Finite Group Databasebeta

← #13140142

Subgroups — #13140142

Order 1728.

713 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 713 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C22×#634⋊C3C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic
3C2×#88449C212yesmaximal
4#11720574C211yesmaximal, characteristic
5#11720334C212yesmaximal
6#10591354—31nomaximal
7#10591346—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2212yes—
10C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×#634⋊C3—22no—
14C2×#634⋊C3—24no—
15#3750⋊C3—22no—
16#88449—24no—
17C22×#634D311yescharacteristic
18C22×C3×Dic6—31no—
19C2×#3750D311yescharacteristic
20C2×C3×Dic3⋊C4—31no—
21C2×Dic3⋊Dic3—32no—
22C2×Dic3⋊Dic3—32no—
23C3×C3⋊#95—31no—
24C3×C3⋊#95—31no—
25C3×C3⋊#95—61no—
26#63579—32no—
27#63579—32no—
28C23×H(3,𝔽3)C2311yescharacteristic
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)D412yes—
31C2×#1084—22no—
32C2×#1084—22no—
33C2×#1084Q814yescomplemented
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—21no—
37C2×(C3⋊Dic3)⋊C3—21no—
38C2×(C3⋊Dic3)⋊C3—22no—
39C2×(C3⋊Dic3)⋊C3—21no—
40C2×(C3⋊Dic3)⋊C3D414yes—
41C2×(C3⋊Dic3)⋊C3Q812yescomplemented
42#634⋊C3—28no—
43#5230—91no—
44#4979—91no—
45#4656—91no—
46C22×C32×C4—21no—
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C2×D311yescharacteristic
49C22×C3⋊Dic3C2×D311yescharacteristic
50C22×C3⋊Dic3C2×D311yescharacteristic
51C22×C3⋊Dic3C2×D312yes—
52C2×#634—22no—
53C2×#634—24no—
54C22×C3×Dic3—31no—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—31no—
59C22×C3×Dic3—62no—
60C2×C3×Dic6—64no—
61C2×C3×Dic6—62no—
62#3750—22no—
63C3×Dic3⋊C4—62no—
64Dic3⋊Dic3—64no—
65Dic3⋊Dic3—64no—
66C22×H(3,𝔽3)C2×D412yes—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C22×H(3,𝔽3)C2×Q811yescharacteristic
69C22×H(3,𝔽3)C2×Q812yes—
70C22×H(3,𝔽3)#4111yescharacteristic
71C4×H(3,𝔽3)—22no—
72#1084—24no—
73(C3⋊Dic3)⋊C3—24no—
74(C3⋊Dic3)⋊C3—22no—
75(C3⋊Dic3)⋊C3—41no—
76C22×C3×Q8—91no—
77C22×Dic6—31no—
78C22×Dic6—91no—
79C2×C3×C4⋊C4—91no—
80C2×C4⋊Dic3—92no—
81C2×Dic3⋊C4—31no—
82C2×Dic3⋊C4—91no—
83C2×Dic3⋊C4—92no—
84C2×Dic3⋊C4—92no—
85C3×#95—91no—
86C3⋊#95—31no—
87C3⋊#95—31no—
88C3⋊#95—61no—
89C3⋊#95—92no—
90C3⋊#95—92no—
91C3⋊#95—92no—
92C23×C32—21no—
93C23×C32C22×D311yescharacteristic
94C23×C32C22×D311yescharacteristic
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4—42no—
99C2×C32×C4#6712yes—
100C2×C32×C4#6712yes—
101C2×C3⋊Dic3—21no—
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3Dic612yescomplemented
108C2×C3⋊Dic3#6714yes—
109#634—28no—
110C2×C3×Dic3—34no—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—34no—
114C2×C3×Dic3—64no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—122no—
126C2×C3×Dic3—122no—
127C3×Dic6—68no—
128#395—271noSylow
129C2×H(3,𝔽3)#12011yescharacteristic
130C2×H(3,𝔽3)C22⋊Q812yes—
131C2×H(3,𝔽3)C22⋊Q811yescharacteristic
132C2×H(3,𝔽3)C4⋊Q812yes—
133C2×H(3,𝔽3)C4⋊Q811yescharacteristic
134C2×C3×Q8—182no—
135C2×C3×Q8—184no—
136C22×C3×C4—31no—
137C22×C3×C4—31no—
138C22×C3×C4—61no—
139C22×C3×C4—91no—
140C22×C3×C4—92no—
141C22×C3×C4—92no—
142C22×C3×C4—91no—
143C22×C3×C4D3211yescharacteristic
144C22×Dic3—32no—
145C22×Dic3—32no—
146C22×Dic3—31no—
147C22×Dic3—32no—
148C22×Dic3—31no—
149C22×Dic3—62no—
150C22×Dic3—91no—
151C22×Dic3—92no—
152C22×Dic3—91no—
153C2×Dic6—64no—
154C2×Dic6—62no—
155C2×Dic6—184no—
156C2×Dic6—182no—
157C3×C4⋊C4—182no—
158C4⋊Dic3—184no—
159Dic3⋊C4—62no—
160Dic3⋊C4—182no—
161Dic3⋊C4—184no—
162Dic3⋊C4—184no—
163C22×C32—21no—
164C22×C32—22no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—22no—
168C22×C32C2×Dic611yescharacteristic
169C22×C32#23511yescharacteristic
170C22×C32D3×D412yes—
171C22×C32D3×Q811yescharacteristic
172C22×C32D3×Q812yes—
173C22×C32D3×Q812yes—
174C22×C32#23911yescharacteristic
175C22×C32C2×#6711yescharacteristic
176C22×C32C2×#6711yescharacteristic
177C22×C32C2×#6712yes—
178C32×C4—22no—
179C32×C4—22no—
180C32×C4—42no—
181C3⋊Dic3—22no—
182C3⋊Dic3—24no—
183C3⋊Dic3—41no—
184C3×Dic3—64no—
185C3×Dic3—64no—
186C3×Dic3—62no—
187C3×Dic3—64no—
188C3×Dic3—124no—
189C3×Dic3—121no—
190C22×Q8—91no—
191C2×C4⋊C4—91no—
192C2×C4⋊C4—272no—
193#95—91no—
194#95—272no—
195H(3,𝔽3)#39511yescharacteristic, complemented, Sylow
196C23×C3—31no—
197C23×C3—31no—
198C23×C3—61no—
199C23×C3C2×D3211yescharacteristic
200C3×Q8—188no—
201C2×C3×C4—22no—
202C2×C3×C4—32no—
203C2×C3×C4—32no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—94no—
208C2×C3×C4—92no—
209C2×C3×C4—94no—
210C2×C3×C4—122no—
211C2×C3×C4—181no—
212C2×C3×C4—181no—
213C2×C3×C4—182no—
214C2×C3×C4—182no—
215C2×C3×C4—182no—
216C2×C3×C4—182no—
217C2×C3×C4—182no—
218C2×C3×C4—181no—
219C2×C3×C4#62512yes—
220C2×Dic3—34no—
221C2×Dic3—34no—
222C2×Dic3—32no—
223C2×Dic3—34no—
224C2×Dic3—64no—
225C2×Dic3—62no—
226C2×Dic3—62no—
227C2×Dic3—62no—
228C2×Dic3—62no—
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—61no—
234C2×Dic3—62no—
235C2×Dic3—92no—
236C2×Dic3—94no—
237C2×Dic3—122no—
238C2×Dic3—122no—
239C2×Dic3—181no—
240C2×Dic3—182no—
241C2×Dic3—182no—
242C2×Dic3—181no—
243C2×Dic3—181no—
244C2×Dic3—182no—
245Dic6—68no—
246Dic6—188no—
247C2×C32—22no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—22no—
251C2×C32—21no—
252C2×C32#88812yes—
253C2×C32#89612yes—
254C2×C32#92411yescharacteristic
255C2×C32#93711yescharacteristic
256C2×C32#94412yes—
257C2×C32#94411yescharacteristic
258C2×C32#94411yescharacteristic
259C2×C32#94611yescharacteristic
260C2×C32#94612yes—
261C2×C32#95311yescharacteristic
262C2×Q8—184no—
263C2×Q8—182no—
264C22×C4—92no—
265C22×C4—91no—
266C22×C4—92no—
267C22×C4—91no—
268C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
269C4⋊C4—182no—
270C4⋊C4—544no—
271C22×C3—31no—
272C22×C3—31no—
273C22×C3—32no—
274C22×C3—32no—
275C22×C3—31no—
276C22×C3—31no—
277C22×C3—31no—
278C22×C3—31no—
279C22×C3—32no—
280C22×C3—32no—
281C22×C3—61no—
282C22×C3—62no—
283C22×C3—61no—
284C22×C3—61no—
285C22×C3—62no—
286C22×C3D3×Dic611yescharacteristic
287C22×C3#379712yes—
288C22×C3#379911yescharacteristic
289C22×C3C2×#62511yescharacteristic
290C22×C3D3×#6712yes—
291C3×C4—22no—
292C3×C4—62no—
293C3×C4—62no—
294C3×C4—122no—
295C3×C4—184no—
296C3×C4—182no—
297C3×C4—184no—
298C3×C4—361no—
299Dic3—64no—
300Dic3—64no—
301Dic3—62no—
302Dic3—64no—
303Dic3—124no—
304Dic3—121no—
305Dic3—184no—
306Dic3—182no—
307Dic3—361no—
308C32—21no—
309C32#497911yescharacteristic, complemented
310C32#523011yescharacteristic, complemented
311C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
312Q8—188no—
313C2×C4—22no—
314C2×C4—92no—
315C2×C4—94no—
316C2×C4—94no—
317C2×C4—182no—
318C2×C4—181no—
319C2×C4—181no—
320C2×C4—182no—
321C2×C4—182no—
322C2×C4—182no—
323C2×C4—181no—
324C2×C4—182no—
325C2×C4#619812yes—
326C2×C3—31no—
327C2×C3—31no—
328C2×C3—32no—
329C2×C3—32no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—32no—
335C2×C3—32no—
336C2×C3—61no—
337C2×C3—62no—
338C2×C3—61no—
339C2×C3—61no—
340C2×C3—62no—
341C2×C3#6383912yes—
342C2×C3#6385611yescharacteristic
343C2×C3#6387311yescharacteristic
344C2×C3#6388812yes—
345C2×C3#6397611yescharacteristic
346C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
347C22#8865112yes—
348C22#8865211yescharacteristic
349C22C2×#619811yescharacteristic
350C22#8867612yes—
351C4—22no—
352C4—184no—
353C4—182no—
354C4—184no—
355C4—361no—
356C3—31no—
357C3—31no—
358C3—61no—
359C3#1059134411yescharacteristic
360C2#1172107912yes—
361C2#1172109911yescharacteristic
362C2#1172110611yescharacteristic
363C2#1172114812yes—
364C2#1172131611yescharacteristic
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)#395
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
33. C2×#1084Q8semidirect
41. C2×(C3⋊Dic3)⋊C3Q8semidirect
107. C2×C3⋊Dic3Dic6semidirect
195. H(3,𝔽3)#395semidirect
309. C32#4979semidirect
310. C32#5230semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
C2×#3750⋊C32
C2×#884492
#117205742
#117203342
#105913543
#105913463