Finite Group Databasebeta

← #13140098

Subgroups — #13140098

Order 1728.

617 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 127 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×C4⋊#1084C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic
3C2×#88448C212yesmaximal
4#63658⋊C3C211yesmaximal, characteristic
5#11720334C212yesmaximal
6#10591358—31nomaximal
7#10591214—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2212yes—
10C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C4⋊#1084—22no—
14#3750⋊C3—22no—
15#88448—24no—
16C2×#3750D311yescharacteristic
17C2×C3×C4⋊Dic3—31no—
18C2×C3×C4⋊Dic3—31no—
19C2×C3×C4⋊Dic3—61no—
20C2×C3×Dic3⋊C4—31no—
21C2×#3724—32no—
22C2×Dic3⋊Dic3—32no—
23#63658D311yescharacteristic
24C3×C3⋊#95—31no—
25#63579—32no—
26#63579—32no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)D412yes—
30C2×#1084—22no—
31C2×#1084—22no—
32C2×#1084—22no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—21no—
35C2×(C3⋊Dic3)⋊C3—21no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—21no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3D412yes—
40C2×(C3⋊Dic3)⋊C3Q814yes—
41#4983—91no—
42#4981—91no—
43#4661—91no—
44C22×C32×C4—21no—
45C22×C32×C4C2×D311yescharacteristic
46C22×C32×C4C2×D311yescharacteristic
47C22×C3⋊Dic3C2×D311yescharacteristic
48C22×C3⋊Dic3C2×D312yes—
49C22×C3⋊Dic3C2×D311yescharacteristic
50C22×C3×Dic3—31no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—31no—
53C22×C3×Dic3—32no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—62no—
56#3750—22no—
57C3×C4⋊Dic3—64no—
58C3×C4⋊Dic3—62no—
59C3×C4⋊Dic3—62no—
60C3×Dic3⋊C4—62no—
61#3724—64no—
62Dic3⋊Dic3—64no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C22×H(3,𝔽3)C2×Q812yes—
66C22×H(3,𝔽3)#4111yescharacteristic
67C22×H(3,𝔽3)#4112yes—
68C4×H(3,𝔽3)—22no—
69#1084—42no—
70(C3⋊Dic3)⋊C3—22no—
71(C3⋊Dic3)⋊C3—24no—
72(C3⋊Dic3)⋊C3—41no—
73C2×C3×C4⋊C4—91no—
74C2×C3×C4⋊C4—91no—
75C2×C4⋊Dic3—31no—
76C2×C4⋊Dic3—31no—
77C2×C4⋊Dic3—61no—
78C2×C4⋊Dic3—92no—
79C2×Dic3⋊C4—31no—
80C2×Dic3⋊C4—91no—
81C2×Dic3⋊C4—92no—
82C2×Dic3⋊C4—92no—
83C3×#95—91no—
84C3⋊#95—31no—
85C3⋊#95—91no—
86C3⋊#95—92no—
87C3⋊#95—92no—
88C3⋊#95—92no—
89C23×C32—21no—
90C23×C32C22×D311yescharacteristic
91C23×C32C22×D311yescharacteristic
92C2×C32×C4—24no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4D1212yes—
97C2×C32×C4#6712yes—
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3—22no—
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3—21no—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3Dic614yes—
105C2×C3⋊Dic3D1212yes—
106C2×C3×Dic3—32no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—64no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—122no—
122C2×C3×Dic3—122no—
123#398—271noSylow
124C2×H(3,𝔽3)#12111yescharacteristic
125C2×H(3,𝔽3)C22⋊Q812yes—
126C2×H(3,𝔽3)#12311yescharacteristic
127C2×H(3,𝔽3)#12512yes—
128C2×H(3,𝔽3)C4⋊Q811yescharacteristic
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4—61no—
132C22×C3×C4—92no—
133C22×C3×C4—91no—
134C22×C3×C4—91no—
135C22×C3×C4—92no—
136C22×C3×C4D3211yescharacteristic
137C22×Dic3—32no—
138C22×Dic3—31no—
139C22×Dic3—32no—
140C22×Dic3—32no—
141C22×Dic3—31no—
142C22×Dic3—62no—
143C22×Dic3—92no—
144C22×Dic3—91no—
145C22×Dic3—91no—
146C3×C4⋊C4—182no—
147C3×C4⋊C4—182no—
148C4⋊Dic3—62no—
149C4⋊Dic3—62no—
150C4⋊Dic3—64no—
151C4⋊Dic3—184no—
152Dic3⋊C4—62no—
153Dic3⋊C4—182no—
154Dic3⋊C4—184no—
155Dic3⋊C4—184no—
156C22×C32—22no—
157C22×C32—21no—
158C22×C32—21no—
159C22×C32—22no—
160C22×C32—21no—
161C22×C32C2×Dic612yes—
162C22×C32C2×D1211yescharacteristic
163C22×C32C2×D1211yescharacteristic
164C22×C32#23511yescharacteristic
165C22×C32D3×D411yescharacteristic
166C22×C32#23712yes—
167C22×C32#23712yes—
168C22×C32D3×Q812yes—
169C22×C32#23911yescharacteristic
170C22×C32C2×#6711yescharacteristic
171C32×C4—24no—
172C32×C4—22no—
173C32×C4—22no—
174C3⋊Dic3—24no—
175C3⋊Dic3—22no—
176C3⋊Dic3—41no—
177C3×Dic3—62no—
178C3×Dic3—64no—
179C3×Dic3—124no—
180C3×Dic3—121no—
181C3×Dic3—122no—
182C3×Dic3—122no—
183C2×C4⋊C4—91no—
184C2×C4⋊C4—91no—
185C2×C4⋊C4—272no—
186#95—91no—
187#95—272no—
188H(3,𝔽3)#39811yescharacteristic, complemented, Sylow
189C23×C3—31no—
190C23×C3—31no—
191C23×C3—61no—
192C23×C3C2×D3211yescharacteristic
193C2×C3×C4—22no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—62no—
197C2×C3×C4—62no—
198C2×C3×C4—64no—
199C2×C3×C4—62no—
200C2×C3×C4—92no—
201C2×C3×C4—94no—
202C2×C3×C4—182no—
203C2×C3×C4—181no—
204C2×C3×C4—181no—
205C2×C3×C4—182no—
206C2×C3×C4—181no—
207C2×C3×C4—182no—
208C2×C3×C4—182no—
209C2×C3×C4—182no—
210C2×C3×C4—182no—
211C2×C3×C4#62612yes—
212C2×Dic3—34no—
213C2×Dic3—32no—
214C2×Dic3—61no—
215C2×Dic3—62no—
216C2×Dic3—62no—
217C2×Dic3—61no—
218C2×Dic3—62no—
219C2×Dic3—62no—
220C2×Dic3—62no—
221C2×Dic3—62no—
222C2×Dic3—62no—
223C2×Dic3—62no—
224C2×Dic3—64no—
225C2×Dic3—62no—
226C2×Dic3—61no—
227C2×Dic3—94no—
228C2×Dic3—92no—
229C2×Dic3—122no—
230C2×Dic3—122no—
231C2×Dic3—182no—
232C2×Dic3—181no—
233C2×Dic3—181no—
234C2×Dic3—182no—
235C2×Dic3—181no—
236C2×Dic3—182no—
237C2×C32—22no—
238C2×C32—21no—
239C2×C32—22no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32#86911yescharacteristic
243C2×C32#87812yes—
244C2×C32#88611yescharacteristic
245C2×C32#88912yes—
246C2×C32#89012yes—
247C2×C32#89511yescharacteristic
248C2×C32#89712yes—
249C2×C32#93011yescharacteristic
250C2×C32#93511yescharacteristic
251C2×C32#94411yescharacteristic
252C22×C4—92no—
253C22×C4—91no—
254C22×C4—91no—
255C22×C4—92no—
256C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
257C4⋊C4—182no—
258C4⋊C4—182no—
259C4⋊C4—544no—
260C22×C3—32no—
261C22×C3—32no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—32no—
267C22×C3—32no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—62no—
271C22×C3—61no—
272C22×C3—61no—
273C22×C3—62no—
274C22×C3—61no—
275C22×C3D3×Dic612yes—
276C22×C3#379911yescharacteristic
277C22×C3D3×D1211yescharacteristic
278C22×C3#380512yes—
279C22×C3C2×#62611yescharacteristic
280C3×C4—22no—
281C3×C4—62no—
282C3×C4—62no—
283C3×C4—64no—
284C3×C4—184no—
285C3×C4—182no—
286C3×C4—361no—
287C3×C4—362no—
288Dic3—64no—
289Dic3—62no—
290Dic3—122no—
291Dic3—122no—
292Dic3—124no—
293Dic3—121no—
294Dic3—184no—
295Dic3—182no—
296Dic3—361no—
297C32—21no—
298C32#466111yescharacteristic, complemented
299C32#498111yescharacteristic, complemented
300C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
301C2×C4—22no—
302C2×C4—94no—
303C2×C4—92no—
304C2×C4—181no—
305C2×C4—182no—
306C2×C4—182no—
307C2×C4—181no—
308C2×C4—182no—
309C2×C4—182no—
310C2×C4—182no—
311C2×C4—182no—
312C2×C4—181no—
313C2×C4#620012yes—
314C2×C3—31no—
315C2×C3—31no—
316C2×C3—32no—
317C2×C3—32no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—32no—
321C2×C3—32no—
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—61no—
325C2×C3—62no—
326C2×C3—61no—
327C2×C3—62no—
328C2×C3—61no—
329C2×C3#6384512yes—
330C2×C3#6386611yescharacteristic
331C2×C3#6389312yes—
332C2×C3#6390911yescharacteristic
333C2×C3#6396211yescharacteristic
334C22((C3⋊D3)⋊C3)⋊Q812yes—
335C22#8865211yescharacteristic
336C22#8865411yescharacteristic
337C22#8867212yes—
338C22C2×#620011yescharacteristic
339C4—22no—
340C4—184no—
341C4—182no—
342C4—361no—
343C4—362no—
344C3—31no—
345C3—31no—
346C3—61no—
347C3#1059135311yescharacteristic
348C2#1172108512yes—
349C2#1172109011yescharacteristic
350C2#1172113312yes—
351C2#1172115611yescharacteristic
352C2#1172130711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#398
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
188. H(3,𝔽3)#398semidirect
298. C32#4661semidirect
299. C32#4981semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
C2×#3750⋊C32
C2×#884482
#63658⋊C32
#117203342
#105913583
#105912143