Finite Group Databasebeta

← #13140124

Subgroups — #13140124

Order 1728.

727 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 727 conjugacy classes of subgroups — too many to draw. Only its 109 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×#88451C212yesmaximal
3C2×#88449C211yesmaximal, characteristic, complemented
4#63658⋊C3C212yesmaximal, complemented
5#11720334C211yesmaximal, characteristic, complemented
6#10591297—32nomaximal
7C23×#272C2211yescharacteristic
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12#88542—22no—
13#88451—24no—
14#88449—22no—
15C2×C3×(C2×D3)⋊C4—32no—
16C2×C3×(C2×D3)⋊C4—61no—
17C2×Dic3⋊Dic3—32no—
18C2×(C2×D3)⋊Dic3—32no—
19C2×(C2×D3)⋊Dic3—32no—
20#63658D312yescomplemented
21C3×C3⋊#95—32no—
22#63579—32no—
23C23×H(3,𝔽3)C2311yescharacteristic
24C22×#272—21no—
25C22×#272—21no—
26C22×#272—22no—
27C22×#272—22no—
28C22×#272D412yes—
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)—22no—
31C2×#1084—22no—
32C2×#1084D412yes—
33C2×(C3⋊Dic3)⋊C3—22no—
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—
39#4953—91no—
40#4675—92no—
41C23×C3×D3—32no—
42C23×C3×D3—61no—
43C22×C32×C4—21no—
44C22×C32×C4C2×D312yes—
45C22×C3⋊Dic3C2×D312yes—
46C22×C3⋊Dic3C2×D312yes—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—32no—
50C22×C3×Dic3—61no—
51C3×(C2×D3)⋊C4—64no—
52C3×(C2×D3)⋊C4—64no—
53Dic3⋊Dic3—64no—
54(C2×D3)⋊Dic3—64no—
55(C2×D3)⋊Dic3—64no—
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C22×H(3,𝔽3)C2×D411yescharacteristic
58C22×H(3,𝔽3)#4111yescharacteristic
59C22×H(3,𝔽3)#4112yes—
60C22×H(3,𝔽3)#4112yes—
61C2×#272—22no—
62C2×#272—24no—
63C2×#272—41no—
64C2×#272—41no—
65C2×#272—42no—
66C4×H(3,𝔽3)—41no—
67#1084—22no—
68(C3⋊Dic3)⋊C3—42no—
69(C3⋊Dic3)⋊C3—42no—
70C2×C3×C22⋊C4—91no—
71C2×#217—92no—
72C2×(C2×D3)⋊C4—32no—
73C2×(C2×D3)⋊C4—61no—
74C2×(C2×D3)⋊C4—92no—
75C2×(C2×D3)⋊C4—92no—
76C2×C4⋊Dic3—91no—
77C2×Dic3⋊C4—92no—
78C3×#95—92no—
79C3⋊#95—32no—
80C3⋊#95—92no—
81C3⋊#95—91no—
82C3⋊#95—92no—
83C23×C32—21no—
84C23×C32C22×D312yes—
85C22×C3×D3—34no—
86C22×C3×D3—62no—
87C22×C3×D3—62no—
88C22×C3×D3—62no—
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—122no—
96C22×C3×D3—122no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—22no—
101C2×C32×C4—42no—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3—22no—
108C2×C3×Dic3—34no—
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—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—122no—
119#397—271noSylow
120C2×H(3,𝔽3)#12111yescharacteristic
121C2×H(3,𝔽3)#12312yes—
122C2×H(3,𝔽3)#12311yescharacteristic
123C2×H(3,𝔽3)#12411yescharacteristic
124C2×H(3,𝔽3)#12612yes—
125#272—42no—
126C24×C3—91no—
127C23×D3—32no—
128C23×D3—61no—
129C22×C3×C4—32no—
130C22×C3×C4—61no—
131C22×C3×C4—91no—
132C22×C3×C4—92no—
133C22×C3×C4—92no—
134C22×C3×C4D3211yescharacteristic
135C22×Dic3—32no—
136C22×Dic3—32no—
137C22×Dic3—32no—
138C22×Dic3—61no—
139C22×Dic3—92no—
140C22×Dic3—92no—
141C3×C22⋊C4—182no—
142#217—184no—
143(C2×D3)⋊C4—64no—
144(C2×D3)⋊C4—64no—
145(C2×D3)⋊C4—184no—
146(C2×D3)⋊C4—184no—
147C4⋊Dic3—182no—
148Dic3⋊C4—184no—
149C22×C32—21no—
150C22×C32—22no—
151C22×C32—21no—
152C22×C32—22no—
153C22×C32—21no—
154C22×C32#23512yes—
155C22×C32#23512yes—
156C22×C32#23512yes—
157C22×C32D3×D412yes—
158C22×C32D3×D412yes—
159C22×C32#23712yes—
160C22×C32#23912yes—
161C2×C3×D3—64no—
162C2×C3×D3—64no—
163C2×C3×D3—64no—
164C2×C3×D3—64no—
165C2×C3×D3—124no—
166C2×C3×D3—124no—
167C2×C3×D3—122no—
168C2×C3×D3—122no—
169C2×C3×D3—122no—
170C2×C3×D3—122no—
171C2×C3×D3—122no—
172C2×C3×D3—122no—
173C32×C4—42no—
174C32×C4—42no—
175C3⋊Dic3—42no—
176C3⋊Dic3—42no—
177C3×Dic3—64no—
178C3×Dic3—122no—
179C3×Dic3—122no—
180C3×Dic3—122no—
181C2×C4⋊C4—271no—
182C2×C22⋊C4—91no—
183C2×C22⋊C4—272no—
184#95—92no—
185#95—271no—
186H(3,𝔽3)#39711yescharacteristic, complemented, Sylow
187C23×C3—32no—
188C23×C3—61no—
189C23×C3—92no—
190C23×C3—181no—
191C23×C3—182no—
192C23×C3—182no—
193C23×C3—181no—
194C23×C3C2×D3211yescharacteristic
195C22×D3—34no—
196C22×D3—62no—
197C22×D3—62no—
198C22×D3—62no—
199C22×D3—62no—
200C22×D3—62no—
201C22×D3—62no—
202C22×D3—62no—
203C22×D3—62no—
204C22×D3—62no—
205C22×D3—122no—
206C22×D3—122no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—62no—
213C2×C3×C4—92no—
214C2×C3×C4—122no—
215C2×C3×C4—182no—
216C2×C3×C4—182no—
217C2×C3×C4—182no—
218C2×C3×C4—182no—
219C2×C3×C4—182no—
220C2×C3×C4—182no—
221C2×C3×C4—182no—
222C2×Dic3—34no—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—62no—
226C2×Dic3—62no—
227C2×Dic3—62no—
228C2×Dic3—62no—
229C2×Dic3—62no—
230C2×Dic3—62no—
231C2×Dic3—62no—
232C2×Dic3—122no—
233C2×Dic3—182no—
234C2×Dic3—182no—
235C2×Dic3—182no—
236C2×Dic3—182no—
237C2×Dic3—182no—
238C2×Dic3—182no—
239C2×C32—22no—
240C2×C32—22no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32#87612yes—
245C2×C32#88312yes—
246C2×C32#88412yes—
247C2×C32#88512yes—
248C2×C32#89412yes—
249C2×C32#89412yes—
250C2×C32#89812yes—
251C3×D3—124no—
252C3×D3—124no—
253C24—91no—
254C22×C4—91no—
255C22×C4—92no—
256C22×C4—92no—
257C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
258C4⋊C4—542no—
259C22⋊C4—182no—
260C22⋊C4—544no—
261C22×C3—32no—
262C22×C3—32no—
263C22×C3—32no—
264C22×C3—32no—
265C22×C3—32no—
266C22×C3—32no—
267C22×C3—32no—
268C22×C3—61no—
269C22×C3—62no—
270C22×C3—61no—
271C22×C3—62no—
272C22×C3—61no—
273C22×C3—184no—
274C22×C3—182no—
275C22×C3—362no—
276C22×C3—361no—
277C22×C3—361no—
278C22×C3#379811yescharacteristic
279C22×C3#379912yes—
280C22×C3#380211yescharacteristic
281C22×C3#380412yes—
282C22×C3#381111yescharacteristic
283C2×D3—64no—
284C2×D3—64no—
285C2×D3—64no—
286C2×D3—64no—
287C2×D3—124no—
288C2×D3—122no—
289C2×D3—122no—
290C2×D3—122no—
291C2×D3—122no—
292C2×D3—122no—
293C2×D3—124no—
294C2×D3—122no—
295C3×C4—41no—
296C3×C4—122no—
297C3×C4—122no—
298C3×C4—182no—
299C3×C4—362no—
300C3×C4—362no—
301Dic3—64no—
302Dic3—122no—
303Dic3—122no—
304Dic3—122no—
305Dic3—362no—
306Dic3—362no—
307C32—21no—
308C32#467512yescomplemented
309C23—92no—
310C23—181no—
311C23—182no—
312C23—182no—
313C23—181no—
314C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—92no—
318C2×C4—182no—
319C2×C4—182no—
320C2×C4—182no—
321C2×C4—182no—
322C2×C4—182no—
323C2×C4—182no—
324C2×C4—182no—
325C2×C3—32no—
326C2×C3—32no—
327C2×C3—32no—
328C2×C3—32no—
329C2×C3—32no—
330C2×C3—32no—
331C2×C3—32no—
332C2×C3—61no—
333C2×C3—62no—
334C2×C3—62no—
335C2×C3—61no—
336C2×C3—61no—
337C2×C3—362no—
338C2×C3#6385412yes—
339C2×C3#6386812yes—
340C2×C3#6389711yescharacteristic
341C2×C3#6391211yescharacteristic
342C2×C3#6395311yescharacteristic
343D3—124no—
344D3—124no—
345C22—184no—
346C22—182no—
347C22—361no—
348C22—362no—
349C22—361no—
350C22#8865011yescharacteristic
351C22#8865212yes—
352C22#8865511yescharacteristic
353C22#8867112yes—
354C22#8867711yescharacteristic
355C4—41no—
356C4—182no—
357C4—362no—
358C4—362no—
359C3—32no—
360C3—61no—
361C3#1059121611yescharacteristic
362C2—362no—
363C2#1172109212yes—
364C2#1172110212yes—
365C2#1172114211yescharacteristic
366C2#1172115111yescharacteristic
367C2#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)#397
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
3. C2×#88449C2semidirect
4. #63658⋊C3C2semidirect
5. #11720334C2semidirect
20. #63658D3semidirect
186. H(3,𝔽3)#397semidirect
308. C32#4675semidirect

Maximal subgroups

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

subgroupindex
C2×#885422
C2×#884512
C2×#884492
#63658⋊C32
#117203342
#105912973