Finite Group Databasebeta

← #13154225

Subgroups — #13154225

Order 1728.

1254 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#11721765C212yesmaximal, complemented
3#11721764C212yesmaximal, complemented
4#11721764C212yesmaximal, complemented
5#11721761C211yesmaximal, characteristic
6#11721761C211yesmaximal, characteristic
7C4⋊C4×#272C212yesmaximal, complemented
8#11721758C211yesmaximal, characteristic, complemented
9#11721756C211yesmaximal, characteristic, complemented
10C4×C4⋊#272C211yesmaximal, characteristic
11C4×#6229C211yesmaximal, characteristic, complemented
12C3×#5690—34nomaximal
13C2×C4⋊#272C2211yescharacteristic
14C2×C4×#272C2212yes—
15C2×C4×#272C2212yes—
16C2×C4×#272C2212yes—
17C2×#6229C2211yescharacteristic
18C2×#6229C2212yes—
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C4⋊C4×H(3,𝔽3)C2212yes—
22C4⋊C4×H(3,𝔽3)C2212yes—
23C42×H(3,𝔽3)C2211yescharacteristic
24#88542C2212yes—
25#88542C2212yes—
26#88542C2212yes—
27C4⋊#1084C2211yescharacteristic
28C4⋊#1084C2212yes—
29C4⋊#1084C2211yescharacteristic
30#88540C2212yes—
31#88540C2212yes—
32#88540C2211yescharacteristic
33#88540C2211yescharacteristic
34C4×#1084C2211yescharacteristic
35C4×#1084C2211yescharacteristic
36C32×#125D314yescomplemented
37C3×#897—38no—
38C3×#896—38no—
39C3×#896—38no—
40C3×#893—34no—
41C3×#893—34no—
42C3×D3×C4⋊C4—38no—
43C3×#890—34no—
44C3×#888—34no—
45C3×C4×D12—34no—
46C3×C4×Dic6—34no—
47C22×#272C2312yes—
48C2×C4×H(3,𝔽3)C2312yes—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×C4×H(3,𝔽3)C2312yes—
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53C2×#1084C2311yescharacteristic
54C2×#1084C2312yes—
55C2×#1084C2311yescharacteristic
56C2×#1084C2311yescharacteristic
57C2×#1084C2311yescharacteristic
58C4⋊#272Q814yescomplemented
59C4×#272—24no—
60C4×#272—42no—
61C4×#272—42no—
62#6229—22no—
63#6229—42no—
64C3×#554—91no—
65#5690—34no—
66C2×C3×D12—34no—
67C2×C3×C4×D3—38no—
68C2×C3×C4×D3—38no—
69C2×C3×C4×D3—38no—
70C2×C3×Dic6—34no—
71C2×C3×Dic6—38no—
72C32×C4⋊C4C2×D314yes—
73C32×C4⋊C4C2×D314yes—
74C32×C4⋊C4C2×D318yes—
75C32×C4⋊C4C2×D318yes—
76C32×C42C2×D314yes—
77C3×(C2×D3)⋊C4—38no—
78C3×(C2×D3)⋊C4—38no—
79C3×(C2×D3)⋊C4—38no—
80C3×C4⋊Dic3—34no—
81C3×C4⋊Dic3—38no—
82C3×C4⋊Dic3—34no—
83C3×Dic3⋊C4—38no—
84C3×Dic3⋊C4—38no—
85C3×Dic3⋊C4—34no—
86C3×Dic3⋊C4—34no—
87C3×C4×Dic3—34no—
88C3×C4×Dic3—34no—
89C22×H(3,𝔽3)C2411yescharacteristic
90C2×#272—22no—
91C2×#272—22no—
92C2×#272C2×Q814yes—
93C4×H(3,𝔽3)—21no—
94C4×H(3,𝔽3)—22no—
95C4×H(3,𝔽3)—22no—
96C4×H(3,𝔽3)—21no—
97C4×H(3,𝔽3)C2×Q812yes—
98#1084—21no—
99#1084—21no—
100#1084—22no—
101#1084—21no—
102#1084#4112yes—
103C3×C4⋊Q8—91no—
104C3×#125—34no—
105C3×#125—91no—
106C3×#125C3⋊D311yescharacteristic
107C3×C22⋊Q8—92no—
108C3×C22⋊Q8—92no—
109C3×C22⋊Q8—92no—
110C3×C4×Q8—91no—
111C3×C4×D4—91no—
112C3×C4×D4—91no—
113C3×C4×D4—91no—
114C2×C3×C4⋊C4—92no—
115#897—38no—
116#896—38no—
117#896—38no—
118#893—34no—
119#893—34no—
120D3×C4⋊C4—38no—
121#890—34no—
122#888—34no—
123C4×D12—34no—
124C4×Dic6—34no—
125C22×C3×D3—38no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D318yes—
128C2×C32×C4C22×D318yes—
129C2×C32×C4C22×D314yes—
130C2×C32×C4C22×D314yes—
131C2×C3×Dic3—34no—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—38no—
136C3×D12—316no—
137C3×C4×D3—616no—
138C3×C4×D3—128no—
139C3×C4×D3—128no—
140C3×Dic6—68no—
141C3×Dic6—128no—
142#554—91noSylow
143C2×H(3,𝔽3)C22×Q811yescharacteristic
144C2×H(3,𝔽3)C2×#4111yescharacteristic
145C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
146#272—24no—
147C2×C3×Q8—91no—
148C2×C3×Q8—92no—
149C2×C3×D4—91no—
150C22×C3×C4—92no—
151C22×C3×C4—92no—
152C22×C3×C4—92no—
153C2×D12—34no—
154C2×C4×D3—38no—
155C2×C4×D3—38no—
156C2×C4×D3—38no—
157C2×Dic6—34no—
158C2×Dic6—38no—
159C3×C4⋊C4—34no—
160C3×C4⋊C4—34no—
161C3×C4⋊C4—38no—
162C3×C4⋊C4—38no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—92no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4—92no—
167C3×C4⋊C4—92no—
168C3×C4⋊C4—91no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4C2×C3⋊D312yes—
171C3×C4⋊C4C2×C3⋊D311yescharacteristic
172C3×C4⋊C4C2×C3⋊D311yescharacteristic
173C3×C4⋊C4C2×C3⋊D312yes—
174C3×C22⋊C4—92no—
175C3×C22⋊C4—92no—
176C3×C22⋊C4—92no—
177C3×C42—34no—
178C3×C42—91no—
179C3×C42—91no—
180C3×C42C2×C3⋊D311yescharacteristic
181(C2×D3)⋊C4—38no—
182(C2×D3)⋊C4—38no—
183(C2×D3)⋊C4—38no—
184C4⋊Dic3—34no—
185C4⋊Dic3—34no—
186C4⋊Dic3—38no—
187Dic3⋊C4—34no—
188Dic3⋊C4—34no—
189Dic3⋊C4—38no—
190Dic3⋊C4—38no—
191C4×Dic3—34no—
192C4×Dic3—34no—
193C22×C32C23×D314yes—
194C2×C3×D3—316no—
195C2×C3×D3—68no—
196C2×C3×D3—68no—
197C32×C4—24no—
198C32×C4—28no—
199C32×C4—28no—
200C32×C4—24no—
201C32×C4D3×Q818yes—
202C3×Dic3—38no—
203C3×Dic3—64no—
204C3×Dic3—64no—
205C3×Dic3—68no—
206C3×Dic3—64no—
207C4⋊Q8—91no—
208#125—91no—
209#125#27211yescharacteristic, complemented
210C22⋊Q8—92no—
211C22⋊Q8—92no—
212C22⋊Q8—92no—
213C4×Q8—91no—
214C4×D4—91no—
215C4×D4—91no—
216C4×D4—91no—
217C2×C4⋊C4—92no—
218H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
219C23×C3—92no—
220C22×D3—38no—
221C3×Q8—182no—
222C3×Q8—362no—
223C3×D4—94no—
224C2×C3×C4—34no—
225C2×C3×C4—34no—
226C2×C3×C4—38no—
227C2×C3×C4—38no—
228C2×C3×C4—34no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C3×C4—92no—
234C2×C3×C4—184no—
235C2×C3×C4—362no—
236C2×C3×C4—362no—
237C2×C3×C4C22×C3⋊D311yescharacteristic
238C2×C3×C4C22×C3⋊D311yescharacteristic
239C2×C3×C4C22×C3⋊D312yes—
240C2×C3×C4C22×C3⋊D312yes—
241C2×C3×C4C22×C3⋊D311yescharacteristic
242C2×Dic3—34no—
243C2×Dic3—34no—
244C2×Dic3—34no—
245C2×Dic3—34no—
246C2×Dic3—38no—
247D12—316no—
248C4×D3—616no—
249C4×D3—128no—
250C4×D3—128no—
251Dic6—68no—
252Dic6—128no—
253C2×C32C2×D3×Q814yes—
254C2×C32D3×#4114yes—
255C2×C32#100914yes—
256C3×D3—616no—
257C2×Q8—91no—
258C2×Q8—92no—
259C2×D4—91no—
260C22×C4—92no—
261C22×C4—92no—
262C22×C4—92no—
263C4⋊C4—91no—
264C4⋊C4—92no—
265C4⋊C4—92no—
266C4⋊C4—91no—
267C4⋊C4—91no—
268C4⋊C4—91no—
269C4⋊C4—92no—
270C4⋊C4C2×#27212yes—
271C4⋊C4C2×#27211yescharacteristic
272C4⋊C4C2×#27211yescharacteristic
273C4⋊C4C2×#27212yes—
274C22⋊C4—92no—
275C22⋊C4—92no—
276C22⋊C4—92no—
277C42—91no—
278C42—91no—
279C42C2×#27211yescharacteristic
280C22×C3—34no—
281C22×C3—94no—
282C22×C3—182no—
283C22×C3—182no—
284C22×C3C23×C3⋊D311yescharacteristic
285C2×D3—316no—
286C2×D3—68no—
287C2×D3—68no—
288C3×C4—22no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—22no—
292C3×C4—38no—
293C3×C4—64no—
294C3×C4—68no—
295C3×C4—68no—
296C3×C4—64no—
297C3×C4—92no—
298C3×C4—181no—
299C3×C4—181no—
300C3×C4—182no—
301C3×C4—181no—
302C3×C4Q8×C3⋊D312yes—
303Dic3—38no—
304Dic3—64no—
305Dic3—64no—
306Dic3—68no—
307Dic3—64no—
308C32#569014yescomplemented
309C23—92no—
310Q8—182no—
311Q8—362no—
312D4—94no—
313C2×C4—91no—
314C2×C4—91no—
315C2×C4—91no—
316C2×C4—91no—
317C2×C4—92no—
318C2×C4—184no—
319C2×C4—362no—
320C2×C4—362no—
321C2×C4C22×#27211yescharacteristic
322C2×C4C22×#27211yescharacteristic
323C2×C4C22×#27212yes—
324C2×C4C22×#27212yes—
325C2×C4C22×#27211yescharacteristic
326C2×C3—34no—
327C2×C3—34no—
328C2×C3—34no—
329C2×C3—184no—
330C2×C3C2×Q8×C3⋊D311yescharacteristic
331C2×C3#41×C3⋊D311yescharacteristic
332C2×C3#6436611yescharacteristic
333D3—616no—
334C22—94no—
335C22—182no—
336C22—182no—
337C22C23×#27211yescharacteristic
338C4—22no—
339C4—21no—
340C4—21no—
341C4—22no—
342C4—92no—
343C4—181no—
344C4—181no—
345C4—182no—
346C4—181no—
347C4Q8×#27212yes—
348C3—34no—
349C3#1059616111yescharacteristic
350C2—184no—
351C2C2×Q8×#27211yescharacteristic
352C2#41×#27211yescharacteristic
353C2#1172267511yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×#125C2
prime radicalC2×C4⋊#272C22
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#554
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
2. #11721765C2semidirect
3. #11721764C2semidirect
4. #11721764C2semidirect
7. C4⋊C4×#272C2semidirect
8. #11721758C2semidirect
9. #11721756C2semidirect
11. C4×#6229C2semidirect
36. C32×#125D3semidirect
58. C4⋊#272Q8semidirect
209. #125#272semidirect
218. H(3,𝔽3)#554semidirect
308. C32#5690semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#117217652
#117217642
#117217612
C4⋊C4×#2722
#117217582
#117217562
C4×C4⋊#2722
C4×#62292
C3×#56903