Finite Group Databasebeta

← #13140228

Subgroups — #13140228

Order 1728.

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

normal subgroups only

This group has 839 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
1C22×#6196C211yesmaximal, characteristic, complemented
2C2×#88542C211yesmaximal, characteristic
3C2×#88451C212yesmaximal
4#63658⋊C3C212yesmaximal, complemented
5#11720334C211yesmaximal, characteristic, complemented
6#10591293—32nomaximal
7C23×#272C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C2×#6196—22no—
13C2×#6196—24no—
14#88542—22no—
15#88451—24no—
16C22×#627—32no—
17C2×C3×(C2×D3)⋊C4—32no—
18C2×C3×(C2×D3)⋊C4—61no—
19C2×(C2×D3)⋊Dic3—32no—
20C2×(C2×D3)⋊Dic3—32no—
21#63658D312yescomplemented
22C3×C3⋊#95—32no—
23#63579—32no—
24C23×H(3,𝔽3)C2311yescharacteristic
25C22×#272—21no—
26C22×#272—22no—
27C22×#272—21no—
28C22×#272—22no—
29C22×#272Q812yescomplemented
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#6196—28no—
41#4947—91no—
42#4673—92no—
43C23×C3×D3—32no—
44C23×C3×D3—61no—
45C22×C32×C4—21no—
46C22×C32×C4C2×D312yes—
47C22×C3⋊Dic3C2×D312yes—
48C22×C3⋊Dic3C2×D312yes—
49C22×C3×Dic3—32no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—61no—
53C2×#627—64no—
54C2×#627—64no—
55C2×#627—64no—
56C3×(C2×D3)⋊C4—64no—
57C3×(C2×D3)⋊C4—64no—
58(C2×D3)⋊Dic3—64no—
59(C2×D3)⋊Dic3—64no—
60C22×H(3,𝔽3)C2×D412yes—
61C22×H(3,𝔽3)C2×D411yescharacteristic
62C22×H(3,𝔽3)C2×Q811yescharacteristic
63C22×H(3,𝔽3)#4111yescharacteristic
64C22×H(3,𝔽3)#4112yes—
65C2×#272—22no—
66C2×#272—24no—
67C2×#272—41no—
68C2×#272—42no—
69C2×#272—41no—
70C4×H(3,𝔽3)—41no—
71#1084—22no—
72(C3⋊Dic3)⋊C3—24no—
73(C3⋊Dic3)⋊C3—42no—
74C22×Dic6—91no—
75C22×Dic6—92no—
76C2×C3×C22⋊C4—91no—
77C2×#217—92no—
78C2×(C2×D3)⋊C4—32no—
79C2×(C2×D3)⋊C4—61no—
80C2×(C2×D3)⋊C4—92no—
81C2×(C2×D3)⋊C4—92no—
82C3×#95—92no—
83C3⋊#95—32no—
84C3⋊#95—92no—
85C3⋊#95—91no—
86C3⋊#95—92no—
87C23×C32—21no—
88C23×C32C22×D312yes—
89C22×C3×D3—34no—
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—62no—
99C22×C3×D3—122no—
100C22×C3×D3—122no—
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—42no—
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3—22no—
111C2×C3⋊Dic3D1214yes—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—34no—
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—62no—
122C2×C3×Dic3—122no—
123#627—616no—
124#393—271noSylow
125C2×H(3,𝔽3)#12011yescharacteristic
126C2×H(3,𝔽3)C22⋊Q811yescharacteristic
127C2×H(3,𝔽3)C22⋊Q812yes—
128C2×H(3,𝔽3)#12412yes—
129C2×H(3,𝔽3)#12411yescharacteristic
130#272—42no—
131C24×C3—91no—
132C23×D3—32no—
133C23×D3—61no—
134C22×C3×C4—32no—
135C22×C3×C4—61no—
136C22×C3×C4—92no—
137C22×C3×C4—91no—
138C22×C3×C4—92no—
139C22×C3×C4D3211yescharacteristic
140C22×Dic3—32no—
141C22×Dic3—32no—
142C22×Dic3—32no—
143C22×Dic3—61no—
144C22×Dic3—92no—
145C22×Dic3—92no—
146C2×Dic6—184no—
147C2×Dic6—184no—
148C2×Dic6—182no—
149C2×Dic6—184no—
150C2×Dic6—184no—
151C3×C22⋊C4—182no—
152#217—184no—
153(C2×D3)⋊C4—64no—
154(C2×D3)⋊C4—64no—
155(C2×D3)⋊C4—184no—
156(C2×D3)⋊C4—184no—
157C22×C32—21no—
158C22×C32—21no—
159C22×C32—22no—
160C22×C32—22no—
161C22×C32—21no—
162C22×C32C2×D1212yes—
163C22×C32#23512yes—
164C22×C32#23512yes—
165C22×C32D3×D412yes—
166C22×C32D3×D412yes—
167C22×C32#23712yes—
168C22×C32D3×Q812yes—
169C2×C3×D3—64no—
170C2×C3×D3—64no—
171C2×C3×D3—64no—
172C2×C3×D3—64no—
173C2×C3×D3—124no—
174C2×C3×D3—124no—
175C2×C3×D3—122no—
176C2×C3×D3—122no—
177C2×C3×D3—122no—
178C2×C3×D3—122no—
179C2×C3×D3—122no—
180C2×C3×D3—122no—
181C32×C4—42no—
182C32×C4—42no—
183C3⋊Dic3—24no—
184C3⋊Dic3—42no—
185C3×Dic3—64no—
186C3×Dic3—64no—
187C3×Dic3—122no—
188C3×Dic3—122no—
189C22×Q8—271no—
190C2×C22⋊C4—91no—
191C2×C22⋊C4—272no—
192#95—92no—
193#95—271no—
194H(3,𝔽3)#39311yescharacteristic, complemented, Sylow
195C23×C3—32no—
196C23×C3—61no—
197C23×C3—92no—
198C23×C3—181no—
199C23×C3—182no—
200C23×C3—182no—
201C23×C3—181no—
202C23×C3C2×D3211yescharacteristic
203C22×D3—34no—
204C22×D3—62no—
205C22×D3—62no—
206C22×D3—62no—
207C22×D3—62no—
208C22×D3—62no—
209C22×D3—62no—
210C22×D3—62no—
211C22×D3—62no—
212C22×D3—62no—
213C22×D3—122no—
214C22×D3—122no—
215C2×C3×C4—22no—
216C2×C3×C4—21no—
217C2×C3×C4—62no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4—62no—
221C2×C3×C4—94no—
222C2×C3×C4—92no—
223C2×C3×C4—122no—
224C2×C3×C4—182no—
225C2×C3×C4—182no—
226C2×C3×C4—182no—
227C2×C3×C4—182no—
228C2×C3×C4—182no—
229C2×C3×C4—182no—
230C2×Dic3—34no—
231C2×Dic3—34no—
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—94no—
241C2×Dic3—122no—
242C2×Dic3—182no—
243C2×Dic3—182no—
244C2×Dic3—182no—
245C2×Dic3—182no—
246C2×Dic3—182no—
247Dic6—188no—
248Dic6—1816no—
249C2×C32—22no—
250C2×C32—21no—
251C2×C32—22no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32#87512yes—
255C2×C32#88212yes—
256C2×C32#88512yes—
257C2×C32#88512yes—
258C2×C32#89612yes—
259C2×C32#89612yes—
260C2×C32#89712yes—
261C3×D3—124no—
262C3×D3—124no—
263C24—91no—
264C2×Q8—544no—
265C2×Q8—542no—
266C22×C4—92no—
267C22×C4—91no—
268C22×C4—92no—
269C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
270C22⋊C4—182no—
271C22⋊C4—544no—
272C22×C3—32no—
273C22×C3—32no—
274C22×C3—32no—
275C22×C3—32no—
276C22×C3—32no—
277C22×C3—32no—
278C22×C3—32no—
279C22×C3—61no—
280C22×C3—61no—
281C22×C3—62no—
282C22×C3—62no—
283C22×C3—61no—
284C22×C3—182no—
285C22×C3—184no—
286C22×C3—362no—
287C22×C3—361no—
288C22×C3—361no—
289C22×C3#379711yescharacteristic
290C22×C3#379811yescharacteristic
291C22×C3D3×D1212yes—
292C22×C3#380412yes—
293C22×C3#381111yescharacteristic
294C2×D3—64no—
295C2×D3—64no—
296C2×D3—64no—
297C2×D3—64no—
298C2×D3—124no—
299C2×D3—122no—
300C2×D3—122no—
301C2×D3—122no—
302C2×D3—122no—
303C2×D3—122no—
304C2×D3—124no—
305C2×D3—122no—
306C3×C4—41no—
307C3×C4—122no—
308C3×C4—122no—
309C3×C4—184no—
310C3×C4—182no—
311C3×C4—362no—
312Dic3—64no—
313Dic3—64no—
314Dic3—122no—
315Dic3—122no—
316Dic3—184no—
317Dic3—362no—
318C32—21no—
319C32#467312yescomplemented
320C23—92no—
321C23—181no—
322C23—182no—
323C23—182no—
324C23—181no—
325C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
326Q8—548no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—94no—
330C2×C4—92no—
331C2×C4—182no—
332C2×C4—182no—
333C2×C4—182no—
334C2×C4—182no—
335C2×C4—182no—
336C2×C4—182no—
337C2×C3—32no—
338C2×C3—32no—
339C2×C3—32no—
340C2×C3—32no—
341C2×C3—32no—
342C2×C3—32no—
343C2×C3—32no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—62no—
347C2×C3—62no—
348C2×C3—61no—
349C2×C3—362no—
350C2×C3#6385212yes—
351C2×C3#6386511yescharacteristic
352C2×C3#6389512yes—
353C2×C3#6392311yescharacteristic
354C2×C3#6395311yescharacteristic
355D3—124no—
356D3—124no—
357C22—182no—
358C22—184no—
359C22—361no—
360C22—362no—
361C22—361no—
362C22#8865011yescharacteristic
363C22#8865111yescharacteristic
364C22#8865412yes—
365C22#8867112yes—
366C22#8867711yescharacteristic
367C4—41no—
368C4—182no—
369C4—184no—
370C4—362no—
371C3—32no—
372C3—61no—
373C3#1059120511yescharacteristic
374C2—362no—
375C2#1172109611yescharacteristic
376C2#1172110312yes—
377C2#1172115012yes—
378C2#1172116311yescharacteristic
379C2#1172129411yescharacteristic
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)#393
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
1. C22×#6196C2semidirect
4. #63658⋊C3C2semidirect
5. #11720334C2semidirect
21. #63658D3semidirect
29. C22×#272Q8semidirect
194. H(3,𝔽3)#393semidirect
319. C32#4673semidirect

Maximal subgroups

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

subgroupindex
C22×#61962
C2×#885422
C2×#884512
#63658⋊C32
#117203342
#105912933