Finite Group Databasebeta

← #104×#272 #13144032

Subgroups — #104×#272 #13144032

Order 1728.

935 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 935 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#41×#272C211yesmaximal, characteristic
2M16×#272C211yesmaximal, characteristic, complemented
3C42×#272C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#104C211yesmaximal, characteristic, complemented
5#11720586C211yesmaximal, characteristic, complemented
6#11720562C211yesmaximal, characteristic, complemented
7#11720522C211yesmaximal, characteristic, complemented
8C3×D3×#104—34nomaximal
9#41×H(3,𝔽3)C2211yescharacteristic
10#88748C411yescharacteristic, complemented
11Q8×#272C411yescharacteristic, complemented
12#88744—21no—
13#88744C411yescharacteristic, complemented
14D4×#272—21no—
15D4×#272C411yescharacteristic, complemented
16#88740—21no—
17#88740C2211yescharacteristic
18C2×C4×#272—21no—
19C2×C4×#272—21no—
20C2×C4×#272C2211yescharacteristic
21M16×H(3,𝔽3)C2211yescharacteristic, complemented
22C42×H(3,𝔽3)C2211yescharacteristic, complemented
23C4×#1084—21no—
24C4×#1084C2211yescharacteristic, complemented
25#88538C2211yescharacteristic, complemented
26#88527—21no—
27C8×#272—21no—
28C3×D3×#41—34no—
29C3×D3×M16—34no—
30C3×C42×D3—34no—
31C32×#104D314yescomplemented
32C3×#837—34no—
33C3×#825—34no—
34C3×#805—34no—
35C22×#272—21no—
36C22×#272D411yescharacteristic
37Q8×H(3,𝔽3)C2×C411yescharacteristic, complemented
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)C2×C411yescharacteristic, complemented
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43#6235—21no—
44#6235—21no—
45#6235—21no—
46C2×#1084—21no—
47C2×#1084—21no—
48C2×#1084D411yescharacteristic
49C4⋊#272—21no—
50C4⋊#272C2×C411yescharacteristic, complemented
51C4×#272—24no—
52C4×#272—21no—
53C4×#272—21no—
54C4×#272—21no—
55C4×#272C2×C411yescharacteristic
56C4×#272C2×C411yescharacteristic
57C4×#272D411yescharacteristic
58C4×#272D411yescharacteristic
59#6229—21no—
60#6229C2×C411yescharacteristic, complemented
61C8×H(3,𝔽3)—21no—
62#6180—21no—
63C2×C3×#104—91no—
64D3×#104—34no—
65C32×#41C2×D314yes—
66C3×#239—34no—
67C3×D3×Q8—34no—
68C3×#237—34no—
69C3×#237—64no—
70C3×D3×D4—34no—
71C3×D3×D4—64no—
72C3×#235—34no—
73C3×#235—64no—
74C2×C3×C4×D3—34no—
75C2×C3×C4×D3—64no—
76C2×C3×C4×D3—64no—
77C32×M16C2×D314yescomplemented
78C32×C42C2×D314yescomplemented
79C3×C4×Dic3—34no—
80C3×C4×Dic3—64no—
81C3×#208—34no—
82C3×ΓL(1,𝔽25)—64no—
83C3×D3×C8—64no—
84C22×H(3,𝔽3)—21no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C2×#272—22no—
87C2×#272—21no—
88C2×#272—42no—
89C2×#272C22⋊C411yescharacteristic
90C2×#272C22⋊C411yescharacteristic
91C4×H(3,𝔽3)—22no—
92C4×H(3,𝔽3)—21no—
93C4×H(3,𝔽3)C22×C411yescharacteristic
94C4×H(3,𝔽3)C2×D411yescharacteristic
95#1084—22no—
96#1084—21no—
97#1084C22⋊C411yescharacteristic
98#1084C22⋊C411yescharacteristic
99C2×C3×#41—91no—
100D3×#41—34no—
101C2×C3×M16—91no—
102C2×C3×C42—91no—
103D3×M16—34no—
104C42×D3—34no—
105C3×#104—34no—
106C3×#104—91no—
107C3×#104—91no—
108C3×#104—91no—
109C3×#104C3⋊D311yescharacteristic
110#837—34no—
111#825—34no—
112#805—34no—
113C22×C3×D3—34no—
114C22×C3×D3—64no—
115C32×Q8C4×D314yescomplemented
116C32×D4—24no—
117C32×D4C4×D314yescomplemented
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4C22×D314yes—
121C3×#67—64no—
122C3×#67—64no—
123C3×#67—64no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3—64no—
127C3×D12—34no—
128C3×D12—64no—
129C3×C4×D3—34no—
130C3×C4×D3—34no—
131C3×C4×D3—34no—
132C3×C4×D3—34no—
133C3×C4×D3—64no—
134C3×C4×D3—64no—
135C3×C4×D3—616no—
136C3×C4×D3—64no—
137C3×Dic6—34no—
138C3×Dic6—64no—
139C32×C8—24no—
140C3×C3⋊C8—64no—
141C2×#104—91noSylow
142C2×H(3,𝔽3)—21no—
143C2×H(3,𝔽3)—41no—
144C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
145#272—21no—
146#272—41no—
147#272#10412yesdirect product factor
148C3×#41—34no—
149C3×#41—91no—
150C3×#41—91no—
151C3×#41—91no—
152C3×#41—181no—
153C3×#41—181no—
154C3×#41C2×C3⋊D311yescharacteristic
155C2×C3×Q8—91no—
156C2×C3×D4—91no—
157C2×C3×D4—181no—
158C22×C3×C4—91no—
159C22×C3×C4—181no—
160C22×C3×C4—181no—
161#239—34no—
162D3×Q8—34no—
163#237—34no—
164#237—64no—
165D3×D4—34no—
166D3×D4—64no—
167#235—34no—
168#235—64no—
169C2×C4×D3—34no—
170C2×C4×D3—64no—
171C2×C4×D3—64no—
172C3×M16—34no—
173C3×M16—91no—
174C3×M16—181no—
175C3×M16C2×C3⋊D311yescharacteristic
176C2×C3×C8—181no—
177C3×C42—34no—
178C3×C42—91no—
179C3×C42—181no—
180C3×C42C2×C3⋊D311yescharacteristic
181C4×Dic3—34no—
182C4×Dic3—64no—
183#208—34no—
184ΓL(1,𝔽25)—64no—
185D3×C8—64no—
186C22×C32—24no—
187C22×C32D3×D414yes—
188C2×C3×D3—34no—
189C2×C3×D3—34no—
190C2×C3×D3—68no—
191C2×C3×D3—64no—
192C2×C3×D3—128no—
193C32×C4—28no—
194C32×C4—24no—
195C32×C4C2×C4×D314yes—
196C32×C4D3×D414yes—
197C3×Dic3—34no—
198C3×Dic3—34no—
199C3×Dic3—68no—
200C3×Dic3—64no—
201C2×#41—91no—
202C2×M16—91no—
203C2×C42—91no—
204#104—91no—
205#104—91no—
206#104—91no—
207#104#27211yescharacteristic, direct product factor
208H(3,𝔽3)C2×#10411yescharacteristic, complemented, Sylow
209C23×C3—91no—
210C23×C3—181no—
211C22×D3—34no—
212C22×D3—64no—
213C3×Q8—34no—
214C3×Q8—91no—
215C3×Q8—181no—
216C3×Q8C4×C3⋊D311yescharacteristic
217C3×D4—21no—
218C3×D4—34no—
219C3×D4—64no—
220C3×D4—91no—
221C3×D4—181no—
222C3×D4—181no—
223C3×D4—181no—
224C3×D4—181no—
225C3×D4C4×C3⋊D311yescharacteristic
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—34no—
229C2×C3×C4—64no—
230C2×C3×C4—64no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C3×C4—91no—
234C2×C3×C4—91no—
235C2×C3×C4—91no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4—184no—
239C2×C3×C4—181no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4C22×C3⋊D311yescharacteristic
243#67—64no—
244#67—64no—
245#67—64no—
246C2×Dic3—34no—
247C2×Dic3—64no—
248C2×Dic3—64no—
249D12—34no—
250D12—64no—
251C4×D3—34no—
252C4×D3—34no—
253C4×D3—34no—
254C4×D3—34no—
255C4×D3—64no—
256C4×D3—64no—
257C4×D3—64no—
258C4×D3—616no—
259Dic6—34no—
260Dic6—64no—
261C3×C8—21no—
262C3×C8—64no—
263C3×C8—181no—
264C3⋊C8—64no—
265C2×C32—24no—
266C2×C32—44no—
267C2×C32D3×C22⋊C414yes—
268C3×D3—38no—
269C3×D3—64no—
270C3×D3—124no—
271#41—91no—
272#41—91no—
273#41—91no—
274#41—181no—
275#41—181no—
276#41C2×#27211yescharacteristic
277C2×Q8—91no—
278C2×D4—91no—
279C2×D4—181no—
280C22×C4—91no—
281C22×C4—181no—
282C22×C4—181no—
283M16—91no—
284M16—181no—
285M16C2×#27211yescharacteristic, complemented
286C2×C8—181no—
287C42—91no—
288C42—181no—
289C42C2×#27211yescharacteristic, complemented
290C22×C3—21no—
291C22×C3—34no—
292C22×C3—64no—
293C22×C3—91no—
294C22×C3—91no—
295C22×C3—182no—
296C22×C3—181no—
297C22×C3—362no—
298C22×C3D4×C3⋊D311yescharacteristic
299C2×D3—34no—
300C2×D3—34no—
301C2×D3—68no—
302C2×D3—64no—
303C2×D3—128no—
304C3×C4—22no—
305C3×C4—21no—
306C3×C4—34no—
307C3×C4—34no—
308C3×C4—64no—
309C3×C4—68no—
310C3×C4—91no—
311C3×C4—91no—
312C3×C4—182no—
313C3×C4—181no—
314C3×C4C2×C4×C3⋊D311yescharacteristic
315C3×C4D4×C3⋊D311yescharacteristic
316Dic3—34no—
317Dic3—34no—
318Dic3—64no—
319Dic3—68no—
320C32D3×#10414yescomplemented
321C23—91no—
322C23—181no—
323Q8—91no—
324Q8—181no—
325Q8C4×#27211yescharacteristic, complemented
326D4—21no—
327D4—91no—
328D4—181no—
329D4—181no—
330D4—181no—
331D4—181no—
332D4C4×#27211yescharacteristic, complemented
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—91no—
336C2×C4—91no—
337C2×C4—91no—
338C2×C4—91no—
339C2×C4—91no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—184no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—181no—
346C2×C4C22×#27211yescharacteristic
347C8—21no—
348C8—181no—
349C2×C3—21no—
350C2×C3—34no—
351C2×C3—41no—
352C2×C3—64no—
353C2×C3—92no—
354C2×C3—124no—
355C2×C3—181no—
356C2×C3—361no—
357C2×C3C22⋊C4×C3⋊D311yescharacteristic
358D3—38no—
359D3—64no—
360D3—124no—
361C22—21no—
362C22—91no—
363C22—91no—
364C22—182no—
365C22—181no—
366C22—362no—
367C22D4×#27211yescharacteristic
368C4—22no—
369C4—21no—
370C4—91no—
371C4—91no—
372C4—182no—
373C4—181no—
374C4C2×C4×#27211yescharacteristic
375C4D4×#27211yescharacteristic
376C3—34no—
377C3C3⋊D3×#10411yescharacteristic
378C2—21no—
379C2—41no—
380C2—92no—
381C2—181no—
382C2—361no—
383C2C22⋊C4×#27211yescharacteristic
subgroupG/H
centerC3×C4D4×C3⋊D3
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#104C2
socleC2×C3C22⋊C4×C3⋊D3
prime radical#41×#272C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×#104C3⋊D3
hypocenterH(3,𝔽3)C2×#104
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×#104

Complements

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

NH ≅ G/Ntype
2. M16×#272C2semidirect
3. C42×#272C2semidirect
4. H(3,𝔽3)×#104C2semidirect
5. #11720586C2semidirect
6. #11720562C2semidirect
7. #11720522C2semidirect
10. #88748C4semidirect
11. Q8×#272C4semidirect
13. #88744C4semidirect
15. D4×#272C4semidirect
21. M16×H(3,𝔽3)C22semidirect
22. C42×H(3,𝔽3)C22semidirect
24. C4×#1084C22semidirect
25. #88538C22semidirect
31. C32×#104D3semidirect
37. Q8×H(3,𝔽3)C2×C4semidirect
39. D4×H(3,𝔽3)C2×C4semidirect
50. C4⋊#272C2×C4semidirect
60. #6229C2×C4semidirect
77. C32×M16C2×D3semidirect
78. C32×C42C2×D3semidirect
115. C32×Q8C4×D3semidirect
117. C32×D4C4×D3semidirect
147. #272#104direct (N × H)
207. #104#272direct (N × H)
208. H(3,𝔽3)C2×#104semidirect
285. M16C2×#272semidirect
289. C42C2×#272semidirect
320. C32D3×#104semidirect
325. Q8C4×#272semidirect
332. D4C4×#272semidirect

Maximal subgroups

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

subgroupindex
#41×#2722
M16×#2722
C42×#2722
H(3,𝔽3)×#1042
#117205862
#117205622
#117205222
C3×D3×#1043