Finite Group Databasebeta

← #13144033

Subgroups — #13144033

Order 1728.

847 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 847 conjugacy classes of subgroups — too many to draw. Only its 111 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
3#11721718C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#104C211yesmaximal, characteristic, complemented
5#11720586C211yesmaximal, characteristic, complemented
6#11720562C211yesmaximal, characteristic, complemented
7#11720522C211yesmaximal, characteristic, complemented
8C3×#4821—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×#272C2211yescharacteristic
20M16×H(3,𝔽3)C2211yescharacteristic, complemented
21C42×H(3,𝔽3)C2211yescharacteristic, complemented
22#88542—21no—
23#88540—21no—
24C4×#1084C2211yescharacteristic, complemented
25#88538C2211yescharacteristic, complemented
26#88527—21no—
27C8×#272—21no—
28C3×D3×#41—34no—
29C3×D3×M16—34no—
30C3×#872—34no—
31C32×#104D314yescomplemented
32C3×#837—34no—
33C3×#825—34no—
34C3×#805—34no—
35C22×#272—21no—
36C22×#272D411yescharacteristic
37Q8×H(3,𝔽3)C2×C411yescharacteristic
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)C2×C411yescharacteristic
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
51C4×#272—21no—
52C4×#272—21no—
53C4×#272—21no—
54C4×#272C2×C411yescharacteristic
55C4×#272C2×C411yescharacteristic
56C4×#272D411yescharacteristic
57C4×#272D411yescharacteristic
58#6229—21no—
59#6229C2×C411yescharacteristic
60C8×H(3,𝔽3)—21no—
61#6180—21no—
62C3×#421—91no—
63#4821—34no—
64C32×#41C2×D314yes—
65C3×#239—34no—
66C3×D3×Q8—34no—
67C3×#237—34no—
68C3×#237—64no—
69C3×D3×D4—34no—
70C3×D3×D4—64no—
71C3×#235—34no—
72C3×#235—64no—
73C2×C3×C4×D3—34no—
74C2×C3×C4×D3—64no—
75C32×M16C2×D314yescomplemented
76C32×C42C2×D314yescomplemented
77C3×(C2×D3)⋊C4—64no—
78C3×Dic3⋊C4—64no—
79C3×C4×Dic3—34no—
80C3×#208—34no—
81C3×ΓL(1,𝔽25)—64no—
82C3×D3×C8—64no—
83C22×H(3,𝔽3)—21no—
84C22×H(3,𝔽3)C2×D411yescharacteristic
85C2×#272—21no—
86C2×#272—41no—
87C2×#272—42no—
88C2×#272C22⋊C411yescharacteristic
89C2×#272C22⋊C411yescharacteristic
90C4×H(3,𝔽3)—21no—
91C4×H(3,𝔽3)—41no—
92C4×H(3,𝔽3)C22×C411yescharacteristic
93C4×H(3,𝔽3)C2×D411yescharacteristic
94#1084—21no—
95#1084—41no—
96#1084C22⋊C411yescharacteristic
97#1084C22⋊C411yescharacteristic
98C2×C3×#41—91no—
99D3×#41—34no—
100C2×C3×M16—91no—
101C3×#117—91no—
102D3×M16—34no—
103#872—34no—
104C3×#104—34no—
105C3×#104—91no—
106C3×#104—91no—
107C3×#104—91no—
108C3×#104C3⋊D311yescharacteristic
109#837—34no—
110#825—34no—
111#805—34no—
112C22×C3×D3—34no—
113C22×C3×D3—64no—
114C32×Q8C4×D314yes—
115C32×D4—24no—
116C32×D4C4×D314yes—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4C22×D314yes—
120C3×#67—64no—
121C3×#67—64no—
122C3×#67—64no—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—64no—
125C2×C3×Dic3—64no—
126C3×D12—34no—
127C3×D12—64no—
128C3×C4×D3—34no—
129C3×C4×D3—34no—
130C3×C4×D3—34no—
131C3×C4×D3—34no—
132C3×C4×D3—64no—
133C3×C4×D3—64no—
134C3×C4×D3—64no—
135C3×Dic6—34no—
136C3×Dic6—64no—
137C32×C8—24no—
138C3×C3⋊C8—64no—
139#421—91noSylow
140C2×H(3,𝔽3)—21no—
141C2×H(3,𝔽3)—41no—
142C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
143#272—21no—
144#272—21no—
145#272—41no—
146C3×#41—34no—
147C3×#41—91no—
148C3×#41—91no—
149C3×#41—91no—
150C3×#41—181no—
151C3×#41—181no—
152C3×#41C2×C3⋊D311yescharacteristic
153C2×C3×Q8—91no—
154C2×C3×D4—91no—
155C2×C3×D4—181no—
156C22×C3×C4—91no—
157C22×C3×C4—181no—
158#239—34no—
159D3×Q8—34no—
160#237—34no—
161#237—64no—
162D3×D4—34no—
163D3×D4—64no—
164#235—34no—
165#235—64no—
166C2×C4×D3—34no—
167C2×C4×D3—64no—
168C3×M16—34no—
169C3×M16—91no—
170C3×M16—181no—
171C3×M16C2×C3⋊D311yescharacteristic
172C2×C3×C8—181no—
173C3×C4⋊C4—181no—
174C3×C22⋊C4—181no—
175C3×C42—34no—
176C3×C42—91no—
177C3×C42C2×C3⋊D311yescharacteristic
178(C2×D3)⋊C4—64no—
179Dic3⋊C4—64no—
180C4×Dic3—34no—
181#208—34no—
182ΓL(1,𝔽25)—64no—
183D3×C8—64no—
184C22×C32—24no—
185C22×C32D3×D414yes—
186C2×C3×D3—34no—
187C2×C3×D3—34no—
188C2×C3×D3—64no—
189C2×C3×D3—124no—
190C2×C3×D3—128no—
191C32×C4—24no—
192C32×C4—44no—
193C32×C4C2×C4×D314yes—
194C32×C4D3×D414yes—
195C3×Dic3—34no—
196C3×Dic3—34no—
197C3×Dic3—64no—
198C3×Dic3—124no—
199C2×#41—91no—
200C2×M16—91no—
201#117—91no—
202#104—91no—
203#104—91no—
204#104—91no—
205#104#27211yescharacteristic, complemented
206H(3,𝔽3)#42111yescharacteristic, complemented, Sylow
207C23×C3—91no—
208C23×C3—181no—
209C22×D3—34no—
210C22×D3—64no—
211C3×Q8—34no—
212C3×Q8—91no—
213C3×Q8—181no—
214C3×Q8C4×C3⋊D311yescharacteristic
215C3×D4—21no—
216C3×D4—34no—
217C3×D4—64no—
218C3×D4—91no—
219C3×D4—181no—
220C3×D4—181no—
221C3×D4—181no—
222C3×D4—181no—
223C3×D4C4×C3⋊D311yescharacteristic
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—34no—
227C2×C3×C4—64no—
228C2×C3×C4—64no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C3×C4—91no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4C22×C3⋊D311yescharacteristic
240#67—64no—
241#67—64no—
242#67—64no—
243C2×Dic3—34no—
244C2×Dic3—64no—
245C2×Dic3—64no—
246D12—34no—
247D12—64no—
248C4×D3—34no—
249C4×D3—34no—
250C4×D3—34no—
251C4×D3—34no—
252C4×D3—64no—
253C4×D3—64no—
254C4×D3—64no—
255Dic6—34no—
256Dic6—64no—
257C3×C8—21no—
258C3×C8—64no—
259C3×C8—181no—
260C3⋊C8—64no—
261C2×C32—24no—
262C2×C32—44no—
263C2×C32D3×C22⋊C414yes—
264C3×D3—64no—
265C3×D3—64no—
266C3×D3—124no—
267#41—91no—
268#41—91no—
269#41—91no—
270#41—181no—
271#41—181no—
272#41C2×#27211yescharacteristic
273C2×Q8—91no—
274C2×D4—91no—
275C2×D4—181no—
276C22×C4—91no—
277C22×C4—181no—
278M16—91no—
279M16—181no—
280M16C2×#27211yescharacteristic, complemented
281C2×C8—181no—
282C4⋊C4—181no—
283C22⋊C4—181no—
284C42—91no—
285C42C2×#27211yescharacteristic, complemented
286C22×C3—21no—
287C22×C3—34no—
288C22×C3—64no—
289C22×C3—91no—
290C22×C3—91no—
291C22×C3—181no—
292C22×C3—361no—
293C22×C3—362no—
294C22×C3D4×C3⋊D311yescharacteristic
295C2×D3—34no—
296C2×D3—34no—
297C2×D3—64no—
298C2×D3—124no—
299C2×D3—128no—
300C3×C4—21no—
301C3×C4—34no—
302C3×C4—34no—
303C3×C4—41no—
304C3×C4—64no—
305C3×C4—91no—
306C3×C4—91no—
307C3×C4—124no—
308C3×C4—181no—
309C3×C4—361no—
310C3×C4C2×C4×C3⋊D311yescharacteristic
311C3×C4D4×C3⋊D311yescharacteristic
312Dic3—34no—
313Dic3—34no—
314Dic3—64no—
315Dic3—124no—
316C32#482114yescomplemented
317C23—91no—
318C23—181no—
319Q8—91no—
320Q8—181no—
321Q8C4×#27211yescharacteristic
322D4—21no—
323D4—91no—
324D4—181no—
325D4—181no—
326D4—181no—
327D4—181no—
328D4C4×#27211yescharacteristic
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—91no—
332C2×C4—91no—
333C2×C4—91no—
334C2×C4—91no—
335C2×C4—91no—
336C2×C4—181no—
337C2×C4—181no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4C22×#27211yescharacteristic
342C8—21no—
343C8—181no—
344C2×C3—21no—
345C2×C3—34no—
346C2×C3—41no—
347C2×C3—64no—
348C2×C3—124no—
349C2×C3—181no—
350C2×C3—181no—
351C2×C3—361no—
352C2×C3C22⋊C4×C3⋊D311yescharacteristic
353D3—64no—
354D3—64no—
355D3—124no—
356C22—21no—
357C22—91no—
358C22—91no—
359C22—181no—
360C22—361no—
361C22—362no—
362C22D4×#27211yescharacteristic
363C4—21no—
364C4—41no—
365C4—91no—
366C4—91no—
367C4—181no—
368C4—361no—
369C4C2×C4×#27211yescharacteristic
370C4D4×#27211yescharacteristic
371C3—34no—
372C3#1059252011yescharacteristic
373C2—21no—
374C2—41no—
375C2—181no—
376C2—181no—
377C2—361no—
378C2C22⋊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)#421
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. #11721718C2semidirect
4. H(3,𝔽3)×#104C2semidirect
5. #11720586C2semidirect
6. #11720562C2semidirect
7. #11720522C2semidirect
10. #88748C4semidirect
11. Q8×#272C4semidirect
13. #88744C4semidirect
15. D4×#272C4semidirect
20. M16×H(3,𝔽3)C22semidirect
21. C42×H(3,𝔽3)C22semidirect
24. C4×#1084C22semidirect
25. #88538C22semidirect
31. C32×#104D3semidirect
75. C32×M16C2×D3semidirect
76. C32×C42C2×D3semidirect
205. #104#272semidirect
206. H(3,𝔽3)#421semidirect
280. M16C2×#272semidirect
285. C42C2×#272semidirect
316. C32#4821semidirect

Maximal subgroups

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

subgroupindex
#41×#2722
M16×#2722
#117217182
H(3,𝔽3)×#1042
#117205862
#117205622
#117205222
C3×#48213