Finite Group Databasebeta

← #13164506

Subgroups — #13164506

Order 1728.

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

normal subgroups only

This group has 757 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
12−1+4⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11722394C211yesmaximal, characteristic, complemented
3#11722393C211yesmaximal, characteristic, complemented
4#11722388C212yesmaximal, complemented
5C2×#88611C212yesmaximal, complemented
6C3×#5926—33nomaximal
7C3×#5837—31nomaximal
8D4×#272—41nomaximal
9#41⋊H(3,𝔽3)C2211yescharacteristic, complemented
10C2×Q8⋊H(3,𝔽3)C2212yescomplemented
11#88611C2211yescharacteristic, complemented
12#88611C2212yescomplemented
13#88610C2211yescharacteristic, complemented
14C32×2−1+4D311yescharacteristic, complemented
15C3×#1009—31no—
16C3×2−1+4⋊C3D313yescomplemented
17C3×#986—33no—
18C3×#985—33no—
19C3×#983—36no—
20C2×C3×GL(2,𝔽3)—36no—
21C3×#950—32no—
22C3×#950—31no—
23C3×#949—32no—
24C3×#949—31no—
25C3×#948—31no—
26C3×#942—31no—
27C3×#942—32no—
28C2×C3×#215—31no—
29C2×C3×#215—32no—
30C22×#272—42no—
31D4×H(3,𝔽3)—41no—
32#6235—42no—
33C4⋊#272—41no—
34C4×#272—41no—
35Q8⋊H(3,𝔽3)C2311yescharacteristic
36#5926—33no—
37C3×#577—91no—
38#5837—31no—
39C32×#41—31no—
40C32×#41—32no—
41C32×#41C2×D311yescharacteristic, complemented
42C2×C32×Q8—31no—
43C2×C32×Q8C2×D312yescomplemented
44C3×#239—61no—
45C3×D3×D4—62no—
46C3×D3×D4—61no—
47C3×D3×D4—123no—
48C3×#235—32no—
49C3×#235—31no—
50C2×C3×D12—32no—
51C2×C3×D12—31no—
52C3×#41⋊C3C2×D313yescomplemented
53C2×C3×SL(2,𝔽3)C2×D316yescomplemented
54C3×GL(2,𝔽3)—36no—
55C3×GL(2,𝔽3)—33no—
56C3×#226—33no—
57C3×#216—32no—
58C3×#216—31no—
59C3×#215—32no—
60C3×#215—32no—
61C3×#215—32no—
62C3×#215—31no—
63C3×#215—32no—
64C3×#214—31no—
65C3×#213—31no—
66C3×#213—32no—
67C3×#208—32no—
68C3×#208—31no—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—31no—
71C22×H(3,𝔽3)—42no—
72C2×#272—41no—
73C2×#272—42no—
74C2×#272—84no—
75C4×H(3,𝔽3)—41no—
76#1084—41no—
77C3×2−1+4—31no—
78C3×2−1+4C3⋊D311yescharacteristic
79C3×H(5,𝔽2)—91no—
80#1009—31no—
812−1+4⋊C3—33no—
82#986—33no—
83#985—33no—
84#983—36no—
85C2×GL(2,𝔽3)—36no—
86C3×#137—91no—
87C3×#137—92no—
88C3×AGL(1,ℤ/8)—92no—
89C3×AGL(1,ℤ/8)—91no—
90C3×#135—92no—
91C3×#135—91no—
92C2×C3×SD16—91no—
93C2×C3×SD16—92no—
94C3×#131—91no—
95#950—32no—
96#950—31no—
97#949—32no—
98#949—31no—
99#948—31no—
100#942—31no—
101#942—32no—
102C2×#215—31no—
103C2×#215—32no—
104C22×C3×D3—61no—
105C22×C3×D3—62no—
106C22×C3×D3—126no—
107C32×Q8—32no—
108C32×Q8—31no—
109C32×Q8C22×D311yescharacteristic
110C32×D4—31no—
111C32×D4—32no—
112C32×D4—43no—
113C32×D4S411yescharacteristic
114C2×C32×C4—32no—
115C2×C32×C4—31no—
116C2×C32×C4—61no—
117C3×#67—61no—
118C3×#67—62no—
119C3×#67—126no—
120C3×D12—32no—
121C3×D12—31no—
122C3×D12—62no—
123C3×D12—61no—
124C3×D12—123no—
125C3×C4×D3—62no—
126C3×C4×D3—61no—
127C3×C4×D3—123no—
128C3×Dic6—31no—
129C3×SL(2,𝔽3)C22×D313yes—
130C3×C3⋊C8—31no—
131C3×C3⋊C8—31no—
132C3×C3⋊C8—32no—
133#577—91noSylow
134C2×H(3,𝔽3)—41no—
135C2×H(3,𝔽3)—82no—
136#272—42no—
137#272—82no—
138C3×#41—31no—
139C3×#41—31no—
140C3×#41—32no—
141C3×#41—32no—
142C3×#41—31no—
143C3×#41—61no—
144C3×#41—62no—
145C3×#41—92no—
146C3×#41—91no—
147C3×#41—181no—
148C3×#41C2×C3⋊D311yescharacteristic
149C2×C3×Q8—31no—
150C2×C3×Q8—31no—
151C2×C3×Q8—32no—
152C2×C3×Q8—61no—
153C2×C3×Q8C2×C3⋊D312yes—
154C2×C3×D4—92no—
155C2×C3×D4—91no—
156C2×C3×D4—182no—
157C2×C3×D4—181no—
158#239—61no—
159D3×D4—62no—
160D3×D4—61no—
161D3×D4—123no—
162#235—32no—
163#235—31no—
164C2×D12—32no—
165C2×D12—31no—
166#41⋊C3—33no—
167C2×SL(2,𝔽3)—36no—
168GL(2,𝔽3)—33no—
169GL(2,𝔽3)—36no—
170#226—33no—
171C3×Q16—92no—
172C3×Q16—91no—
173C3×SD16—91no—
174C3×SD16—92no—
175C3×SD16—92no—
176C3×SD16—92no—
177C3×SD16—91no—
178C3×SD16—92no—
179C3×D8—92no—
180C3×D8—91no—
181C3×M16—92no—
182C3×M16—91no—
183C2×C3×C8—92no—
184C2×C3×C8—91no—
185#216—32no—
186#216—31no—
187#215—32no—
188#215—31no—
189#215—32no—
190#215—32no—
191#215—32no—
192#214—31no—
193#213—32no—
194#213—31no—
195#208—32no—
196#208—31no—
197C2×C3⋊C8—32no—
198C2×C3⋊C8—31no—
199C22×C32—31no—
200C22×C32—46no—
201C22×C32C2×S412yes—
202C2×C3×D3—61no—
203C2×C3×D3—62no—
204C2×C3×D3—123no—
205C2×C3×D3—126no—
206C2×C3×D3—242no—
207C2×C3×D3—241no—
208C2×C3×D3—2412no—
209C32×C4—31no—
210C32×C4—32no—
211C32×C4—43no—
212C32×C4C2×S411yescharacteristic
213C3×Dic3—61no—
214C3×Dic3—123no—
2152−1+4#27211yescharacteristic, complemented
216H(5,𝔽2)—91no—
217#137—91no—
218#137—92no—
219AGL(1,ℤ/8)—92no—
220AGL(1,ℤ/8)—91no—
221#135—92no—
222#135—91no—
223C2×SD16—91no—
224C2×SD16—92no—
225#131—91no—
226H(3,𝔽3)—41noSylow
227C23×C3—181no—
228C23×C3—182no—
229C22×D3—61no—
230C22×D3—62no—
231C22×D3—126no—
232C3×Q8—32no—
233C3×Q8—31no—
234C3×Q8—31no—
235C3×Q8—32no—
236C3×Q8—31no—
237C3×Q8—62no—
238C3×Q8—61no—
239C3×Q8—91no—
240C3×Q8C22×C3⋊D311yescharacteristic
241C3×D4—31no—
242C3×D4—32no—
243C3×D4—32no—
244C3×D4—31no—
245C3×D4—31no—
246C3×D4—61no—
247C3×D4—62no—
248C3×D4—92no—
249C3×D4—91no—
250C3×D4—123no—
251C3×D4—182no—
252C3×D4—181no—
253C3×D4—182no—
254C3×D4—181no—
255C3×D4C3⋊S411yescharacteristic
256C2×C3×C4—32no—
257C2×C3×C4—31no—
258C2×C3×C4—32no—
259C2×C3×C4—31no—
260C2×C3×C4—61no—
261C2×C3×C4—62no—
262C2×C3×C4—61no—
263C2×C3×C4—62no—
264C2×C3×C4—61no—
265C2×C3×C4—182no—
266C2×C3×C4—181no—
267#67—61no—
268#67—62no—
269#67—126no—
270D12—32no—
271D12—31no—
272D12—62no—
273D12—61no—
274D12—123no—
275C4×D3—62no—
276C4×D3—61no—
277C4×D3—123no—
278Dic6—31no—
279SL(2,𝔽3)—33no—
280C3×C8—91no—
281C3×C8—91no—
282C3×C8—92no—
283C3⋊C8—31no—
284C3⋊C8—31no—
285C3⋊C8—32no—
286C2×C32—22no—
287C2×C32—43no—
288C2×C32—61no—
289C2×C32—86no—
290C2×C32C22×S411yescharacteristic
291C3×D3—126no—
292C3×D3—121no—
293C3×D3—122no—
294C3×D3—246no—
295#41—31no—
296#41—32no—
297#41—92no—
298#41—91no—
299#41—181no—
300#41C2×#27211yescharacteristic, complemented
301C2×Q8—31no—
302C2×Q8C2×#27212yescomplemented
303C2×D4—92no—
304C2×D4—91no—
305C2×D4—182no—
306C2×D4—181no—
307Q16—92no—
308Q16—91no—
309SD16—91no—
310SD16—92no—
311SD16—92no—
312SD16—92no—
313SD16—91no—
314SD16—92no—
315D8—92no—
316D8—91no—
317M16—92no—
318M16—91no—
319C2×C8—92no—
320C2×C8—91no—
321C22×C3—31no—
322C22×C3—31no—
323C22×C3—32no—
324C22×C3—61no—
325C22×C3—126no—
326C22×C3—181no—
327C22×C3—182no—
328C22×C3—722no—
329C22×C3—721no—
330C22×C3C2×C3⋊S412yes—
331C2×D3—61no—
332C2×D3—62no—
333C2×D3—123no—
334C2×D3—126no—
335C2×D3—242no—
336C2×D3—241no—
337C2×D3—2412no—
338C3×C4—31no—
339C3×C4—32no—
340C3×C4—31no—
341C3×C4—32no—
342C3×C4—31no—
343C3×C4—61no—
344C3×C4—62no—
345C3×C4—123no—
346C3×C4—181no—
347C3×C4C2×C3⋊S411yescharacteristic
348Dic3—61no—
349Dic3—123no—
350C32—43no—
351C32#592611yescharacteristic, complemented
352C23—181no—
353C23—182no—
354Q8—32no—
355Q8—31no—
356Q8—91no—
357Q8C22×#27211yescharacteristic
358D4—31no—
359D4—32no—
360D4—92no—
361D4—91no—
362D4—182no—
363D4—181no—
364D4—182no—
365D4—181no—
366D4#625811yescharacteristic
367C2×C4—32no—
368C2×C4—31no—
369C2×C4—61no—
370C2×C4—182no—
371C2×C4—181no—
372C8—91no—
373C8—91no—
374C8—92no—
375C2×C3—22no—
376C2×C3—31no—
377C2×C3—62no—
378C2×C3—61no—
379C2×C3—61no—
380C2×C3—121no—
381C2×C3—123no—
382C2×C3—246no—
383C2×C3—361no—
384C2×C3—362no—
385C2×C3C22×C3⋊S411yescharacteristic
386D3—121no—
387D3—122no—
388D3—126no—
389D3—246no—
390C22—31no—
391C22—181no—
392C22—182no—
393C22—722no—
394C22—721no—
395C22C2×#625812yes—
396C4—31no—
397C4—32no—
398C4—181no—
399C4C2×#625811yescharacteristic
400C3—31no—
401C3—123no—
402C3#1059658911yescharacteristic
403C2—22no—
404C2—61no—
405C2—361no—
406C2—362no—
407C2C22×#625811yescharacteristic
subgroupG/H
center, Frattini, socleC2×C3C22×C3⋊S4
commutator, cosocle, hypocenterQ8⋊H(3,𝔽3)C23
FittingC32×2−1+4D3
hypercenterC3×D4C3⋊S4
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. 2−1+4⋊H(3,𝔽3)C2semidirect
2. #11722394C2semidirect
3. #11722393C2semidirect
4. #11722388C2semidirect
5. C2×#88611C2semidirect
9. #41⋊H(3,𝔽3)C22semidirect
10. C2×Q8⋊H(3,𝔽3)C22semidirect
11. #88611C22semidirect
12. #88611C22semidirect
13. #88610C22semidirect
14. C32×2−1+4D3semidirect
16. C3×2−1+4⋊C3D3semidirect
41. C32×#41C2×D3semidirect
43. C2×C32×Q8C2×D3semidirect
52. C3×#41⋊C3C2×D3semidirect
53. C2×C3×SL(2,𝔽3)C2×D3semidirect
215. 2−1+4#272semidirect
300. #41C2×#272semidirect
302. C2×Q8C2×#272semidirect
351. C32#5926semidirect

Maximal subgroups

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

subgroupindex
2−1+4⋊H(3,𝔽3)2
#117223942
#117223932
#117223882
C2×#886112
C3×#59263
C3×#58373
D4×#2724