Finite Group Databasebeta

← #13164499

Subgroups — #13164499

Order 1728.

707 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 707 conjugacy classes of subgroups — too many to draw. Only its 64 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×#41⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11722394C211yesmaximal, characteristic, complemented
3#11722393C212yesmaximal, complemented
4#11722392C211yesmaximal, characteristic, complemented
5#11722388C212yesmaximal, complemented
6C3×#5923—33nomaximal
7C3×#5822—31nomaximal
8#88740—41nomaximal
9#41⋊H(3,𝔽3)C2211yescharacteristic
10#41⋊H(3,𝔽3)C2211yescharacteristic, complemented
11C2×Q8⋊H(3,𝔽3)C2211yescharacteristic, complemented
12#88611C2212yescomplemented
13#88610C2212yescomplemented
14C2×C32×#41D311yescharacteristic, complemented
15C2×C3×#235—31no—
16C2×C3×#41⋊C3D313yescomplemented
17C3×#986—33no—
18C3×#985—36no—
19C3×#984—33no—
20C3×#983—36no—
21C3×#951—31no—
22C3×#951—31no—
23C3×#950—32no—
24C3×#950—32no—
25C3×#949—31no—
26C3×#949—31no—
27C3×#942—32no—
28C3×#932—32no—
29C2×C3×#208—31no—
30C2×C4×H(3,𝔽3)—41no—
31#6235—42no—
32C4⋊#272—41no—
33C4×#272—42no—
34#6229—41no—
35Q8⋊H(3,𝔽3)C2311yescharacteristic
36#5923—33no—
37C3×#575—91no—
38#5822—31no—
39C32×#41—31no—
40C32×#41—31no—
41C32×#41C2×D311yescharacteristic
42C32×#41C2×D311yescharacteristic
43C2×C32×Q8C2×D311yescharacteristic
44C2×C32×D4—31no—
45C22×C32×C4—31no—
46C2×C3×#67—61no—
47C3×#235—32no—
48C3×#235—32no—
49C3×#235—61no—
50C3×#235—61no—
51C3×#235—123no—
52C2×C3×D12—31no—
53C2×C3×C4×D3—61no—
54C2×C3×Dic6—31no—
55C3×#41⋊C3C2×D313yes—
56C3×#41⋊C3C2×D313yescomplemented
57C2×C3×SL(2,𝔽3)C2×D313yescomplemented
58C3×GL(2,𝔽3)—36no—
59C3×#226—36no—
60C3×#216—32no—
61C3×#216—32no—
62C3×#215—32no—
63C3×#215—32no—
64C3×#214—32no—
65C3×#214—32no—
66C3×#213—32no—
67C3×#213—32no—
68C3×#208—32no—
69C3×#208—31no—
70C3×#208—31no—
71C2×C3×C3⋊C8—32no—
72C22×H(3,𝔽3)—41no—
73C2×#272—42no—
74C4×H(3,𝔽3)—41no—
75C4×H(3,𝔽3)—41no—
76#1084—42no—
77C2×C3×#41—31no—
78C2×C3×#41—91no—
79C2×C3×#41C3⋊D311yescharacteristic
80C2×#235—31no—
81C2×#41⋊C3—33no—
82#986—33no—
83#985—36no—
84#984—33no—
85#983—36no—
86C3×#137—91no—
87C3×#137—91no—
88C3×#137—92no—
89C3×AGL(1,ℤ/8)—92no—
90C3×AGL(1,ℤ/8)—91no—
91C3×AGL(1,ℤ/8)—91no—
92C3×#135—92no—
93C3×#135—92no—
94C2×C3×M16—91no—
95#951—31no—
96#951—31no—
97#950—32no—
98#950—32no—
99#949—31no—
100#949—31no—
101#942—32no—
102#932—32no—
103C2×#208—31no—
104C23×C32—31no—
105C22×C3×D3—61no—
106C32×Q8—31no—
107C32×Q8C22×D311yescharacteristic
108C32×D4—31no—
109C32×D4—31no—
110C32×D4—61no—
111C2×C32×C4—31no—
112C2×C32×C4—31no—
113C2×C32×C4—31no—
114C2×C32×C4—31no—
115C2×C32×C4—31no—
116C2×C32×C4—43no—
117C2×C32×C4S411yescharacteristic
118C3×#67—62no—
119C3×#67—62no—
120C3×#67—126no—
121C2×C3×Dic3—61no—
122C3×D12—32no—
123C3×D12—61no—
124C3×D12—123no—
125C3×C4×D3—62no—
126C3×C4×D3—62no—
127C3×C4×D3—126no—
128C3×Dic6—32no—
129C3×Dic6—61no—
130C3×Dic6—123no—
131C3×SL(2,𝔽3)C22×D313yes—
132C3×C3⋊C8—32no—
133C3×C3⋊C8—32no—
134#575—91noSylow
135C2×H(3,𝔽3)—41no—
136C2×H(3,𝔽3)—81no—
137#272—82no—
138C3×#41—31no—
139C3×#41—31no—
140C3×#41—31no—
141C3×#41—31no—
142C3×#41—31no—
143C3×#41—31no—
144C3×#41—61no—
145C3×#41—61no—
146C3×#41—92no—
147C3×#41—92no—
148C3×#41—181no—
149C3×#41—181no—
150C3×#41C2×C3⋊D311yescharacteristic
151C3×#41C2×C3⋊D311yescharacteristic
152C2×C3×Q8—31no—
153C2×C3×Q8—91no—
154C2×C3×Q8C2×C3⋊D311yescharacteristic
155C2×C3×D4—31no—
156C2×C3×D4—31no—
157C2×C3×D4—61no—
158C2×C3×D4—91no—
159C2×C3×D4—181no—
160C22×C3×C4—31no—
161C22×C3×C4—31no—
162C22×C3×C4—61no—
163C22×C3×C4—181no—
164C2×#67—61no—
165#235—32no—
166#235—32no—
167#235—61no—
168#235—61no—
169#235—123no—
170C2×D12—31no—
171C2×C4×D3—61no—
172C2×Dic6—31no—
173#41⋊C3—33no—
174#41⋊C3—33no—
175C2×SL(2,𝔽3)—33no—
176GL(2,𝔽3)—36no—
177#226—36no—
178C3×Q16—92no—
179C3×Q16—92no—
180C3×SD16—92no—
181C3×SD16—92no—
182C3×SD16—92no—
183C3×SD16—92no—
184C3×D8—92no—
185C3×D8—92no—
186C3×M16—92no—
187C3×M16—91no—
188C3×M16—91no—
189C2×C3×C8—92no—
190#216—32no—
191#216—32no—
192#215—32no—
193#215—32no—
194#214—32no—
195#214—32no—
196#213—32no—
197#213—32no—
198#208—32no—
199#208—31no—
200#208—31no—
201C2×C3⋊C8—32no—
202C22×C32—31no—
203C22×C32—31no—
204C22×C32—43no—
205C22×C32—121no—
206C22×C32C2×S411yescharacteristic
207C2×C3×D3—62no—
208C2×C3×D3—126no—
209C2×C3×D3—122no—
210C32×C4—31no—
211C32×C4—31no—
212C32×C4—43no—
213C32×C4—43no—
214C32×C4C2×S411yescharacteristic
215C32×C4C2×S411yescharacteristic
216C3×Dic3—62no—
217C3×Dic3—126no—
218C2×#41—91no—
219C2×#41#27211yescharacteristic, complemented
220#137—91no—
221#137—91no—
222#137—92no—
223AGL(1,ℤ/8)—92no—
224AGL(1,ℤ/8)—91no—
225AGL(1,ℤ/8)—91no—
226#135—92no—
227#135—92no—
228C2×M16—91no—
229H(3,𝔽3)—41noSylow
230C23×C3—31no—
231C23×C3—31no—
232C23×C3—61no—
233C23×C3—181no—
234C22×D3—61no—
235C3×Q8—31no—
236C3×Q8—31no—
237C3×Q8—31no—
238C3×Q8—61no—
239C3×Q8—92no—
240C3×Q8—181no—
241C3×Q8C22×C3⋊D311yescharacteristic
242C3×D4—31no—
243C3×D4—31no—
244C3×D4—31no—
245C3×D4—31no—
246C3×D4—61no—
247C3×D4—61no—
248C3×D4—61no—
249C3×D4—61no—
250C3×D4—61no—
251C3×D4—61no—
252C3×D4—92no—
253C3×D4—182no—
254C3×D4—182no—
255C3×D4—181no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—123no—
273C2×C3×C4—182no—
274C2×C3×C4—181no—
275C2×C3×C4—182no—
276C2×C3×C4C3⋊S411yescharacteristic
277#67—62no—
278#67—62no—
279#67—126no—
280C2×Dic3—61no—
281D12—32no—
282D12—61no—
283D12—123no—
284C4×D3—62no—
285C4×D3—62no—
286C4×D3—126no—
287Dic6—32no—
288Dic6—61no—
289Dic6—123no—
290SL(2,𝔽3)—33no—
291C3×C8—92no—
292C3×C8—92no—
293C3⋊C8—32no—
294C3⋊C8—32no—
295C2×C32—21no—
296C2×C32—43no—
297C2×C32—61no—
298C2×C32—61no—
299C2×C32—83no—
300C2×C32C22×S411yescharacteristic
301C3×D3—122no—
302C3×D3—246no—
303#41—31no—
304#41—31no—
305#41—92no—
306#41—92no—
307#41—181no—
308#41—181no—
309#41C2×#27211yescharacteristic
310#41C2×#27211yescharacteristic
311C2×Q8—91no—
312C2×Q8C2×#27211yescharacteristic
313C2×D4—31no—
314C2×D4—91no—
315C2×D4—181no—
316C22×C4—31no—
317C22×C4—181no—
318Q16—92no—
319Q16—92no—
320SD16—92no—
321SD16—92no—
322SD16—92no—
323SD16—92no—
324D8—92no—
325D8—92no—
326M16—92no—
327M16—91no—
328M16—91no—
329C2×C8—92no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—61no—
336C22×C3—61no—
337C22×C3—121no—
338C22×C3—121no—
339C22×C3—122no—
340C22×C3—123no—
341C22×C3—182no—
342C22×C3—362no—
343C22×C3C2×C3⋊S411yescharacteristic
344C2×D3—62no—
345C2×D3—126no—
346C2×D3—122no—
347C3×C4—31no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—31no—
353C3×C4—61no—
354C3×C4—61no—
355C3×C4—123no—
356C3×C4—123no—
357C3×C4—182no—
358C3×C4C2×C3⋊S411yescharacteristic
359C3×C4C2×C3⋊S411yescharacteristic
360Dic3—62no—
361Dic3—126no—
362C32—43no—
363C32#592311yescharacteristic, complemented
364C23—31no—
365C23—181no—
366Q8—31no—
367Q8—92no—
368Q8—181no—
369Q8C22×#27211yescharacteristic
370D4—31no—
371D4—31no—
372D4—61no—
373D4—92no—
374D4—182no—
375D4—182no—
376D4—181no—
377C2×C4—31no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—31no—
382C2×C4—182no—
383C2×C4—181no—
384C2×C4—182no—
385C2×C4#625811yescharacteristic
386C8—92no—
387C8—92no—
388C2×C3—21no—
389C2×C3—31no—
390C2×C3—61no—
391C2×C3—61no—
392C2×C3—61no—
393C2×C3—61no—
394C2×C3—61no—
395C2×C3—121no—
396C2×C3—121no—
397C2×C3—123no—
398C2×C3—243no—
399C2×C3—362no—
400C2×C3C22×C3⋊S411yescharacteristic
401D3—122no—
402D3—246no—
403C22—31no—
404C22—31no—
405C22—121no—
406C22—182no—
407C22—362no—
408C22C2×#625811yescharacteristic
409C4—31no—
410C4—31no—
411C4—182no—
412C4C2×#625811yescharacteristic
413C4C2×#625811yescharacteristic
414C3—31no—
415C3—123no—
416C3#1059658611yescharacteristic
417C2—21no—
418C2—61no—
419C2—61no—
420C2—362no—
421C2C22×#625811yescharacteristic
subgroupG/H
centerC3×C4C2×C3⋊S4
commutator, cosocle, hypocenterQ8⋊H(3,𝔽3)C23
Frattini, socleC2×C3C22×C3⋊S4
FittingC2×C32×#41D3
hypercenterC2×C3×C4C3⋊S4
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×#41⋊H(3,𝔽3)C2semidirect
2. #11722394C2semidirect
3. #11722393C2semidirect
4. #11722392C2semidirect
5. #11722388C2semidirect
10. #41⋊H(3,𝔽3)C22semidirect
11. C2×Q8⋊H(3,𝔽3)C22semidirect
12. #88611C22semidirect
13. #88610C22semidirect
14. C2×C32×#41D3semidirect
16. C2×C3×#41⋊C3D3semidirect
56. C3×#41⋊C3C2×D3semidirect
57. C2×C3×SL(2,𝔽3)C2×D3semidirect
219. C2×#41#272semidirect
363. C32#5923semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
#117223942
#117223932
#117223922
#117223882
C3×#59233
C3×#58223
#887404