Finite Group Databasebeta

← #13143944

Subgroups — #13143944

Order 1728.

1089 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1089 conjugacy classes of subgroups — too many to draw. Only its 115 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×D4×#272C211yesmaximal, characteristic, complemented
2C2×#88545C211yesmaximal, characteristic, complemented
3C2×#88529C211yesmaximal, characteristic, complemented
4#11721763C211yesmaximal, characteristic, complemented
5H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
6#11720553C211yesmaximal, characteristic, complemented
7#11720531C211yesmaximal, characteristic, complemented
8C3×#4773—34nomaximal
9C23×#272—21no—
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12D4×#272—22no—
13D4×#272—22no—
14C2×C4⋊#272—21no—
15C2×C4⋊#272C2211yescharacteristic
16C2×C4⋊#272C2211yescharacteristic, complemented
17C2×C4×#272C2211yescharacteristic
18C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
20#88545—22no—
21#88542—21no—
22C2×#6180C2211yescharacteristic, complemented
23#88529—22no—
24C2×C3×D3×D4—34no—
25C2×C3×#213—34no—
26C2×C3×D24—34no—
27C3×#895—34no—
28C32×#102D314yescomplemented
29C3×(C2×D3)⋊C8—34no—
30C3×#809—34no—
31C23×H(3,𝔽3)—21no—
32C22×#272—24no—
33C22×#272—21no—
34C22×#272—21no—
35C22×#272—42no—
36C22×#272—42no—
37C22×#272D411yescharacteristic
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)D412yescomplemented
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42#6235—24no—
43C2×#1084D411yescharacteristic
44C4⋊#272—21no—
45C4⋊#272—21no—
46C4⋊#272—21no—
47C4⋊#272—42no—
48C4⋊#272D412yescomplemented
49C4×#272—21no—
50C4×#272—21no—
51C8×H(3,𝔽3)—21no—
52#6180—21no—
53C3×#447—91no—
54#4773—34no—
55C23×C3×D3—64no—
56C2×C32×D4C2×D314yes—
57C2×C3×#67—64no—
58C3×D3×D4—68no—
59C3×D3×D4—68no—
60C2×C3×D12—34no—
61C2×C3×D12—34no—
62C2×C3×D12—64no—
63C2×C3×C4×D3—34no—
64C2×C32×C8C2×D314yescomplemented
65C32×C4⋊C4C2×D314yescomplemented
66C3×#213—68no—
67C3×(C2×D3)⋊C4—64no—
68C2×C3×C3⋊C8—34no—
69C3×D24—68no—
70C22×H(3,𝔽3)—22no—
71C22×H(3,𝔽3)—41no—
72C22×H(3,𝔽3)—41no—
73C22×H(3,𝔽3)C2×D411yescharacteristic
74C2×#272—21no—
75C2×#272—21no—
76C2×#272—22no—
77C2×#272—48no—
78C2×#272—41no—
79C2×#272—41no—
80C2×#272—41no—
81C2×#272—41no—
82C2×#272—41no—
83C2×#272D812yescomplemented
84C4×H(3,𝔽3)—41no—
85C4×H(3,𝔽3)C2×D411yescharacteristic
86C4×H(3,𝔽3)C2×D411yescharacteristic
87#1084—21no—
88C22×C3×D4—91no—
89C2×D3×D4—34no—
90C2×C3×D8—91no—
91C2×C3×D8—91no—
92C3×#121—91no—
93C2×#213—34no—
94C2×D24—34no—
95#895—34no—
96C3×#102—34no—
97C3×#102—91no—
98C3×#102C3⋊D311yescharacteristic
99C3×C22⋊C8—91no—
100(C2×D3)⋊C8—34no—
101#809—34no—
102C23×C32—24no—
103C22×C3×D3—34no—
104C22×C3×D3—616no—
105C22×C3×D3—64no—
106C22×C3×D3—64no—
107C22×C3×D3—128no—
108C22×C3×D3—128no—
109C32×D4—24no—
110C32×D4D1218yescomplemented
111C2×C32×C4—24no—
112C2×C32×C4C22×D314yes—
113C3×#67—616no—
114C2×C3×Dic3—34no—
115C3×D12—38no—
116C3×D12—64no—
117C3×D12—64no—
118C3×D12—64no—
119C3×D12—128no—
120C3×C4×D3—64no—
121C3×C4×D3—64no—
122C32×C8—24no—
123C3×C3⋊C8—64no—
124#447—91noSylow
125C2×H(3,𝔽3)—42no—
126C2×H(3,𝔽3)#12011yescharacteristic
127C2×H(3,𝔽3)C2×D811yescharacteristic
128C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
129#272—22no—
130#272—42no—
131#272—81no—
132C24×C3—181no—
133C23×D3—64no—
134C2×C3×D4—34no—
135C2×C3×D4—91no—
136C2×C3×D4—91no—
137C2×C3×D4—181no—
138C2×C3×D4—181no—
139C2×C3×D4—182no—
140C2×C3×D4—182no—
141C2×C3×D4C2×C3⋊D311yescharacteristic
142C22×C3×C4—91no—
143C2×#67—64no—
144D3×D4—68no—
145D3×D4—68no—
146C2×D12—34no—
147C2×D12—34no—
148C2×D12—64no—
149C2×C4×D3—34no—
150C3×D8—182no—
151C3×D8—182no—
152C2×C3×C8—34no—
153C2×C3×C8—91no—
154C2×C3×C8C2×C3⋊D311yescharacteristic
155C3×C4⋊C4—34no—
156C3×C4⋊C4C2×C3⋊D311yescharacteristic
157C3×C22⋊C4—181no—
158#213—68no—
159(C2×D3)⋊C4—64no—
160C2×C3⋊C8—34no—
161D24—68no—
162C22×C32—28no—
163C22×C32—44no—
164C22×C32—44no—
165C22×C32D3×D414yes—
166C2×C3×D3—38no—
167C2×C3×D3—64no—
168C2×C3×D3—64no—
169C2×C3×D3—68no—
170C2×C3×D3—1232no—
171C2×C3×D3—124no—
172C2×C3×D3—124no—
173C2×C3×D3—124no—
174C2×C3×D3—124no—
175C2×C3×D3—124no—
176C32×C4—44no—
177C32×C4C2×D1214yes—
178C32×C4D3×D414yes—
179C3×Dic3—64no—
180C22×D4—91no—
181C2×D8—91no—
182C2×D8—91no—
183#121—91no—
184#102—91no—
185#102#27211yescharacteristic, complemented
186C22⋊C8—91no—
187H(3,𝔽3)#44711yescharacteristic, complemented, Sylow
188C23×C3—21no—
189C23×C3—64no—
190C23×C3—91no—
191C23×C3—184no—
192C23×C3—181no—
193C23×C3—181no—
194C23×C3—362no—
195C23×C3—362no—
196C22×D3—34no—
197C22×D3—64no—
198C22×D3—616no—
199C22×D3—64no—
200C22×D3—128no—
201C22×D3—128no—
202C3×D4—21no—
203C3×D4—38no—
204C3×D4—64no—
205C3×D4—92no—
206C3×D4—184no—
207C3×D4—181no—
208C3×D4—181no—
209C3×D4—181no—
210C3×D4—362no—
211C3×D4C4⋊(C3⋊D3)12yes—
212C2×C3×C4—21no—
213C2×C3×C4—34no—
214C2×C3×C4—64no—
215C2×C3×C4—91no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218C2×C3×C4C22×C3⋊D311yescharacteristic
219#67—616no—
220C2×Dic3—34no—
221D12—38no—
222D12—64no—
223D12—64no—
224D12—64no—
225D12—128no—
226C4×D3—64no—
227C4×D3—64no—
228C3×C8—21no—
229C3×C8—64no—
230C3×C8—181no—
231C3⋊C8—64no—
232C2×C32—48no—
233C2×C32#88214yes—
234C2×C32D3×D814yes—
235C2×C32#91414yes—
236C3×D3—68no—
237C3×D3—128no—
238C3×D3—244no—
239C24—181no—
240C2×D4—91no—
241C2×D4—91no—
242C2×D4—181no—
243C2×D4—181no—
244C2×D4—182no—
245C2×D4—182no—
246C2×D4C2×#27211yescharacteristic
247C22×C4—91no—
248D8—182no—
249D8—182no—
250C2×C8—91no—
251C2×C8C2×#27211yescharacteristic, complemented
252C4⋊C4C2×#27211yescharacteristic, complemented
253C22⋊C4—181no—
254C22×C3—22no—
255C22×C3—34no—
256C22×C3—41no—
257C22×C3—41no—
258C22×C3—68no—
259C22×C3—92no—
260C22×C3—124no—
261C22×C3—124no—
262C22×C3—181no—
263C22×C3—181no—
264C22×C3—182no—
265C22×C3—368no—
266C22×C3—361no—
267C22×C3—361no—
268C22×C3—361no—
269C22×C3—361no—
270C22×C3—361no—
271C22×C3D4×C3⋊D311yescharacteristic
272C2×D3—38no—
273C2×D3—64no—
274C2×D3—64no—
275C2×D3—68no—
276C2×D3—1232no—
277C2×D3—124no—
278C2×D3—124no—
279C2×D3—124no—
280C2×D3—124no—
281C2×D3—124no—
282C3×C4—34no—
283C3×C4—34no—
284C3×C4—41no—
285C3×C4—124no—
286C3×C4—181no—
287C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
288C3×C4D4×C3⋊D311yescharacteristic
289Dic3—64no—
290C32#477314yescomplemented
291C23—21no—
292C23—91no—
293C23—184no—
294C23—181no—
295C23—181no—
296C23—362no—
297C23—362no—
298D4—21no—
299D4—92no—
300D4—184no—
301D4—181no—
302D4—181no—
303D4—181no—
304D4—362no—
305D4C4⋊#27212yescomplemented
306C2×C4—21no—
307C2×C4—91no—
308C2×C4—181no—
309C2×C4—181no—
310C2×C4C22×#27211yescharacteristic
311C8—21no—
312C8—181no—
313C2×C3—34no—
314C2×C3—34no—
315C2×C3—34no—
316C2×C3—42no—
317C2×C3—128no—
318C2×C3—182no—
319C2×C3—362no—
320C2×C3—721no—
321C2×C3#6409111yescharacteristic
322C2×C3D8×C3⋊D311yescharacteristic
323C2×C3#6412311yescharacteristic
324D3—68no—
325D3—128no—
326D3—244no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—92no—
331C22—181no—
332C22—181no—
333C22—182no—
334C22—368no—
335C22—361no—
336C22—361no—
337C22—361no—
338C22—361no—
339C22—361no—
340C22D4×#27211yescharacteristic
341C4—41no—
342C4—181no—
343C4C2×C4⋊#27211yescharacteristic
344C4D4×#27211yescharacteristic
345C3—34no—
346C3#1059247211yescharacteristic
347C2—42no—
348C2—182no—
349C2—362no—
350C2—721no—
351C2#1172173811yescharacteristic
352C2D8×#27211yescharacteristic
353C2#1172180011yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#102C2
hypercenterC3×#102C3⋊D3
hypocenterH(3,𝔽3)#447
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
1. C2×D4×#272C2semidirect
2. C2×#88545C2semidirect
3. C2×#88529C2semidirect
4. #11721763C2semidirect
5. H(3,𝔽3)×#102C2semidirect
6. #11720553C2semidirect
7. #11720531C2semidirect
16. C2×C4⋊#272C22semidirect
18. C2×C8×H(3,𝔽3)C22semidirect
19. C4⋊C4×H(3,𝔽3)C22semidirect
22. C2×#6180C22semidirect
28. C32×#102D3semidirect
39. D4×H(3,𝔽3)D4semidirect
48. C4⋊#272D4semidirect
64. C2×C32×C8C2×D3semidirect
65. C32×C4⋊C4C2×D3semidirect
83. C2×#272D8semidirect
110. C32×D4D12semidirect
185. #102#272semidirect
187. H(3,𝔽3)#447semidirect
251. C2×C8C2×#272semidirect
252. C4⋊C4C2×#272semidirect
290. C32#4773semidirect
305. D4C4⋊#272semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
C2×#885452
C2×#885292
#117217632
H(3,𝔽3)×#1022
#117205532
#117205312
C3×#47733