Finite Group Databasebeta

← #13140211

Subgroups — #13140211

Order 1728.

783 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 117 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×#88542C211yesmaximal, characteristic
2C2×#3750⋊C3C212yesmaximal, complemented
3C2×#88451C212yesmaximal
4C2×#88449C211yesmaximal, characteristic, complemented
5#11720334C211yesmaximal, characteristic, complemented
6#10591303—32nomaximal
7C23×#272C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12#88542—22no—
13#3750⋊C3—24no—
14#88451—24no—
15#88449—22no—
16C2×#3750D312yescomplemented
17C2×C3×(C2×D3)⋊C4—32no—
18C2×C3×(C2×D3)⋊C4—61no—
19C2×C3×Dic3⋊C4—32no—
20C2×Dic3⋊Dic3—32no—
21C2×(C2×D3)⋊Dic3—32no—
22C2×(C2×D3)⋊Dic3—32no—
23#63579—32no—
24C23×H(3,𝔽3)C2311yescharacteristic
25C22×#272—21no—
26C22×#272—22no—
27C22×#272—21no—
28C22×#272—22no—
29C22×#272Q812yes—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—22no—
32C2×#1084—22no—
33C2×#1084D412yes—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3D414yes—
40#4993—92no—
41#4949—91no—
42C23×C3×D3—32no—
43C23×C3×D3—61no—
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—61no—
52#3750—24no—
53C3×(C2×D3)⋊C4—64no—
54C3×(C2×D3)⋊C4—64no—
55C3×Dic3⋊C4—64no—
56Dic3⋊Dic3—64no—
57(C2×D3)⋊Dic3—64no—
58(C2×D3)⋊Dic3—64no—
59C22×H(3,𝔽3)C2×D411yescharacteristic
60C22×H(3,𝔽3)C2×D412yes—
61C22×H(3,𝔽3)C2×Q811yescharacteristic
62C22×H(3,𝔽3)#4111yescharacteristic
63C22×H(3,𝔽3)#4112yes—
64C2×#272—22no—
65C2×#272—24no—
66C2×#272—41no—
67C2×#272—42no—
68C2×#272—41no—
69C4×H(3,𝔽3)—41no—
70#1084—22no—
71(C3⋊Dic3)⋊C3—24no—
72(C3⋊Dic3)⋊C3—42no—
73C2×C3×C4⋊C4—92no—
74C2×C3×C22⋊C4—91no—
75C2×#217—92no—
76C2×(C2×D3)⋊C4—32no—
77C2×(C2×D3)⋊C4—61no—
78C2×(C2×D3)⋊C4—92no—
79C2×(C2×D3)⋊C4—92no—
80C2×C4⋊Dic3—91no—
81C2×Dic3⋊C4—32no—
82C2×Dic3⋊C4—92no—
83C2×Dic3⋊C4—92no—
84C3⋊#95—91no—
85C3⋊#95—92no—
86C23×C32—21no—
87C23×C32C22×D312yes—
88C22×C3×D3—34no—
89C22×C3×D3—62no—
90C22×C3×D3—62no—
91C22×C3×D3—62no—
92C22×C3×D3—62no—
93C22×C3×D3—62no—
94C22×C3×D3—62no—
95C22×C3×D3—62no—
96C22×C3×D3—62no—
97C22×C3×D3—62no—
98C22×C3×D3—122no—
99C22×C3×D3—122no—
100C2×C32×C4—22no—
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—42no—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3#6714yes—
111C2×C3×Dic3—34no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—122no—
122#399—271noSylow
123C2×H(3,𝔽3)C22⋊Q811yescharacteristic
124C2×H(3,𝔽3)C22⋊Q812yes—
125C2×H(3,𝔽3)#12311yescharacteristic
126C2×H(3,𝔽3)#12312yes—
127C2×H(3,𝔽3)#12711yescharacteristic
128#272—42no—
129C24×C3—91no—
130C23×D3—32no—
131C23×D3—61no—
132C22×C3×C4—32no—
133C22×C3×C4—61no—
134C22×C3×C4—92no—
135C22×C3×C4—91no—
136C22×C3×C4—92no—
137C22×C3×C4D3211yescharacteristic
138C22×Dic3—32no—
139C22×Dic3—32no—
140C22×Dic3—32no—
141C22×Dic3—61no—
142C22×Dic3—92no—
143C22×Dic3—92no—
144C3×C4⋊C4—184no—
145C3×C22⋊C4—182no—
146#217—184no—
147(C2×D3)⋊C4—64no—
148(C2×D3)⋊C4—64no—
149(C2×D3)⋊C4—184no—
150(C2×D3)⋊C4—184no—
151C4⋊Dic3—182no—
152Dic3⋊C4—64no—
153Dic3⋊C4—184no—
154Dic3⋊C4—184no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32—22no—
158C22×C32—22no—
159C22×C32—21no—
160C22×C32#23512yes—
161C22×C32#23512yes—
162C22×C32D3×D412yes—
163C22×C32D3×D412yes—
164C22×C32D3×Q812yes—
165C22×C32#23912yes—
166C22×C32C2×#6712yes—
167C2×C3×D3—64no—
168C2×C3×D3—64no—
169C2×C3×D3—64no—
170C2×C3×D3—64no—
171C2×C3×D3—124no—
172C2×C3×D3—124no—
173C2×C3×D3—122no—
174C2×C3×D3—122no—
175C2×C3×D3—122no—
176C2×C3×D3—122no—
177C2×C3×D3—122no—
178C2×C3×D3—122no—
179C32×C4—42no—
180C32×C4—42no—
181C3⋊Dic3—24no—
182C3⋊Dic3—42no—
183C3×Dic3—64no—
184C3×Dic3—64no—
185C3×Dic3—122no—
186C3×Dic3—122no—
187C2×C4⋊C4—92no—
188C2×C4⋊C4—271no—
189C2×C22⋊C4—91no—
190C2×C22⋊C4—272no—
191#95—271no—
192H(3,𝔽3)#39911yescharacteristic, complemented, Sylow
193C23×C3—32no—
194C23×C3—61no—
195C23×C3—92no—
196C23×C3—181no—
197C23×C3—182no—
198C23×C3—182no—
199C23×C3—181no—
200C23×C3C2×D3211yescharacteristic
201C22×D3—34no—
202C22×D3—62no—
203C22×D3—62no—
204C22×D3—62no—
205C22×D3—62no—
206C22×D3—62no—
207C22×D3—62no—
208C22×D3—62no—
209C22×D3—62no—
210C22×D3—62no—
211C22×D3—122no—
212C22×D3—122no—
213C2×C3×C4—22no—
214C2×C3×C4—21no—
215C2×C3×C4—62no—
216C2×C3×C4—62no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4—94no—
220C2×C3×C4—92no—
221C2×C3×C4—122no—
222C2×C3×C4—182no—
223C2×C3×C4—182no—
224C2×C3×C4—182no—
225C2×C3×C4—182no—
226C2×C3×C4—182no—
227C2×C3×C4—182no—
228C2×Dic3—34no—
229C2×Dic3—34no—
230C2×Dic3—62no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—94no—
239C2×Dic3—122no—
240C2×Dic3—182no—
241C2×Dic3—182no—
242C2×Dic3—182no—
243C2×Dic3—182no—
244C2×Dic3—182no—
245C2×C32—22no—
246C2×C32—21no—
247C2×C32—22no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32#89412yes—
251C2×C32#89412yes—
252C2×C32#89612yes—
253C2×C32#89612yes—
254C2×C32#92912yes—
255C2×C32#94012yes—
256C2×C32#94612yes—
257C3×D3—124no—
258C3×D3—124no—
259C24—91no—
260C22×C4—91no—
261C22×C4—92no—
262C22×C4—92no—
263C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
264C4⋊C4—184no—
265C4⋊C4—542no—
266C22⋊C4—182no—
267C22⋊C4—544no—
268C22×C3—32no—
269C22×C3—32no—
270C22×C3—32no—
271C22×C3—32no—
272C22×C3—32no—
273C22×C3—32no—
274C22×C3—32no—
275C22×C3—61no—
276C22×C3—61no—
277C22×C3—62no—
278C22×C3—62no—
279C22×C3—61no—
280C22×C3—182no—
281C22×C3—184no—
282C22×C3—362no—
283C22×C3—361no—
284C22×C3—361no—
285C22×C3#379711yescharacteristic
286C22×C3#379811yescharacteristic
287C22×C3#379912yes—
288C22×C3D3×#6712yes—
289C22×C3#381111yescharacteristic
290C2×D3—64no—
291C2×D3—64no—
292C2×D3—64no—
293C2×D3—64no—
294C2×D3—124no—
295C2×D3—122no—
296C2×D3—122no—
297C2×D3—122no—
298C2×D3—122no—
299C2×D3—122no—
300C2×D3—124no—
301C2×D3—122no—
302C3×C4—41no—
303C3×C4—122no—
304C3×C4—122no—
305C3×C4—184no—
306C3×C4—182no—
307C3×C4—362no—
308Dic3—64no—
309Dic3—64no—
310Dic3—122no—
311Dic3—122no—
312Dic3—184no—
313Dic3—362no—
314C32—21no—
315C32#499312yescomplemented
316C23—92no—
317C23—181no—
318C23—182no—
319C23—182no—
320C23—181no—
321C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—94no—
325C2×C4—92no—
326C2×C4—182no—
327C2×C4—182no—
328C2×C4—182no—
329C2×C4—182no—
330C2×C4—182no—
331C2×C4—182no—
332C2×C3—32no—
333C2×C3—32no—
334C2×C3—32no—
335C2×C3—32no—
336C2×C3—32no—
337C2×C3—32no—
338C2×C3—32no—
339C2×C3—61no—
340C2×C3—61no—
341C2×C3—62no—
342C2×C3—62no—
343C2×C3—61no—
344C2×C3—362no—
345C2×C3#6385012yes—
346C2×C3#6386511yescharacteristic
347C2×C3#6388812yes—
348C2×C3#6389711yescharacteristic
349C2×C3#6397911yescharacteristic
350D3—124no—
351D3—124no—
352C22—182no—
353C22—184no—
354C22—361no—
355C22—362no—
356C22—361no—
357C22#8865011yescharacteristic
358C22#8865111yescharacteristic
359C22#8865212yes—
360C22#8867612yes—
361C22#8867711yescharacteristic
362C4—41no—
363C4—184no—
364C4—182no—
365C4—362no—
366C3—32no—
367C3—61no—
368C3#1059120311yescharacteristic
369C2—362no—
370C2#1172109611yescharacteristic
371C2#1172110012yes—
372C2#1172114812yes—
373C2#1172115111yescharacteristic
374C2#1172131711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×#272C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#399
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#3750⋊C3C2semidirect
4. C2×#88449C2semidirect
5. #11720334C2semidirect
16. C2×#3750D3semidirect
192. H(3,𝔽3)#399semidirect
315. C32#4993semidirect

Maximal subgroups

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

subgroupindex
C2×#885422
C2×#3750⋊C32
C2×#884512
C2×#884492
#117203342
#105913033