Finite Group Databasebeta

← #13140095

Subgroups — #13140095

Order 1728.

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

normal subgroups only

This group has 743 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, complemented
2C2×#3752⋊C3C211yesmaximal, characteristic
3C2×#88450C212yesmaximal
4C2×#88448C212yesmaximal, complemented
5#63658⋊C3C211yesmaximal, characteristic, complemented
6#10591350—31nomaximal
7#10591324—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C4⋊#1084—22no—
14#3752⋊C3—22no—
15#88450—24no—
16#88448—24no—
17C2×#3752D311yescharacteristic
18C2×C3×(C2×D3)⋊C4—31no—
19C2×C3×C4⋊Dic3—31no—
20C2×C3×C4⋊Dic3—31no—
21C2×C3×C4⋊Dic3—61no—
22C2×#3724—32no—
23C2×Dic3⋊Dic3—32no—
24C2×#3722—32no—
25C2×(C2×D3)⋊Dic3—32no—
26#63658D311yescharacteristic, complemented
27C3×C3⋊#95—31no—
28C23×H(3,𝔽3)C2311yescharacteristic
29C22×(C3⋊D3)⋊C3—22no—
30C22×(C3⋊D3)⋊C3—21no—
31C22×(C3⋊D3)⋊C3—21no—
32C22×(C3⋊D3)⋊C3—22no—
33C22×(C3⋊D3)⋊C3Q812yes—
34C2×C4×H(3,𝔽3)—22no—
35C2×C4×H(3,𝔽3)D412yes—
36C2×#1084—22no—
37C2×#1084—22no—
38C2×#1084—22no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D414yes—
44#4993—91no—
45#4949—91no—
46#4677—91no—
47C23×C3⋊D3C2×D311yescharacteristic
48C23×C3×D3—31no—
49C22×C32×C4—21no—
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C2×D311yescharacteristic
52C22×C3⋊Dic3C2×D311yescharacteristic
53C22×C3⋊Dic3C2×D312yes—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—31no—
58C22×C3×Dic3—62no—
59#3752—22no—
60C3×(C2×D3)⋊C4—62no—
61C3×C4⋊Dic3—64no—
62C3×C4⋊Dic3—62no—
63C3×C4⋊Dic3—62no—
64#3724—64no—
65Dic3⋊Dic3—64no—
66#3722—64no—
67(C2×D3)⋊Dic3—64no—
68C22×H(3,𝔽3)C2×D412yes—
69C22×H(3,𝔽3)C2×D411yescharacteristic
70C22×H(3,𝔽3)C2×Q811yescharacteristic
71C22×H(3,𝔽3)#4111yescharacteristic
72C22×H(3,𝔽3)#4112yes—
73C2×(C3⋊D3)⋊C3—24no—
74C2×(C3⋊D3)⋊C3—22no—
75C2×(C3⋊D3)⋊C3—42no—
76C2×(C3⋊D3)⋊C3—41no—
77C2×(C3⋊D3)⋊C3—41no—
78C4×H(3,𝔽3)—22no—
79#1084—42no—
80(C3⋊Dic3)⋊C3—24no—
81(C3⋊Dic3)⋊C3—41no—
82C2×C3×C4⋊C4—91no—
83C2×C3×C22⋊C4—91no—
84C2×#217—92no—
85C2×(C2×D3)⋊C4—31no—
86C2×(C2×D3)⋊C4—91no—
87C2×(C2×D3)⋊C4—92no—
88C2×(C2×D3)⋊C4—92no—
89C2×C4⋊Dic3—31no—
90C2×C4⋊Dic3—31no—
91C2×C4⋊Dic3—61no—
92C2×C4⋊Dic3—92no—
93C2×Dic3⋊C4—92no—
94C2×Dic3⋊C4—92no—
95C3×#95—91no—
96C3⋊#95—31no—
97C3⋊#95—91no—
98C23×C32—21no—
99C23×C32C22×D311yescharacteristic
100C23×C32C22×D311yescharacteristic
101C22×C3⋊D3—21no—
102C22×C3⋊D3—22no—
103C22×C3⋊D3—22no—
104C22×C3⋊D3—21no—
105C22×C3⋊D3Dic612yes—
106C22×C3×D3—32no—
107C22×C3×D3—62no—
108C22×C3×D3—61no—
109C22×C3×D3—61no—
110C22×C3×D3—62no—
111C2×C32×C4—24no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4D1212yes—
116C2×C32×C4#6712yes—
117C2×C3⋊Dic3—22no—
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3—22no—
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3D1214yes—
122C2×C3×Dic3—34no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—122no—
135C2×C3×Dic3—122no—
136#399—271noSylow
137C2×H(3,𝔽3)C22⋊Q812yes—
138C2×H(3,𝔽3)C22⋊Q811yescharacteristic
139C2×H(3,𝔽3)#12312yes—
140C2×H(3,𝔽3)#12311yescharacteristic
141C2×H(3,𝔽3)#12711yescharacteristic
142(C3⋊D3)⋊C3—42no—
143C24×C3—91no—
144C23×D3—31no—
145C23×D3—91no—
146C22×C3×C4—31no—
147C22×C3×C4—31no—
148C22×C3×C4—61no—
149C22×C3×C4—92no—
150C22×C3×C4—91no—
151C22×C3×C4—92no—
152C22×C3×C4D3211yescharacteristic
153C22×Dic3—32no—
154C22×Dic3—32no—
155C22×Dic3—32no—
156C22×Dic3—31no—
157C22×Dic3—62no—
158C22×Dic3—92no—
159C22×Dic3—91no—
160C3×C4⋊C4—182no—
161C3×C22⋊C4—182no—
162#217—184no—
163(C2×D3)⋊C4—62no—
164(C2×D3)⋊C4—182no—
165(C2×D3)⋊C4—184no—
166(C2×D3)⋊C4—184no—
167C4⋊Dic3—62no—
168C4⋊Dic3—62no—
169C4⋊Dic3—64no—
170C4⋊Dic3—184no—
171Dic3⋊C4—184no—
172Dic3⋊C4—184no—
173C22×C32—21no—
174C22×C32—21no—
175C22×C32—22no—
176C22×C32—22no—
177C22×C32—21no—
178C22×C32C2×Dic611yescharacteristic
179C22×C32C2×D1212yes—
180C22×C32C2×D1211yescharacteristic
181C22×C32#23511yescharacteristic
182C22×C32D3×D412yes—
183C22×C32#23711yescharacteristic
184C22×C32#23712yes—
185C22×C32#23712yes—
186C22×C32D3×Q811yescharacteristic
187C22×C32C2×#6711yescharacteristic
188C2×C3⋊D3—24no—
189C2×C3⋊D3—22no—
190C2×C3⋊D3—42no—
191C2×C3⋊D3—41no—
192C2×C3⋊D3—41no—
193C2×C3×D3—62no—
194C2×C3×D3—64no—
195C2×C3×D3—121no—
196C2×C3×D3—122no—
197C2×C3×D3—121no—
198C32×C4—24no—
199C32×C4—22no—
200C32×C4—22no—
201C3⋊Dic3—24no—
202C3⋊Dic3—41no—
203C3×Dic3—64no—
204C3×Dic3—124no—
205C3×Dic3—121no—
206C3×Dic3—122no—
207C3×Dic3—122no—
208C2×C4⋊C4—91no—
209C2×C4⋊C4—272no—
210C2×C22⋊C4—91no—
211C2×C22⋊C4—272no—
212#95—91no—
213H(3,𝔽3)#39911yescharacteristic, complemented, Sylow
214C23×C3—31no—
215C23×C3—31no—
216C23×C3—61no—
217C23×C3—92no—
218C23×C3—182no—
219C23×C3—181no—
220C23×C3—181no—
221C23×C3—182no—
222C23×C3C2×D3211yescharacteristic
223C22×D3—32no—
224C22×D3—61no—
225C22×D3—62no—
226C22×D3—62no—
227C22×D3—61no—
228C22×D3—92no—
229C22×D3—182no—
230C22×D3—181no—
231C22×D3—181no—
232C22×D3—182no—
233C2×C3×C4—22no—
234C2×C3×C4—32no—
235C2×C3×C4—32no—
236C2×C3×C4—62no—
237C2×C3×C4—62no—
238C2×C3×C4—64no—
239C2×C3×C4—62no—
240C2×C3×C4—94no—
241C2×C3×C4—182no—
242C2×C3×C4—182no—
243C2×C3×C4—182no—
244C2×C3×C4—181no—
245C2×C3×C4—182no—
246C2×C3×C4—182no—
247C2×C3×C4—182no—
248C2×C3×C4#62612yes—
249C2×Dic3—34no—
250C2×Dic3—62no—
251C2×Dic3—62no—
252C2×Dic3—62no—
253C2×Dic3—62no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—62no—
257C2×Dic3—62no—
258C2×Dic3—64no—
259C2×Dic3—62no—
260C2×Dic3—61no—
261C2×Dic3—94no—
262C2×Dic3—122no—
263C2×Dic3—122no—
264C2×Dic3—182no—
265C2×Dic3—182no—
266C2×Dic3—182no—
267C2×Dic3—181no—
268C2×C32—21no—
269C2×C32—22no—
270C2×C32—22no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32#87411yescharacteristic
274C2×C32#87812yes—
275C2×C32#88312yes—
276C2×C32#88612yes—
277C2×C32#88611yescharacteristic
278C2×C32#89711yescharacteristic
279C2×C32#89712yes—
280C2×C32#92411yescharacteristic
281C2×C32#93511yescharacteristic
282C2×C32#94011yescharacteristic
283C3⋊D3—42no—
284C3×D3—122no—
285C24—91no—
286C22×C4—92no—
287C22×C4—91no—
288C22×C4—92no—
289C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
290C4⋊C4—182no—
291C4⋊C4—544no—
292C22⋊C4—182no—
293C22⋊C4—544no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—32no—
299C22×C3—32no—
300C22×C3—32no—
301C22×C3—32no—
302C22×C3—31no—
303C22×C3—31no—
304C22×C3—61no—
305C22×C3—61no—
306C22×C3—62no—
307C22×C3—62no—
308C22×C3—61no—
309C22×C3—182no—
310C22×C3—184no—
311C22×C3—361no—
312C22×C3—362no—
313C22×C3—361no—
314C22×C3D3×Dic611yescharacteristic
315C22×C3#379511yescharacteristic
316C22×C3D3×D1212yes—
317C22×C3#380512yes—
318C22×C3C2×#62611yescharacteristic
319C2×D3—62no—
320C2×D3—64no—
321C2×D3—122no—
322C2×D3—121no—
323C2×D3—121no—
324C2×D3—182no—
325C2×D3—184no—
326C2×D3—361no—
327C2×D3—362no—
328C2×D3—361no—
329C3×C4—22no—
330C3×C4—62no—
331C3×C4—62no—
332C3×C4—64no—
333C3×C4—184no—
334C3×C4—361no—
335C3×C4—362no—
336Dic3—64no—
337Dic3—122no—
338Dic3—122no—
339Dic3—124no—
340Dic3—121no—
341Dic3—184no—
342Dic3—361no—
343C32—21no—
344C32#467711yescharacteristic, complemented
345C32#494911yescharacteristic, complemented
346C23—92no—
347C23—181no—
348C23—182no—
349C23—182no—
350C23—181no—
351C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
352C2×C4—22no—
353C2×C4—94no—
354C2×C4—182no—
355C2×C4—182no—
356C2×C4—182no—
357C2×C4—182no—
358C2×C4—182no—
359C2×C4—181no—
360C2×C4—182no—
361C2×C4#620012yes—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—32no—
367C2×C3—32no—
368C2×C3—32no—
369C2×C3—32no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—61no—
373C2×C3—61no—
374C2×C3—62no—
375C2×C3—62no—
376C2×C3—61no—
377C2×C3—362no—
378C2×C3#6385711yescharacteristic
379C2×C3#6389112yes—
380C2×C3#6389312yes—
381C2×C3#6391111yescharacteristic
382C2×C3#6396211yescharacteristic
383D3—122no—
384D3—362no—
385C22—182no—
386C22—184no—
387C22—361no—
388C22—362no—
389C22—361no—
390C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
391C22#8864811yescharacteristic
392C22#8865412yes—
393C22#8867212yes—
394C22C2×#620011yescharacteristic
395C4—22no—
396C4—184no—
397C4—362no—
398C4—361no—
399C3—31no—
400C3—31no—
401C3—61no—
402C3#1059131711yescharacteristic
403C2—362no—
404C2#1172110711yescharacteristic
405C2#1172113112yes—
406C2#1172113312yes—
407C2#1172115811yescharacteristic
408C2#1172130711yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#399
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
1. C2×C4⋊#1084C2semidirect
4. C2×#88448C2semidirect
5. #63658⋊C3C2semidirect
26. #63658D3semidirect
213. H(3,𝔽3)#399semidirect
344. C32#4677semidirect
345. C32#4949semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
C2×#3752⋊C32
C2×#884502
C2×#884482
#63658⋊C32
#105913503
#105913243