Finite Group Databasebeta

← #13140099

Subgroups — #13140099

Order 1728.

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

normal subgroups only

This group has 793 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×#6229C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic
3C2×#88448C212yesmaximal
4#63658⋊C3C211yesmaximal, characteristic
5#11720334C212yesmaximal
6#10591346—31nomaximal
7#10591344—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2212yes—
10C2×#6229—22no—
11C2×#6229—24no—
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15#3750⋊C3—22no—
16#88448—24no—
17C22×C3×Dic6—31no—
18C22×C3×Dic6—31no—
19C22×C3×Dic6—61no—
20C2×#3750D311yescharacteristic
21C2×C3×Dic3⋊C4—31no—
22C2×#3724—32no—
23C2×Dic3⋊Dic3—32no—
24#63658D311yescharacteristic
25C3×C3⋊#95—31no—
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×#1084D414yes—
34#6229—28no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—21no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—21no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3Q814yescomplemented
42C2×(C3⋊Dic3)⋊C3Q812yescomplemented
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—
52C22×C3×Dic3—31no—
53C22×C3×Dic3—32no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—31no—
57C22×C3×Dic3—62no—
58C2×C3×Dic6—64no—
59C2×C3×Dic6—64no—
60C2×C3×Dic6—68no—
61C2×C3×Dic6—64no—
62C2×C3×Dic6—62no—
63C2×C3×Dic6—62no—
64#3750—22no—
65C3×Dic3⋊C4—62no—
66#3724—64no—
67Dic3⋊Dic3—64no—
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C22×H(3,𝔽3)C2×D412yes—
70C22×H(3,𝔽3)C2×Q812yes—
71C22×H(3,𝔽3)C2×Q811yescharacteristic
72C22×H(3,𝔽3)#4111yescharacteristic
73C4×H(3,𝔽3)—22no—
74#1084—24no—
75(C3⋊Dic3)⋊C3—24no—
76(C3⋊Dic3)⋊C3—22no—
77(C3⋊Dic3)⋊C3—41no—
78C22×C3×Q8—91no—
79C22×Dic6—31no—
80C22×Dic6—31no—
81C22×Dic6—61no—
82C2×C3×C4⋊C4—91no—
83C2×C4⋊Dic3—92no—
84C2×Dic3⋊C4—31no—
85C2×Dic3⋊C4—91no—
86C2×Dic3⋊C4—92no—
87C2×Dic3⋊C4—92no—
88C3×#95—91no—
89C3⋊#95—31no—
90C3⋊#95—91no—
91C3⋊#95—92no—
92C3⋊#95—92no—
93C3⋊#95—92no—
94C23×C32—21no—
95C23×C32C22×D311yescharacteristic
96C23×C32C22×D311yescharacteristic
97C2×C32×C4—24no—
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—22no—
101C2×C32×C4D1212yes—
102C2×C32×C4#6712yes—
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3Dic614yescomplemented
110C2×C3⋊Dic3Dic612yescomplemented
111C2×C3×Dic3—34no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—34no—
114C2×C3×Dic3—32no—
115C2×C3×Dic3—64no—
116C2×C3×Dic3—64no—
117C2×C3×Dic3—64no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—62no—
128C3×Dic6—616no—
129C3×Dic6—68no—
130C3×Dic6—68no—
131#395—271noSylow
132C2×H(3,𝔽3)#12011yescharacteristic
133C2×H(3,𝔽3)C22⋊Q812yes—
134C2×H(3,𝔽3)C22⋊Q811yescharacteristic
135C2×H(3,𝔽3)C4⋊Q811yescharacteristic
136C2×H(3,𝔽3)C4⋊Q812yes—
137C2×C3×Q8—184no—
138C2×C3×Q8—182no—
139C22×C3×C4—31no—
140C22×C3×C4—31no—
141C22×C3×C4—61no—
142C22×C3×C4—91no—
143C22×C3×C4—92no—
144C22×C3×C4—92no—
145C22×C3×C4—91no—
146C22×C3×C4D3211yescharacteristic
147C22×Dic3—32no—
148C22×Dic3—32no—
149C22×Dic3—32no—
150C22×Dic3—31no—
151C22×Dic3—31no—
152C22×Dic3—62no—
153C22×Dic3—92no—
154C22×Dic3—91no—
155C22×Dic3—91no—
156C2×Dic6—64no—
157C2×Dic6—64no—
158C2×Dic6—62no—
159C2×Dic6—62no—
160C2×Dic6—68no—
161C2×Dic6—64no—
162C3×C4⋊C4—182no—
163C4⋊Dic3—184no—
164Dic3⋊C4—62no—
165Dic3⋊C4—182no—
166Dic3⋊C4—184no—
167Dic3⋊C4—184no—
168C22×C32—21no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—21no—
172C22×C32—22no—
173C22×C32C2×Dic612yes—
174C22×C32C2×Dic611yescharacteristic
175C22×C32C2×D1211yescharacteristic
176C22×C32#23511yescharacteristic
177C22×C32D3×D412yes—
178C22×C32D3×D412yes—
179C22×C32#23711yescharacteristic
180C22×C32D3×Q811yescharacteristic
181C22×C32D3×Q812yes—
182C22×C32C2×#6711yescharacteristic
183C32×C4—24no—
184C32×C4—22no—
185C32×C4—22no—
186C3⋊Dic3—24no—
187C3⋊Dic3—22no—
188C3⋊Dic3—41no—
189C3×Dic3—68no—
190C3×Dic3—64no—
191C3×Dic3—64no—
192C3×Dic3—62no—
193C3×Dic3—64no—
194C3×Dic3—121no—
195C22×Q8—91no—
196C2×C4⋊C4—91no—
197C2×C4⋊C4—272no—
198#95—91no—
199#95—272no—
200H(3,𝔽3)#39511yescharacteristic, complemented, Sylow
201C23×C3—31no—
202C23×C3—31no—
203C23×C3—61no—
204C23×C3C2×D3211yescharacteristic
205C3×Q8—188no—
206C2×C3×C4—22no—
207C2×C3×C4—32no—
208C2×C3×C4—32no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—64no—
212C2×C3×C4—62no—
213C2×C3×C4—92no—
214C2×C3×C4—94no—
215C2×C3×C4—94no—
216C2×C3×C4—182no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4—182no—
220C2×C3×C4—182no—
221C2×C3×C4—182no—
222C2×C3×C4—182no—
223C2×C3×C4—181no—
224C2×C3×C4#62612yes—
225C2×Dic3—34no—
226C2×Dic3—34no—
227C2×Dic3—32no—
228C2×Dic3—34no—
229C2×Dic3—64no—
230C2×Dic3—62no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—64no—
235C2×Dic3—64no—
236C2×Dic3—61no—
237C2×Dic3—62no—
238C2×Dic3—61no—
239C2×Dic3—62no—
240C2×Dic3—61no—
241C2×Dic3—62no—
242C2×Dic3—92no—
243C2×Dic3—94no—
244C2×Dic3—181no—
245C2×Dic3—182no—
246C2×Dic3—182no—
247C2×Dic3—181no—
248C2×Dic3—181no—
249C2×Dic3—182no—
250Dic6—68no—
251Dic6—68no—
252Dic6—616no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32—22no—
257C2×C32—21no—
258C2×C32#86911yescharacteristic
259C2×C32#87812yes—
260C2×C32#88211yescharacteristic
261C2×C32#88812yes—
262C2×C32#88812yes—
263C2×C32#89612yes—
264C2×C32#89711yescharacteristic
265C2×C32#92411yescharacteristic
266C2×C32#93711yescharacteristic
267C2×C32#94411yescharacteristic
268C2×Q8—184no—
269C2×Q8—182no—
270C22×C4—92no—
271C22×C4—91no—
272C22×C4—92no—
273C22×C4—91no—
274C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
275C4⋊C4—182no—
276C4⋊C4—544no—
277C22×C3—31no—
278C22×C3—32no—
279C22×C3—31no—
280C22×C3—32no—
281C22×C3—31no—
282C22×C3—31no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—32no—
286C22×C3—32no—
287C22×C3—61no—
288C22×C3—62no—
289C22×C3—61no—
290C22×C3—61no—
291C22×C3—62no—
292C22×C3D3×Dic611yescharacteristic
293C22×C3D3×Dic612yes—
294C22×C3#379511yescharacteristic
295C22×C3C2×#62611yescharacteristic
296C22×C3#381112yes—
297C3×C4—22no—
298C3×C4—62no—
299C3×C4—62no—
300C3×C4—64no—
301C3×C4—182no—
302C3×C4—184no—
303C3×C4—184no—
304C3×C4—361no—
305Dic3—64no—
306Dic3—64no—
307Dic3—68no—
308Dic3—62no—
309Dic3—64no—
310Dic3—121no—
311Dic3—182no—
312Dic3—184no—
313Dic3—361no—
314C32—21no—
315C32#465611yescharacteristic, complemented
316C32#497911yescharacteristic, complemented
317C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
318Q8—188no—
319C2×C4—22no—
320C2×C4—92no—
321C2×C4—94no—
322C2×C4—94no—
323C2×C4—181no—
324C2×C4—182no—
325C2×C4—181no—
326C2×C4—182no—
327C2×C4—182no—
328C2×C4—182no—
329C2×C4—181no—
330C2×C4—182no—
331C2×C4#620012yes—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—32no—
335C2×C3—32no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—32no—
341C2×C3—32no—
342C2×C3—61no—
343C2×C3—62no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—62no—
347C2×C3#6384012yes—
348C2×C3#6385711yescharacteristic
349C2×C3#6386611yescharacteristic
350C2×C3#6389612yes—
351C2×C3#6398111yescharacteristic
352C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
353C22((C3⋊D3)⋊C3)⋊Q812yes—
354C22#8864811yescharacteristic
355C22C2×#620011yescharacteristic
356C22#8867712yes—
357C4—22no—
358C4—184no—
359C4—182no—
360C4—184no—
361C4—361no—
362C3—31no—
363C3—31no—
364C3—61no—
365C3#1059135411yescharacteristic
366C2#1172108012yes—
367C2#1172109011yescharacteristic
368C2#1172110711yescharacteristic
369C2#1172113512yes—
370C2#1172132311yescharacteristic
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
41. C2×(C3⋊Dic3)⋊C3Q8semidirect
42. C2×(C3⋊Dic3)⋊C3Q8semidirect
109. C2×C3⋊Dic3Dic6semidirect
110. C2×C3⋊Dic3Dic6semidirect
200. H(3,𝔽3)#395semidirect
315. C32#4656semidirect
316. C32#4979semidirect

Maximal subgroups

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

subgroupindex
C22×#62292
C2×#3750⋊C32
C2×#884482
#63658⋊C32
#117203342
#105913463
#105913443