Finite Group Databasebeta

← #13147768

Subgroups — #13147768

Order 1728.

885 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 885 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C42×#272C211yesmaximal, characteristic
2C4×#634⋊C3C212yesmaximal, complemented
3#11721117C211yesmaximal, characteristic, complemented
4C4×#6196C212yesmaximal, complemented
5#11721099C211yesmaximal, characteristic, complemented
6#11721098C212yesmaximal, complemented
7#11721097C212yesmaximal
8#11721096C212yesmaximal
9#11721094C212yesmaximal
10#10593873—32nomaximal
11C2×C4×#272C2211yescharacteristic
12C2×C4×#272C2211yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C2×#634⋊C3C2212yes—
15C2×#6196C2212yes—
16C42×H(3,𝔽3)C2211yescharacteristic
17C4×#1084C2211yescharacteristic
18C4×#1084C2211yescharacteristic
19C4×#1084C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2212yes—
21#3750⋊C3C2214yes—
22C4×(C3⋊Dic3)⋊C3C2214yes—
23#88451C2214yes—
24#88449C2212yes—
25#88449C2212yes—
26#88449C2212yes—
27#88448C2214yes—
28C4×#634D312yescomplemented
29C3×C42×D3—32no—
30C3×C42×D3—61no—
31C3×C4×Dic6—32no—
32#63918—32no—
33C4×#627—34no—
34#63867—34no—
35#63866—32no—
36#63857—32no—
37#63856—32no—
38#63855—34no—
39#63851—34no—
40C22×#272C2311yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×#1084C2311yescharacteristic
45C2×#1084C2311yescharacteristic
46C2×#1084C2311yescharacteristic
47C4×#272—22no—
48C4×#272—22no—
49C4×#272Q814yescomplemented
50C2×(C3⋊Dic3)⋊C3C2314yes—
51C2×(C3⋊Dic3)⋊C3C2314yes—
52#634⋊C3—24no—
53#6196—24no—
54#5578—92no—
55#5470—91no—
56C2×#634C2×D312yes—
57C2×C3×C4×D3—32no—
58C2×C3×C4×D3—32no—
59C2×C3×C4×D3—32no—
60C2×C3×C4×D3—61no—
61C2×C3×C4×D3—61no—
62C2×C3×C4×D3—61no—
63C2×C3×Dic6—32no—
64C2×#627—34no—
65C32×C42—21no—
66C32×C42C2×D312yes—
67C4⋊(C3⋊Dic3)C2×D312yes—
68#3750C2×D314yes—
69C4×C3⋊Dic3C2×D314yes—
70C3×C4⋊Dic3—32no—
71C3×Dic3⋊C4—34no—
72C3×C4×Dic3—34no—
73C3×C4×Dic3—32no—
74C3×C4×Dic3—32no—
75C3×C4×Dic3—32no—
76C3×C4×Dic3—61no—
77C3×C4×Dic3—61no—
78C3×C4×Dic3—61no—
79#3724—34no—
80Dic3⋊Dic3—34no—
81Dic3⋊Dic3—34no—
82Dic3⋊Dic3—34no—
83Dic3⋊Dic3—34no—
84(C2×D3)⋊Dic3—34no—
85(C2×D3)⋊Dic3—34no—
86C22×H(3,𝔽3)C2411yescharacteristic
87C2×#272—21no—
88C2×#272—21no—
89C2×#272C2×Q812yes—
90C4×H(3,𝔽3)—21no—
91C4×H(3,𝔽3)C2×Q812yes—
92C4×H(3,𝔽3)#4112yes—
93#1084—21no—
94#1084C2×Q812yes—
95#1084#4112yes—
96(C3⋊Dic3)⋊C3—24no—
97(C3⋊Dic3)⋊C3—24no—
98C3×C4×Q8—92no—
99C2×C3×C42—91no—
100#946—92no—
101#944—92no—
102#926—92no—
103#924—92no—
104#924—92no—
105#897—92no—
106#896—94no—
107#889—94no—
108#888—92no—
109#872—94no—
110C42×D3—32no—
111C42×D3—61no—
112#870—92no—
113#869—91no—
114#869—91no—
115C4×Dic6—32no—
116C4×Dic6—92no—
117C4×Dic6—92no—
118C4×Dic6—94no—
119C22×C3×D3—32no—
120C22×C3×D3—61no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4C22×D312yes—
125C2×C32×C4C22×D312yes—
126C2×C32×C4C22×D312yes—
127C2×C3⋊Dic3C22×D314yes—
128C2×C3⋊Dic3C22×D314yes—
129#634—24no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C3×C4×D3—38no—
139C3×C4×D3—64no—
140C3×C4×D3—64no—
141C3×C4×D3—64no—
142C3×C4×D3—64no—
143C3×C4×D3—64no—
144C3×Dic6—64no—
145#627—68no—
146#533—271noSylow
147C2×H(3,𝔽3)C22×Q811yescharacteristic
148C2×H(3,𝔽3)C2×#4111yescharacteristic
149C2×H(3,𝔽3)C2×#4111yescharacteristic
150#272—22no—
151C2×C3×Q8—92no—
152C22×C3×C4—91no—
153C22×C3×C4—91no—
154C22×C3×C4—91no—
155C2×C4×D3—32no—
156C2×C4×D3—32no—
157C2×C4×D3—32no—
158C2×C4×D3—61no—
159C2×C4×D3—61no—
160C2×C4×D3—61no—
161C2×Dic6—32no—
162C2×Dic6—92no—
163C2×Dic6—92no—
164C2×Dic6—94no—
165C3×C4⋊C4—92no—
166C3×C4⋊C4—94no—
167C3×C42—32no—
168C3×C42—61no—
169C3×C42—94no—
170C3×C42—91no—
171C3×C42—91no—
172C3×C42—91no—
173C3×C42D3211yescharacteristic
174#217—94no—
175(C2×D3)⋊C4—94no—
176(C2×D3)⋊C4—94no—
177C4⋊Dic3—32no—
178C4⋊Dic3—92no—
179C4⋊Dic3—92no—
180C4⋊Dic3—92no—
181C4⋊Dic3—92no—
182C4⋊Dic3—94no—
183Dic3⋊C4—34no—
184Dic3⋊C4—94no—
185Dic3⋊C4—94no—
186Dic3⋊C4—94no—
187Dic3⋊C4—94no—
188Dic3⋊C4—94no—
189Dic3⋊C4—94no—
190C4×Dic3—32no—
191C4×Dic3—32no—
192C4×Dic3—32no—
193C4×Dic3—34no—
194C4×Dic3—61no—
195C4×Dic3—61no—
196C4×Dic3—61no—
197C4×Dic3—94no—
198C22×C32—21no—
199C22×C32C23×D312yes—
200C2×C3×D3—34no—
201C2×C3×D3—62no—
202C2×C3×D3—62no—
203C2×C3×D3—62no—
204C2×C3×D3—62no—
205C2×C3×D3—62no—
206C32×C4—22no—
207C32×C4—22no—
208C32×C4—22no—
209C32×C4—22no—
210C32×C4#23514yes—
211C32×C4D3×Q814yes—
212C3⋊Dic3—24no—
213C3⋊Dic3—24no—
214C3×Dic3—34no—
215C3×Dic3—34no—
216C3×Dic3—62no—
217C3×Dic3—62no—
218C3×Dic3—62no—
219C3×Dic3—64no—
220C3×Dic3—62no—
221C3×Dic3—64no—
222C4⋊Q8—271no—
223C4⋊Q8—271no—
224#125—272no—
225C22⋊Q8—272no—
226C22⋊Q8—272no—
227C4×Q8—92no—
228C4×Q8—272no—
229#117—272no—
230C2×C42—91no—
231H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
232C23×C3—91no—
233C22×D3—32no—
234C22×D3—61no—
235C3×Q8—184no—
236C2×C3×C4—32no—
237C2×C3×C4—32no—
238C2×C3×C4—32no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—91no—
243C2×C3×C4—91no—
244C2×C3×C4—94no—
245C2×C3×C4—94no—
246C2×C3×C4—94no—
247C2×C3×C4—91no—
248C2×C3×C4—182no—
249C2×C3×C4—182no—
250C2×C3×C4C2×D3211yescharacteristic
251C2×C3×C4C2×D3211yescharacteristic
252C2×C3×C4C2×D3211yescharacteristic
253C2×Dic3—32no—
254C2×Dic3—32no—
255C2×Dic3—32no—
256C2×Dic3—34no—
257C2×Dic3—34no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—94no—
262C2×Dic3—94no—
263C4×D3—38no—
264C4×D3—64no—
265C4×D3—64no—
266C4×D3—64no—
267C4×D3—64no—
268C4×D3—64no—
269Dic6—64no—
270Dic6—184no—
271Dic6—188no—
272Dic6—184no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32C2×#23512yes—
277C2×C32C2×D3×Q812yes—
278C2×C32D3×#4112yes—
279C3×D3—64no—
280C3×D3—64no—
281C2×Q8—92no—
282C2×Q8—272no—
283C22×C4—91no—
284C22×C4—91no—
285C22×C4—91no—
286C4⋊C4—94no—
287C4⋊C4—92no—
288C4⋊C4—272no—
289C4⋊C4—272no—
290C4⋊C4—274no—
291C4⋊C4—272no—
292C22⋊C4—274no—
293C42—91no—
294C42—91no—
295C42—91no—
296C42—94no—
297C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
298C22×C3—32no—
299C22×C3—61no—
300C22×C3—92no—
301C22×C3—181no—
302C22×C3—181no—
303C22×C3C22×D3211yescharacteristic
304C2×D3—34no—
305C2×D3—62no—
306C2×D3—62no—
307C2×D3—62no—
308C2×D3—62no—
309C2×D3—62no—
310C3×C4—21no—
311C3×C4—34no—
312C3×C4—34no—
313C3×C4—62no—
314C3×C4—62no—
315C3×C4—62no—
316C3×C4—62no—
317C3×C4—92no—
318C3×C4—92no—
319C3×C4—184no—
320C3×C4—184no—
321C3×C4—181no—
322C3×C4#379712yes—
323C3×C4#379812yes—
324Dic3—34no—
325Dic3—34no—
326Dic3—62no—
327Dic3—62no—
328Dic3—62no—
329Dic3—62no—
330Dic3—64no—
331Dic3—64no—
332Dic3—184no—
333Dic3—184no—
334C32—21no—
335C32#557812yescomplemented
336C23—91no—
337Q8—184no—
338Q8—544no—
339C2×C4—91no—
340C2×C4—91no—
341C2×C4—94no—
342C2×C4—91no—
343C2×C4—94no—
344C2×C4—94no—
345C2×C4—182no—
346C2×C4—182no—
347C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
348C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
349C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
350C2×C3—32no—
351C2×C3—32no—
352C2×C3—32no—
353C2×C3—61no—
354C2×C3—61no—
355C2×C3—61no—
356C2×C3—182no—
357C2×C3C2×#379711yescharacteristic
358C2×C3C2×#379811yescharacteristic
359C2×C3#6430611yescharacteristic
360D3—64no—
361D3—64no—
362C22—92no—
363C22—181no—
364C22—181no—
365C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
366C4—21no—
367C4—92no—
368C4—92no—
369C4—181no—
370C4—184no—
371C4—184no—
372C4#8865012yes—
373C4#8865112yes—
374C3—32no—
375C3—61no—
376C3#1059388211yescharacteristic
377C2—182no—
378C2C2×#8865011yescharacteristic
379C2C2×#8865111yescharacteristic
380C2#1172249111yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×#272C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#533
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×#634⋊C3C2semidirect
3. #11721117C2semidirect
4. C4×#6196C2semidirect
5. #11721099C2semidirect
6. #11721098C2semidirect
28. C4×#634D3semidirect
49. C4×#272Q8semidirect
231. H(3,𝔽3)#533semidirect
335. C32#5578semidirect

Maximal subgroups

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

subgroupindex
C42×#2722
C4×#634⋊C32
#117211172
C4×#61962
#117210992
#117210982
#117210972
#117210962
#117210942
#105938733