×C4Dic614yescomplemented
93C2×C3⋊Dic3—22no—
94C2×C3⋊Dic3—22no—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—22no—
97C2×C3⋊Dic3—22no—
98C2×C3⋊Dic3C4×D314yes—
99C2×C3×Dic3—34no—
100C2×C3×Dic3—34no—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—34no—
103C2×C3×Dic3—34no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117#627—616no—
118#391—271noSylow
119C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
120C2×H(3,𝔽3)C4×Q811yescharacteristic
121C2×H(3,𝔽3)C4×Q811yescharacteristic
122C2×H(3,𝔽3)C22⋊Q811yescharacteristic
123C2×H(3,𝔽3)C22⋊Q811yescharacteristic
124C2×H(3,𝔽3)#12411yescharacteristic
125C2×H(3,𝔽3)C4⋊Q811yescharacteristic
126C22×C3×C4—32no—
127C22×C3×C4—61no—
128C22×C3×C4—91no—
129C22×C3×C4—92no—
130C22×C3×C4—91no—
131C22×C3×C4—92no—
132C22×C3×C4D3211yescharacteristic
133C22×Dic3—32no—
134C22×Dic3—32no—
135C22×Dic3—32no—
136C22×Dic3—32no—
137C22×Dic3—61no—
138C22×Dic3—61no—
139C22×Dic3—92no—
140C22×Dic3—92no—
141C2×Dic6—94no—
142C2×Dic6—98no—
143C2×Dic6—184no—
144C2×Dic6—184no—
145C2×Dic6—182no—
146C2×Dic6—182no—
147C3×C42—182no—
148C4⋊Dic3—182no—
149Dic3⋊C4—184no—
150C4×Dic3—64no—
151C4×Dic3—64no—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32—21no—
158C22×C32—21no—
159C22×C32C2×Dic612yes—
160C22×C32C2×C4×D312yes—
161C22×C32#23512yes—
162C22×C32D3×D412yes—
163C22×C32D3×D412yes—
164C22×C32#23712yes—
165C22×C32D3×Q812yes—
166C32×C4—24no—
167C32×C4—24no—
168C3⋊Dic3—24no—
169C3⋊Dic3—42no—
170C3×Dic3—38no—
171C3×Dic3—64no—
172C3×Dic3—64no—
173C3×Dic3—64no—
174C3×Dic3—64no—
175C3×Dic3—122no—
176C22×Q8—271no—
177C2×C4⋊C4—271no—
178C2×C42—91no—
179#95—92no—
180#95—272no—
181H(3,𝔽3)#39111yescharacteristic, complemented, Sylow
182C23×C3—32no—
183C23×C3—61no—
184C23×C3C2×D3211yescharacteristic
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—34no—
188C2×C3×C4—62no—
189C2×C3×C4—62no—
190C2×C3×C4—62no—
191C2×C3×C4—62no—
192C2×C3×C4—62no—
193C2×C3×C4—92no—
194C2×C3×C4—92no—
195C2×C3×C4—92no—
196C2×C3×C4—94no—
197C2×C3×C4—92no—
198C2×C3×C4—182no—
199C2×C3×C4—182no—
200C2×C3×C4—181no—
201C2×C3×C4—181no—
202C2×C3×C4—182no—
203C2×C3×C4—182no—
204C2×C3×C4—182no—
205C2×C3×C4#62712yes—
206C2×Dic3—34no—
207C2×Dic3—34no—
208C2×Dic3—34no—
209C2×Dic3—34no—
210C2×Dic3—34no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—62no—
214C2×Dic3—62no—
215C2×Dic3—62no—
216C2×Dic3—62no—
217C2×Dic3—62no—
218C2×Dic3—62no—
219C2×Dic3—62no—
220C2×Dic3—62no—
221C2×Dic3—62no—
222C2×Dic3—62no—
223C2×Dic3—62no—
224C2×Dic3—94no—
225C2×Dic3—182no—
226C2×Dic3—182no—
227C2×Dic3—182no—
228C2×Dic3—182no—
229C2×Dic3—182no—
230Dic6—1816no—
231Dic6—188no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32C4×Dic612yes—
240C2×C32#87812yes—
241C2×C32D3×C22⋊C412yes—
242C2×C32#88512yes—
243C2×C32#88712yes—
244C2×C32#88812yes—
245C2×C32#89612yes—
246C2×Q8—274no—
247C2×Q8—542no—
248C2×Q8—542no—
249C22×C4—91no—
250C22×C4—92no—
251C22×C4—91no—
252C22×C4—92no—
253C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
254C4⋊C4—542no—
255C42—182no—
256C22×C3—32no—
257C22×C3—32no—
258C22×C3—32no—
259C22×C3—32no—
260C22×C3—32no—
261C22×C3—32no—
262C22×C3—32no—
263C22×C3—61no—
264C22×C3—61no—
265C22×C3—61no—
266C22×C3—61no—
267C22×C3—61no—
268C22×C3—61no—
269C22×C3—61no—
270C22×C3#379711yescharacteristic
271C22×C3#379811yescharacteristic
272C22×C3C4×D3211yescharacteristic
273C22×C3#380211yescharacteristic
274C22×C3#380511yescharacteristic
275C22×C3C2×#62711yescharacteristic
276C22×C3#381111yescharacteristic
277C3×C4—22no—
278C3×C4—64no—
279C3×C4—64no—
280C3×C4—94no—
281C3×C4—184no—
282C3×C4—182no—
283C3×C4—362no—
284Dic3—38no—
285Dic3—64no—
286Dic3—64no—
287Dic3—64no—
288Dic3—64no—
289Dic3—122no—
290Dic3—184no—
291Dic3—362no—
292C32—21no—
293C32#464612yescomplemented
294C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
295Q8—548no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—92no—
299C2×C4—92no—
300C2×C4—92no—
301C2×C4—92no—
302C2×C4—94no—
303C2×C4—181no—
304C2×C4—182no—
305C2×C4—181no—
306C2×C4—182no—
307C2×C4—182no—
308C2×C4—182no—
309C2×C4—182no—
310C2×C4#619612yescomplemented
311C2×C3—32no—
312C2×C3—32no—
313C2×C3—32no—
314C2×C3—32no—
315C2×C3—32no—
316C2×C3—32no—
317C2×C3—32no—
318C2×C3—61no—
319C2×C3—61no—
320C2×C3—61no—
321C2×C3—61no—
322C2×C3—61no—
323C2×C3—61no—
324C2×C3—61no—
325C2×C3#6383811yescharacteristic
326C2×C3#6386511yescharacteristic
327C2×C3#6391511yescharacteristic
328C2×C3C4×#62711yescharacteristic
329C2×C3#6391911yescharacteristic
330C2×C3#6392111yescharacteristic
331C2×C3#6398211yescharacteristic
332C22#8865011yescharacteristic
333C22#8865111yescharacteristic
334C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
335C22#8865511yescharacteristic
336C22C2×#619611yescharacteristic
337C22#8867211yescharacteristic
338C22#8867711yescharacteristic
339C4—22no—
340C4—94no—
341C4—182no—
342C4—184no—
343C4—362no—
344C3—32no—
345C3—61no—
346C3#1059139611yescharacteristic
347C2#1172107811yescharacteristic
348C2#1172109611yescharacteristic
349C2C4×#619611yescharacteristic
350C2#1172111711yescharacteristic
351C2#1172114511yescharacteristic
352C2#1172116111yescharacteristic
353C2#1172130911yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical, cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#391
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
14. C2×#6196C4semidirect
28. C2×C4×H(3,𝔽3)Q8semidirect
34. C2×#1084Q8semidirect
92. C2×C32×C4Dic6semidirect
181. H(3,𝔽3)#391semidirect
293. C32#4646semidirect
310. C2×C4#6196semidirect

Maximal subgroups

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

subgroupindex
C22×#61962
C2×C4×#10842
C2×#884492
#63658⋊C32
#117203342
#105911913