Finite Group Databasebeta

← #13140586

Subgroups — #13140586

Order 1728.

717 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 81 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×#88744C211yesmaximal, characteristic
2#11721298C212yesmaximal, complemented
3#63661⋊C3C212yesmaximal, complemented
4#11720335C212yesmaximal, complemented
5#10591528—32nomaximal
6C2×D4×H(3,𝔽3)C2211yescharacteristic
7C2×#6235C411yescharacteristic, complemented
8C2×#6235C2211yescharacteristic
9C22×#1084C411yescharacteristic, complemented
10C22×#1084C2211yescharacteristic
11#88744—22no—
12#88744—22no—
13C2×C4×#272C411yescharacteristic, complemented
14C2×#6229C411yescharacteristic, complemented
15#3757⋊C3C2214yes—
16#88449—22no—
17#88447—22no—
18C2×C3×#237—32no—
19C2×C3×#237—61no—
20#63955—34no—
21#63661D312yescomplemented
22C3×#834—32no—
23#63580—34no—
24C23×H(3,𝔽3)C2×C411yescharacteristic
25C23×H(3,𝔽3)C2311yescharacteristic
26C22×#272C2×C411yescharacteristic
27D4×H(3,𝔽3)—22no—
28C2×C4×H(3,𝔽3)C2×C411yescharacteristic
29#6235—22no—
30#6235—22no—
31C2×#1084—21no—
32C2×#1084—21no—
33C2×#1084C2×C411yescharacteristic
34C2×#1084C2×C411yescharacteristic
35C2×#1084C2×C411yescharacteristic
36C2×#1084D412yes—
37C2×#1084D412yes—
38C4×#272—21no—
39C4×#272—21no—
40#6229—22no—
41C2×(C3⋊Dic3)⋊C3—24no—
42#5240—91no—
43#4740—92no—
44C2×C32×D4—21no—
45C2×C32×D4C2×D312yes—
46C2×C3×#67—32no—
47C2×C3×#67—32no—
48C2×C3×#67—61no—
49C2×C3×#67—61no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—61no—
53C22×C3×Dic3—61no—
54C3×#237—64no—
55C3×#237—64no—
56C3×#237—64no—
57C3×#237—64no—
58C2×C3×C4×D3—32no—
59C2×C3×C4×D3—61no—
60C2×C3×Dic6—32no—
61C2×C3×Dic6—61no—
62#3757C2×D314yes—
63C3×#217—34no—
64Dic3⋊Dic3—64no—
65Dic32—64no—
66C22×H(3,𝔽3)—21no—
67C22×H(3,𝔽3)—41no—
68C22×H(3,𝔽3)—41no—
69C22×H(3,𝔽3)C22×C411yescharacteristic
70C22×H(3,𝔽3)C2×D412yes—
71C2×#272—21no—
72C2×#272—41no—
73C4×H(3,𝔽3)—21no—
74#1084—21no—
75#1084C22⋊C412yes—
76#1084C22⋊C411yescharacteristic
77#1084C22⋊C411yescharacteristic
78(C3⋊Dic3)⋊C3—44no—
79C2×C3×#41—91no—
80C2×#237—32no—
81C2×#237—61no—
82#926—92no—
83#877—94no—
84C3×#99—92no—
85#834—32no—
86#834—92no—
87#834—92no—
88#806—94no—
89C23×C32—21no—
90C23×C32—21no—
91C23×C32C4×D312yes—
92C23×C32C22×D312yes—
93C22×C3×D3—32no—
94C22×C3×D3—61no—
95C32×D4—24no—
96C32×D4—24no—
97C2×C32×C4—21no—
98C2×C32×C4C4×D312yes—
99C2×C3⋊Dic3—24no—
100C3×#67—64no—
101C3×#67—64no—
102C3×#67—64no—
103C3×#67—64no—
104C2×C3×Dic3—32no—
105C2×C3×Dic3—32no—
106C2×C3×Dic3—34no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—32no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
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—64no—
119C3×C4×D3—62no—
120C3×C4×D3—62no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×Dic6—64no—
124C3×Dic6—64no—
125#410—271noSylow
126C2×H(3,𝔽3)—21no—
127C2×H(3,𝔽3)—22no—
128C2×H(3,𝔽3)—41no—
129C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
130#272—41no—
131C3×#41—182no—
132C3×#41—182no—
133C2×C3×Q8—91no—
134C2×C3×D4—32no—
135C2×C3×D4—61no—
136C2×C3×D4—91no—
137C2×C3×D4—91no—
138C2×C3×D4D3211yescharacteristic
139C22×C3×C4—91no—
140C22×C3×C4—91no—
141C22×C3×C4—91no—
142C2×#67—32no—
143C2×#67—32no—
144C2×#67—61no—
145C2×#67—61no—
146C22×Dic3—32no—
147C22×Dic3—32no—
148C22×Dic3—61no—
149C22×Dic3—61no—
150#237—64no—
151#237—64no—
152#237—64no—
153#237—64no—
154C2×C4×D3—32no—
155C2×C4×D3—61no—
156C2×Dic6—32no—
157C2×Dic6—61no—
158C3×C22⋊C4—94no—
159#217—34no—
160#217—94no—
161C4⋊Dic3—182no—
162Dic3⋊C4—184no—
163C4×Dic3—184no—
164C4×Dic3—182no—
165C22×C32—22no—
166C22×C32—22no—
167C22×C32—21no—
168C22×C32—22no—
169C22×C32—42no—
170C22×C32—42no—
171C22×C32—42no—
172C22×C32—42no—
173C22×C32C2×C4×D312yes—
174C22×C32D3×D414yes—
175C2×C3×D3—62no—
176C2×C3×D3—62no—
177C2×C3×D3—122no—
178C2×C3×D3—122no—
179C32×C4—22no—
180C32×C4—22no—
181C3⋊Dic3—44no—
182C3×Dic3—34no—
183C3×Dic3—32no—
184C3×Dic3—32no—
185C3×Dic3—62no—
186C3×Dic3—61no—
187C3×Dic3—61no—
188C3×Dic3—62no—
189C3×Dic3—62no—
190C3×Dic3—124no—
191C2×#41—91no—
192#117—272no—
193#99—92no—
194#99—272no—
195H(3,𝔽3)#41011yescharacteristic, complemented, Sylow
196C23×C3—32no—
197C23×C3—32no—
198C23×C3—61no—
199C23×C3—61no—
200C23×C3—91no—
201C23×C3#62411yescharacteristic
202C23×C3C2×D3211yescharacteristic
203C22×D3—32no—
204C22×D3—61no—
205C3×Q8—182no—
206C3×D4—22no—
207C3×D4—64no—
208C3×D4—64no—
209C3×D4—182no—
210C3×D4—182no—
211C2×C3×C4—32no—
212C2×C3×C4—61no—
213C2×C3×C4—91no—
214C2×C3×C4—91no—
215C2×C3×C4—92no—
216C2×C3×C4—92no—
217C2×C3×C4—91no—
218C2×C3×C4—181no—
219C2×C3×C4—181no—
220C2×C3×C4—181no—
221C2×C3×C4—181no—
222C2×C3×C4—184no—
223C2×C3×C4#62411yescharacteristic
224#67—64no—
225#67—64no—
226#67—64no—
227#67—64no—
228C2×Dic3—34no—
229C2×Dic3—34no—
230C2×Dic3—32no—
231C2×Dic3—32no—
232C2×Dic3—32no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—62no—
241C2×Dic3—62no—
242C2×Dic3—64no—
243C2×Dic3—184no—
244C4×D3—62no—
245C4×D3—62no—
246C4×D3—62no—
247C4×D3—62no—
248Dic6—64no—
249Dic6—64no—
250C2×C32—22no—
251C2×C32—24no—
252C2×C32—22no—
253C2×C32—21no—
254C2×C32—42no—
255C2×C32—42no—
256C2×C32—42no—
257C2×C32D3×C22⋊C412yes—
258C3×D3—122no—
259C3×D3—122no—
260#41—182no—
261#41—182no—
262C2×Q8—91no—
263C2×D4—91no—
264C2×D4—91no—
265C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
266C22×C4—91no—
267C22×C4—91no—
268C22×C4—91no—
269C4⋊C4—542no—
270C22⋊C4—94no—
271C42—542no—
272C22×C3—21no—
273C22×C3—32no—
274C22×C3—34no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—62no—
278C22×C3—62no—
279C22×C3—61no—
280C22×C3—62no—
281C22×C3—122no—
282C22×C3—122no—
283C22×C3—122no—
284C22×C3—122no—
285C22×C3—181no—
286C22×C3—361no—
287C22×C3C2×#62411yescharacteristic
288C22×C3#381112yes—
289C2×D3—62no—
290C2×D3—62no—
291C2×D3—122no—
292C2×D3—122no—
293C3×C4—21no—
294C3×C4—62no—
295C3×C4—62no—
296C3×C4—91no—
297C3×C4—91no—
298C3×C4—92no—
299C3×C4—181no—
300C3×C4—364no—
301Dic3—34no—
302Dic3—32no—
303Dic3—32no—
304Dic3—62no—
305Dic3—61no—
306Dic3—61no—
307Dic3—62no—
308Dic3—62no—
309Dic3—124no—
310Dic3—364no—
311C32—21no—
312C32#474012yescomplemented
313C23—91no—
314C23#619911yescharacteristic
315C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
316Q8—182no—
317D4—22no—
318D4—182no—
319D4—182no—
320C2×C4—91no—
321C2×C4—91no—
322C2×C4—92no—
323C2×C4—92no—
324C2×C4—91no—
325C2×C4—181no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4—181no—
329C2×C4—184no—
330C2×C4#619911yescharacteristic
331C2×C3—22no—
332C2×C3—21no—
333C2×C3—32no—
334C2×C3—41no—
335C2×C3—64no—
336C2×C3—62no—
337C2×C3—62no—
338C2×C3—61no—
339C2×C3—122no—
340C2×C3—122no—
341C2×C3—122no—
342C2×C3—361no—
343C2×C3#6397411yescharacteristic
344D3—122no—
345D3—122no—
346C22—21no—
347C22—41no—
348C22—41no—
349C22—181no—
350C22—361no—
351C22C2×#619911yescharacteristic
352C22#8867712yes—
353C4—21no—
354C4—91no—
355C4—91no—
356C4—92no—
357C4—181no—
358C4—364no—
359C3—32no—
360C3—61no—
361C3#1059152911yescharacteristic
362C2—22no—
363C2—21no—
364C2—41no—
365C2—361no—
366C2#1172132011yescharacteristic
subgroupG/H
centerC2#11721320
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC23×C3C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC2×C3#63974
prime radicalC2×#88744C2
cosocleC23×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#410
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11721298C2semidirect
3. #63661⋊C3C2semidirect
4. #11720335C2semidirect
7. C2×#6235C4semidirect
9. C22×#1084C4semidirect
13. C2×C4×#272C4semidirect
14. C2×#6229C4semidirect
21. #63661D3semidirect
195. H(3,𝔽3)#410semidirect
312. C32#4740semidirect

Maximal subgroups

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

subgroupindex
C2×#887442
#117212982
#63661⋊C32
#117203352
#105915283