Finite Group Databasebeta

← #13140203

Subgroups — #13140203

Order 1728.

951 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 951 conjugacy classes of subgroups — too many to draw. Only its 135 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×#6199C211yesmaximal, characteristic
2C2×#88542C211yesmaximal, characteristic
3C2×#88451C211yesmaximal, characteristic
4C2×#88448C212yesmaximal, complemented
5#63658⋊C3C212yesmaximal, complemented
6#10591373—32nomaximal
7C23×#272C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C2×#6199—22no—
13C2×#6199—24no—
14#88542—22no—
15#88451C414yescomplemented
16#88448—24no—
17C22×D3×Dic3—32no—
18C2×C3×(C2×D3)⋊C4—32no—
19C2×C3×(C2×D3)⋊C4—61no—
20C2×#3724—32no—
21C2×Dic3⋊Dic3—32no—
22C2×(C2×D3)⋊Dic3—32no—
23#63658D312yescomplemented
24C3×C3⋊#95—32no—
25C23×H(3,𝔽3)C2311yescharacteristic
26C22×#272—22no—
27C22×#272—22no—
28C22×#272C2×C412yes—
29C22×#272D412yes—
30C22×#272Q812yes—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—22no—
33C2×#1084—21no—
34C2×#1084—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3C2×C414yescomplemented
40C2×(C3⋊Dic3)⋊C3D414yes—
41#6199—28no—
42#4946—91no—
43#4668—92no—
44C23×C3×D3—32no—
45C23×C3×D3—61no—
46C22×C32×C4—21no—
47C22×C32×C4C2×D312yes—
48C22×C3⋊Dic3C2×D312yes—
49C22×C3⋊Dic3C2×D312yes—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—61no—
54C2×D3×Dic3—64no—
55C2×D3×Dic3—64no—
56C2×D3×Dic3—64no—
57C3×(C2×D3)⋊C4—64no—
58C3×(C2×D3)⋊C4—64no—
59#3724—64no—
60Dic3⋊Dic3—64no—
61(C2×D3)⋊Dic3—38no—
62C22×H(3,𝔽3)C22×C411yescharacteristic
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C22×H(3,𝔽3)C2×D412yes—
65C22×H(3,𝔽3)C2×Q811yescharacteristic
66C22×H(3,𝔽3)#4112yes—
67C2×#272—24no—
68C2×#272—22no—
69C2×#272—24no—
70C2×#272—22no—
71C2×#272C4⋊C414yescomplemented
72C4×H(3,𝔽3)—41no—
73#1084—41no—
74(C3⋊Dic3)⋊C3—24no—
75(C3⋊Dic3)⋊C3—24no—
76C23×Dic3—91no—
77C22×C4×D3—92no—
78C2×C3×C22⋊C4—91no—
79C2×#217—91no—
80C2×(C2×D3)⋊C4—32no—
81C2×(C2×D3)⋊C4—61no—
82C2×(C2×D3)⋊C4—92no—
83C2×C4⋊Dic3—92no—
84C2×Dic3⋊C4—92no—
85C2×Dic3⋊C4—92no—
86C3×#95—92no—
87C3⋊#95—32no—
88C3⋊#95—92no—
89C23×C32—21no—
90C23×C32C22×D312yes—
91C22×C3×D3—34no—
92C22×C3×D3—34no—
93C22×C3×D3—34no—
94C22×C3×D3—62no—
95C22×C3×D3—62no—
96C22×C3×D3—62no—
97C22×C3×D3—62no—
98C22×C3×D3—62no—
99C22×C3×D3—62no—
100C22×C3×D3—62no—
101C22×C3×D3—122no—
102C22×C3×D3—122no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—42no—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3—22no—
111C2×C3⋊Dic3—22no—
112C2×C3⋊Dic3C4×D314yescomplemented
113C2×C3⋊Dic3D1214yes—
114C2×C3×Dic3—34no—
115C2×C3×Dic3—34no—
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—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—122no—
125D3×Dic3—616no—
126#385—271noSylow
127C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
128C2×H(3,𝔽3)C4×D412yes—
129C2×H(3,𝔽3)#12011yescharacteristic
130C2×H(3,𝔽3)C22⋊Q812yes—
131C2×H(3,𝔽3)#12311yescharacteristic
132#272—24no—
133C24×C3—91no—
134C23×D3—32no—
135C23×D3—61no—
136C22×C3×C4—32no—
137C22×C3×C4—61no—
138C22×C3×C4—92no—
139C22×C3×C4—92no—
140C22×C3×C4—91no—
141C22×C3×C4D3211yescharacteristic
142C22×Dic3—32no—
143C22×Dic3—32no—
144C22×Dic3—32no—
145C22×Dic3—61no—
146C22×Dic3—92no—
147C22×Dic3—92no—
148C22×Dic3—184no—
149C22×Dic3—182no—
150C2×C4×D3—184no—
151C2×C4×D3—184no—
152C2×C4×D3—184no—
153C3×C22⋊C4—182no—
154#217—94no—
155(C2×D3)⋊C4—64no—
156(C2×D3)⋊C4—64no—
157(C2×D3)⋊C4—98no—
158C4⋊Dic3—184no—
159Dic3⋊C4—184no—
160Dic3⋊C4—184no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—22no—
164C22×C32—22no—
165C22×C32—21no—
166C22×C32C2×C4×D312yes—
167C22×C32C2×D1212yes—
168C22×C32#23512yes—
169C22×C32D3×D412yes—
170C22×C32D3×D412yes—
171C22×C32#23712yes—
172C22×C32D3×Q812yes—
173C2×C3×D3—38no—
174C2×C3×D3—64no—
175C2×C3×D3—64no—
176C2×C3×D3—64no—
177C2×C3×D3—64no—
178C2×C3×D3—64no—
179C2×C3×D3—64no—
180C2×C3×D3—64no—
181C2×C3×D3—124no—
182C2×C3×D3—124no—
183C2×C3×D3—122no—
184C2×C3×D3—122no—
185C32×C4—42no—
186C32×C4—42no—
187C3⋊Dic3—24no—
188C3⋊Dic3—24no—
189C3×Dic3—64no—
190C3×Dic3—64no—
191C3×Dic3—122no—
192C3×Dic3—122no—
193C23×C4—271no—
194C2×C4⋊C4—272no—
195C2×C22⋊C4—91no—
196C2×C22⋊C4—271no—
197#95—92no—
198H(3,𝔽3)#38511yescharacteristic, complemented, Sylow
199C23×C3—32no—
200C23×C3—61no—
201C23×C3—92no—
202C23×C3—92no—
203C23×C3—92no—
204C23×C3—182no—
205C23×C3—182no—
206C23×C3C2×D3211yescharacteristic
207C22×D3—34no—
208C22×D3—34no—
209C22×D3—34no—
210C22×D3—62no—
211C22×D3—62no—
212C22×D3—62no—
213C22×D3—62no—
214C22×D3—62no—
215C22×D3—62no—
216C22×D3—62no—
217C22×D3—122no—
218C22×D3—122no—
219C2×C3×C4—22no—
220C2×C3×C4—21no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—62no—
225C2×C3×C4—94no—
226C2×C3×C4—94no—
227C2×C3×C4—122no—
228C2×C3×C4—182no—
229C2×C3×C4—182no—
230C2×C3×C4—182no—
231C2×C3×C4—182no—
232C2×C3×C4—182no—
233C2×C3×C4—181no—
234C2×Dic3—34no—
235C2×Dic3—34no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—62no—
241C2×Dic3—62no—
242C2×Dic3—62no—
243C2×Dic3—62no—
244C2×Dic3—94no—
245C2×Dic3—94no—
246C2×Dic3—122no—
247C2×Dic3—182no—
248C2×Dic3—182no—
249C2×Dic3—182no—
250C2×Dic3—182no—
251C2×Dic3—188no—
252C4×D3—1816no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—22no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32C4×D1212yes—
259C2×C32#67⋊C412yes—
260C2×C32#88212yes—
261C2×C32#88312yes—
262C2×C32D3×C4⋊C412yes—
263C2×C32#89612yes—
264C2×C32#89712yes—
265C3×D3—68no—
266C3×D3—124no—
267C24—91no—
268C22×C4—92no—
269C22×C4—92no—
270C22×C4—91no—
271C22×C4—544no—
272C22×C4—542no—
273C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
274C4⋊C4—544no—
275C22⋊C4—182no—
276C22⋊C4—274no—
277C22×C3—32no—
278C22×C3—32no—
279C22×C3—32no—
280C22×C3—32no—
281C22×C3—32no—
282C22×C3—32no—
283C22×C3—32no—
284C22×C3—61no—
285C22×C3—61no—
286C22×C3—62no—
287C22×C3—62no—
288C22×C3—61no—
289C22×C3—94no—
290C22×C3—184no—
291C22×C3—184no—
292C22×C3—182no—
293C22×C3—182no—
294C22×C3#379711yescharacteristic
295C22×C3C4×D3211yescharacteristic
296C22×C3D3×D1212yes—
297C22×C3#380412yes—
298C22×C3#381111yescharacteristic
299C2×D3—38no—
300C2×D3—64no—
301C2×D3—64no—
302C2×D3—64no—
303C2×D3—64no—
304C2×D3—64no—
305C2×D3—64no—
306C2×D3—64no—
307C2×D3—122no—
308C2×D3—124no—
309C2×D3—124no—
310C2×D3—122no—
311C3×C4—41no—
312C3×C4—122no—
313C3×C4—122no—
314C3×C4—184no—
315C3×C4—184no—
316C3×C4—361no—
317Dic3—64no—
318Dic3—64no—
319Dic3—122no—
320Dic3—122no—
321Dic3—184no—
322Dic3—184no—
323C32—21no—
324C32#466812yescomplemented
325C23—92no—
326C23—92no—
327C23—92no—
328C23—182no—
329C23—182no—
330C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—94no—
334C2×C4—94no—
335C2×C4—182no—
336C2×C4—182no—
337C2×C4—182no—
338C2×C4—182no—
339C2×C4—182no—
340C2×C4—181no—
341C2×C4—548no—
342C2×C3—32no—
343C2×C3—32no—
344C2×C3—32no—
345C2×C3—32no—
346C2×C3—32no—
347C2×C3—32no—
348C2×C3—32no—
349C2×C3—61no—
350C2×C3—62no—
351C2×C3—61no—
352C2×C3—62no—
353C2×C3—61no—
354C2×C3—184no—
355C2×C3#6388012yes—
356C2×C3#6388311yescharacteristic
357C2×C3#6389512yes—
358C2×C3#6392311yescharacteristic
359C2×C3#6397511yescharacteristic
360D3—68no—
361D3—124no—
362C22—94no—
363C22—182no—
364C22—184no—
365C22—184no—
366C22—182no—
367C22#8865111yescharacteristic
368C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
369C22#8865412yes—
370C22#8867112yes—
371C22#8867711yescharacteristic
372C4—41no—
373C4—184no—
374C4—184no—
375C4—361no—
376C3—32no—
377C3—61no—
378C3#1059137511yescharacteristic
379C2—184no—
380C2#1172112312yes—
381C2#1172114711yescharacteristic
382C2#1172115012yes—
383C2#1172116311yescharacteristic
384C2#1172131911yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×#272C22
cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#385
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
4. C2×#88448C2semidirect
5. #63658⋊C3C2semidirect
15. #88451C4semidirect
23. #63658D3semidirect
39. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
71. C2×#272C4⋊C4semidirect
112. C2×C3⋊Dic3C4×D3semidirect
198. H(3,𝔽3)#385semidirect
324. C32#4668semidirect

Maximal subgroups

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

subgroupindex
C22×#61992
C2×#885422
C2×#884512
C2×#884482
#63658⋊C32
#105913733