Finite Group Databasebeta

← #13141174

Subgroups — #13141174

Order 1728.

839 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 839 conjugacy classes of subgroups — too many to draw. Only its 125 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×#88552C211yesmaximal, characteristic
3C2×#3757⋊C3C212yesmaximal
4#11720334C211yesmaximal, characteristic, complemented
5#11720334C212yesmaximal, complemented
6#10591692—32nomaximal
7C24×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C2×#6196—22no—
13C2×#6196—24no—
14#88552—22no—
15#3757⋊C3—24no—
16C22×#627—32no—
17C2×#3757D312yes—
18C2×C3×#217—32no—
19C2×C3×#217—32no—
20C2×C3×#217—61no—
21#63579—32no—
22#63579—32no—
23#63579—32no—
24C23×H(3,𝔽3)—21no—
25C23×H(3,𝔽3)—22no—
26C23×H(3,𝔽3)—21no—
27C23×H(3,𝔽3)—22no—
28C23×H(3,𝔽3)Q812yescomplemented
29C23×H(3,𝔽3)C2311yescharacteristic
30C2×#1084—22no—
31C2×#1084—21no—
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#4947—92no—
43C24×C32—21no—
44C24×C32C2×D312yes—
45C22×C3⋊Dic3C2×D312yes—
46C22×C3⋊Dic3C2×D312yes—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—32no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—61no—
52C22×C3×Dic3—61no—
53C2×#627—64no—
54C2×#627—64no—
55C2×#627—64no—
56#3757—24no—
57C3×#217—64no—
58C3×#217—64no—
59C3×#217—64no—
60C22×H(3,𝔽3)—22no—
61C22×H(3,𝔽3)—24no—
62C22×H(3,𝔽3)—41no—
63C22×H(3,𝔽3)—41no—
64C22×H(3,𝔽3)—42no—
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C22×H(3,𝔽3)C2×D412yes—
67C22×H(3,𝔽3)C2×Q811yescharacteristic
68C22×H(3,𝔽3)#4112yes—
69C22×H(3,𝔽3)#4111yescharacteristic
70#1084—22no—
71#1084—41no—
72(C3⋊Dic3)⋊C3—24no—
73(C3⋊Dic3)⋊C3—42no—
74C22×Dic6—91no—
75C22×Dic6—92no—
76C2×C3×C22⋊C4—92no—
77C2×C3×C22⋊C4—91no—
78C2×#217—32no—
79C2×#217—32no—
80C2×#217—61no—
81C2×#217—92no—
82C3⋊#95—92no—
83C3⋊#95—91no—
84C3⋊#95—92no—
85C3⋊#95—92no—
86C3⋊#95—92no—
87C23×C32—22no—
88C23×C32—22no—
89C23×C32—22no—
90C23×C32—22no—
91C23×C32—22no—
92C23×C32—22no—
93C23×C32—22no—
94C23×C32—22no—
95C23×C32—22no—
96C23×C32—21no—
97C23×C32—42no—
98C23×C32—42no—
99C23×C32Dic614yescomplemented
100C23×C32C22×D312yes—
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3#6714yes—
107C2×C3×Dic3—34no—
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—62no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—122no—
122C2×C3×Dic3—122no—
123#627—616no—
124#393—271noSylow
125C2×H(3,𝔽3)—42no—
126C2×H(3,𝔽3)#12011yescharacteristic
127C2×H(3,𝔽3)C22⋊Q811yescharacteristic
128C2×H(3,𝔽3)C22⋊Q812yes—
129C2×H(3,𝔽3)#12411yescharacteristic
130C2×H(3,𝔽3)#12412yes—
131C24×C3—32no—
132C24×C3—61no—
133C24×C3D3211yescharacteristic
134C22×C3×C4—92no—
135C22×C3×C4—91no—
136C22×C3×C4—92no—
137C22×C3×C4—91no—
138C22×Dic3—32no—
139C22×Dic3—32no—
140C22×Dic3—32no—
141C22×Dic3—32no—
142C22×Dic3—61no—
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—
152C3×C22⋊C4—184no—
153#217—64no—
154#217—64no—
155#217—64no—
156#217—184no—
157C22×C32—24no—
158C22×C32—24no—
159C22×C32—24no—
160C22×C32—24no—
161C22×C32—22no—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32—22no—
165C22×C32—21no—
166C22×C32—44no—
167C22×C32—42no—
168C22×C32—44no—
169C22×C32—42no—
170C22×C32—42no—
171C22×C32—42no—
172C22×C32—42no—
173C22×C32—42no—
174C22×C32C2×Dic612yes—
175C22×C32#23512yes—
176C22×C32D3×D412yes—
177C22×C32D3×D412yes—
178C22×C32#23712yes—
179C22×C32#23712yes—
180C22×C32C2×#6712yes—
181C3⋊Dic3—24no—
182C3⋊Dic3—42no—
183C3×Dic3—64no—
184C3×Dic3—64no—
185C3×Dic3—122no—
186C3×Dic3—122no—
187C3×Dic3—122no—
188C3×Dic3—122no—
189C22×Q8—271no—
190C2×C22⋊C4—91no—
191C2×C22⋊C4—92no—
192#95—271no—
193#95—272no—
194H(3,𝔽3)#39311yescharacteristic, complemented, Sylow
195C23×C3—22no—
196C23×C3—21no—
197C23×C3—22no—
198C23×C3—21no—
199C23×C3—34no—
200C23×C3—32no—
201C23×C3—62no—
202C23×C3—62no—
203C23×C3—62no—
204C23×C3—62no—
205C23×C3—62no—
206C23×C3—62no—
207C23×C3—62no—
208C23×C3—62no—
209C23×C3—62no—
210C23×C3—61no—
211C23×C3—122no—
212C23×C3—122no—
213C23×C3#62712yes—
214C23×C3C2×D3211yescharacteristic
215C2×C3×C4—94no—
216C2×C3×C4—92no—
217C2×C3×C4—182no—
218C2×C3×C4—182no—
219C2×C3×C4—182no—
220C2×C3×C4—182no—
221C2×C3×C4—182no—
222C2×C3×C4—182no—
223C2×C3×C4—182no—
224C2×C3×C4—181no—
225C2×Dic3—34no—
226C2×Dic3—34no—
227C2×Dic3—62no—
228C2×Dic3—62no—
229C2×Dic3—62no—
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—62no—
239C2×Dic3—94no—
240C2×Dic3—122no—
241C2×Dic3—122no—
242C2×Dic3—182no—
243C2×Dic3—182no—
244C2×Dic3—182no—
245C2×Dic3—182no—
246C2×Dic3—182no—
247Dic6—188no—
248Dic6—1816no—
249C2×C32—21no—
250C2×C32—22no—
251C2×C32—22no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—44no—
255C2×C32—44no—
256C2×C32#87812yes—
257C2×C32#87812yes—
258C2×C32#88512yes—
259C2×C32#88512yes—
260C2×C32#92412yes—
261C2×C32#93612yes—
262C2×C32#93712yes—
263C24((C3⋊D3)⋊C3)⋊C211yescharacteristic
264C2×Q8—544no—
265C2×Q8—542no—
266C22×C4—91no—
267C22×C4—92no—
268C22×C4—91no—
269C22×C4—92no—
270C22⋊C4—182no—
271C22⋊C4—184no—
272C22×C3—24no—
273C22×C3—22no—
274C22×C3—32no—
275C22×C3—32no—
276C22×C3—32no—
277C22×C3—32no—
278C22×C3—32no—
279C22×C3—32no—
280C22×C3—32no—
281C22×C3—41no—
282C22×C3—42no—
283C22×C3—41no—
284C22×C3—64no—
285C22×C3—64no—
286C22×C3—64no—
287C22×C3—64no—
288C22×C3—61no—
289C22×C3—62no—
290C22×C3—61no—
291C22×C3—62no—
292C22×C3—61no—
293C22×C3—124no—
294C22×C3—122no—
295C22×C3—122no—
296C22×C3—122no—
297C22×C3—122no—
298C22×C3—122no—
299C22×C3—124no—
300C22×C3—122no—
301C22×C3#380412yes—
302C22×C3#380511yescharacteristic
303C22×C3C2×#62711yescharacteristic
304C22×C3D3×#6712yes—
305C22×C3#381111yescharacteristic
306C3×C4—184no—
307C3×C4—182no—
308C3×C4—362no—
309C3×C4—361no—
310Dic3—64no—
311Dic3—64no—
312Dic3—122no—
313Dic3—122no—
314Dic3—122no—
315Dic3—122no—
316Dic3—184no—
317Dic3—362no—
318C32—21no—
319C32#494712yescomplemented
320C23—22no—
321C23—21no—
322C23—22no—
323C23—21no—
324C23#619612yescomplemented
325C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
326Q8—548no—
327C2×C4—94no—
328C2×C4—92no—
329C2×C4—182no—
330C2×C4—182no—
331C2×C4—182no—
332C2×C4—182no—
333C2×C4—182no—
334C2×C4—182no—
335C2×C4—182no—
336C2×C4—181no—
337C2×C3—32no—
338C2×C3—32no—
339C2×C3—32no—
340C2×C3—32no—
341C2×C3—32no—
342C2×C3—32no—
343C2×C3—32no—
344C2×C3—42no—
345C2×C3—61no—
346C2×C3—62no—
347C2×C3—62no—
348C2×C3—61no—
349C2×C3—61no—
350C2×C3—124no—
351C2×C3—124no—
352C2×C3#6395311yescharacteristic
353C2×C3#6395912yes—
354C2×C3#6396412yes—
355C2×C3#6398211yescharacteristic
356C2×C3#6398311yescharacteristic
357C22—24no—
358C22—22no—
359C22—41no—
360C22—42no—
361C22—41no—
362C22C2×#619611yescharacteristic
363C22#8867112yes—
364C22#8867211yescharacteristic
365C22#8867612yes—
366C22#8867711yescharacteristic
367C4—182no—
368C4—184no—
369C4—361no—
370C4—362no—
371C3—32no—
372C3—61no—
373C3#1059169211yescharacteristic
374C2—42no—
375C2#1172129411yescharacteristic
376C2#1172129912yes—
377C2#1172130412yes—
378C2#1172130911yescharacteristic
379C2#1172132411yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
Fitting, prime radicalC24×H(3,𝔽3)C22
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#393
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#6196C2semidirect
4. #11720334C2semidirect
5. #11720334C2semidirect
28. C23×H(3,𝔽3)Q8semidirect
99. C23×C32Dic6semidirect
194. H(3,𝔽3)#393semidirect
319. C32#4947semidirect
324. C23#6196semidirect

Maximal subgroups

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

subgroupindex
C22×#61962
C2×#885522
C2×#3757⋊C32
#117203342
#105916923