Finite Group Databasebeta

← #13140144

Subgroups — #13140144

Order 1728.

601 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 123 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⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic
3C2×#88449C212yesmaximal
4#11720574C211yesmaximal, characteristic
5#11720334C212yesmaximal
6#10591358—31nomaximal
7#10591353—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2212yes—
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)—22no—
14#3750⋊C3—22no—
15#88449—24no—
16C2×C4⋊(C3⋊Dic3)D311yescharacteristic
17C2×#3750D311yescharacteristic
18C2×C3×C4⋊Dic3—31no—
19C2×C3×Dic3⋊C4—31no—
20C2×Dic3⋊Dic3—32no—
21C2×Dic3⋊Dic3—32no—
22C3×C3⋊#95—31no—
23C3×C3⋊#95—31no—
24C3×C3⋊#95—61no—
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×#1084Q814yes—
33C2×(C3⋊Dic3)⋊C3—21no—
34C2×(C3⋊Dic3)⋊C3—21no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—21no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3D412yes—
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—32no—
53C22×C3×Dic3—31no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—62no—
56C4⋊(C3⋊Dic3)—22no—
57#3750—22no—
58C3×C4⋊Dic3—62no—
59C3×Dic3⋊C4—62no—
60Dic3⋊Dic3—64no—
61Dic3⋊Dic3—64no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C22×H(3,𝔽3)C2×Q812yes—
65C22×H(3,𝔽3)#4111yescharacteristic
66C22×H(3,𝔽3)#4112yes—
67C4×H(3,𝔽3)—22no—
68#1084—24no—
69(C3⋊Dic3)⋊C3—22no—
70(C3⋊Dic3)⋊C3—42no—
71(C3⋊Dic3)⋊C3—41no—
72C2×C3×C4⋊C4—91no—
73C2×C3×C4⋊C4—91no—
74C2×C4⋊Dic3—31no—
75C2×C4⋊Dic3—91no—
76C2×C4⋊Dic3—92no—
77C2×Dic3⋊C4—31no—
78C2×Dic3⋊C4—91no—
79C2×Dic3⋊C4—92no—
80C2×Dic3⋊C4—92no—
81C3×#95—91no—
82C3⋊#95—31no—
83C3⋊#95—31no—
84C3⋊#95—61no—
85C3⋊#95—92no—
86C3⋊#95—92no—
87C3⋊#95—92no—
88C23×C32—21no—
89C23×C32C22×D311yescharacteristic
90C23×C32C22×D311yescharacteristic
91C2×C32×C4—22no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—42no—
95C2×C32×C4#6712yes—
96C2×C32×C4#6712yes—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3—22no—
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3D1212yes—
105C2×C3×Dic3—32no—
106C2×C3×Dic3—34no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—64no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—122no—
121C2×C3×Dic3—122no—
122#398—271noSylow
123C2×H(3,𝔽3)#12111yescharacteristic
124C2×H(3,𝔽3)C22⋊Q812yes—
125C2×H(3,𝔽3)#12311yescharacteristic
126C2×H(3,𝔽3)#12512yes—
127C2×H(3,𝔽3)C4⋊Q811yescharacteristic
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C22×C3×C4—61no—
131C22×C3×C4—91no—
132C22×C3×C4—92no—
133C22×C3×C4—91no—
134C22×C3×C4—92no—
135C22×C3×C4D3211yescharacteristic
136C22×Dic3—32no—
137C22×Dic3—32no—
138C22×Dic3—31no—
139C22×Dic3—31no—
140C22×Dic3—32no—
141C22×Dic3—62no—
142C22×Dic3—91no—
143C22×Dic3—91no—
144C22×Dic3—92no—
145C3×C4⋊C4—182no—
146C3×C4⋊C4—182no—
147C4⋊Dic3—62no—
148C4⋊Dic3—182no—
149C4⋊Dic3—184no—
150Dic3⋊C4—62no—
151Dic3⋊C4—182no—
152Dic3⋊C4—184no—
153Dic3⋊C4—184no—
154C22×C32—21no—
155C22×C32—22no—
156C22×C32—21no—
157C22×C32—22no—
158C22×C32—21no—
159C22×C32C2×D1211yescharacteristic
160C22×C32#23511yescharacteristic
161C22×C32#23512yes—
162C22×C32D3×D411yescharacteristic
163C22×C32#23711yescharacteristic
164C22×C32#23712yes—
165C22×C32D3×Q812yes—
166C22×C32D3×Q812yes—
167C22×C32C2×#6711yescharacteristic
168C22×C32C2×#6711yescharacteristic
169C32×C4—22no—
170C32×C4—22no—
171C32×C4—42no—
172C3⋊Dic3—22no—
173C3⋊Dic3—41no—
174C3⋊Dic3—42no—
175C3×Dic3—64no—
176C3×Dic3—64no—
177C3×Dic3—62no—
178C3×Dic3—124no—
179C3×Dic3—121no—
180C3×Dic3—122no—
181C2×C4⋊C4—91no—
182C2×C4⋊C4—91no—
183C2×C4⋊C4—272no—
184#95—91no—
185#95—272no—
186H(3,𝔽3)#39811yescharacteristic, complemented, Sylow
187C23×C3—31no—
188C23×C3—31no—
189C23×C3—61no—
190C23×C3C2×D3211yescharacteristic
191C2×C3×C4—22no—
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—62no—
195C2×C3×C4—62no—
196C2×C3×C4—62no—
197C2×C3×C4—92no—
198C2×C3×C4—94no—
199C2×C3×C4—122no—
200C2×C3×C4—181no—
201C2×C3×C4—181no—
202C2×C3×C4—182no—
203C2×C3×C4—182no—
204C2×C3×C4—182no—
205C2×C3×C4—181no—
206C2×C3×C4—182no—
207C2×C3×C4—182no—
208C2×C3×C4—182no—
209C2×C3×C4#62512yes—
210C2×Dic3—34no—
211C2×Dic3—34no—
212C2×Dic3—32no—
213C2×Dic3—64no—
214C2×Dic3—62no—
215C2×Dic3—62no—
216C2×Dic3—62no—
217C2×Dic3—62no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C2×Dic3—62no—
221C2×Dic3—62no—
222C2×Dic3—62no—
223C2×Dic3—61no—
224C2×Dic3—62no—
225C2×Dic3—92no—
226C2×Dic3—122no—
227C2×Dic3—122no—
228C2×Dic3—181no—
229C2×Dic3—181no—
230C2×Dic3—182no—
231C2×Dic3—181no—
232C2×Dic3—182no—
233C2×Dic3—182no—
234C2×Dic3—182no—
235C2×C32—22no—
236C2×C32—21no—
237C2×C32—22no—
238C2×C32—21no—
239C2×C32—21no—
240C2×C32#88912yes—
241C2×C32#88912yes—
242C2×C32#89612yes—
243C2×C32#89712yes—
244C2×C32#92911yescharacteristic
245C2×C32#93011yescharacteristic
246C2×C32#93511yescharacteristic
247C2×C32#93811yescharacteristic
248C2×C32#94411yescharacteristic
249C2×C32#94411yescharacteristic
250C22×C4—92no—
251C22×C4—91no—
252C22×C4—91no—
253C22×C4—92no—
254C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
255C4⋊C4—182no—
256C4⋊C4—182no—
257C4⋊C4—544no—
258C22×C3—31no—
259C22×C3—31no—
260C22×C3—32no—
261C22×C3—32no—
262C22×C3—31no—
263C22×C3—31no—
264C22×C3—32no—
265C22×C3—32no—
266C22×C3—31no—
267C22×C3—31no—
268C22×C3—61no—
269C22×C3—62no—
270C22×C3—61no—
271C22×C3—62no—
272C22×C3—61no—
273C22×C3#379511yescharacteristic
274C22×C3#379712yes—
275C22×C3D3×D1211yescharacteristic
276C22×C3#380412yes—
277C22×C3C2×#62511yescharacteristic
278C3×C4—22no—
279C3×C4—62no—
280C3×C4—62no—
281C3×C4—122no—
282C3×C4—182no—
283C3×C4—184no—
284C3×C4—361no—
285C3×C4—362no—
286Dic3—64no—
287Dic3—64no—
288Dic3—62no—
289Dic3—124no—
290Dic3—122no—
291Dic3—121no—
292Dic3—182no—
293Dic3—361no—
294Dic3—362no—
295C32—21no—
296C32#498111yescharacteristic, complemented
297C32#498311yescharacteristic, complemented
298C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
299C2×C4—22no—
300C2×C4—94no—
301C2×C4—92no—
302C2×C4—181no—
303C2×C4—182no—
304C2×C4—182no—
305C2×C4—181no—
306C2×C4—182no—
307C2×C4—182no—
308C2×C4—182no—
309C2×C4—181no—
310C2×C4—182no—
311C2×C4#619812yes—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—32no—
315C2×C3—32no—
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—32no—
321C2×C3—32no—
322C2×C3—61no—
323C2×C3—62no—
324C2×C3—61no—
325C2×C3—61no—
326C2×C3—62no—
327C2×C3#6384712yes—
328C2×C3#6387311yescharacteristic
329C2×C3#6389512yes—
330C2×C3#6390811yescharacteristic
331C2×C3#6396111yescharacteristic
332C22#8864811yescharacteristic
333C22#8865112yes—
334C22#8865411yescharacteristic
335C22#8867112yes—
336C22C2×#619811yescharacteristic
337C4—22no—
338C4—182no—
339C4—184no—
340C4—361no—
341C4—362no—
342C3—31no—
343C3—31no—
344C3—61no—
345C3#1059121411yescharacteristic
346C2#1172108612yes—
347C2#1172109911yescharacteristic
348C2#1172114111yescharacteristic
349C2#1172115012yes—
350C2#1172130611yescharacteristic
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
186. H(3,𝔽3)#398semidirect
296. C32#4981semidirect
297. C32#4983semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#3750⋊C32
C2×#884492
#117205742
#117203342
#105913583
#105913533