Finite Group Databasebeta

← #13140037

Subgroups — #13140037

Order 1728.

603 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 93 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, complemented
2#64139⋊C3C211yesmaximal, characteristic
3C2×#88431C212yesmaximal, complemented
4#11720573C211yesmaximal, characteristic, complemented
5#11720313C212yesmaximal, complemented
6#10591181—31nomaximal
7#10591174—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3—21no—
10C22×(C3⋊Dic3)⋊C3—21no—
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3—21no—
13C2×#634⋊C3—24no—
14C2×#634⋊C3C2212yes—
15C2×#6180C2212yes—
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#88431—24no—
19C22×#634D311yescharacteristic, complemented
20C22×C3×Dic6—31no—
21#64139D311yescharacteristic
22C3×#930—31no—
23C2×#3717—32no—
24C2×#3716—32no—
25C3×#830—31no—
26C3×#830—31no—
27C3×#830—61no—
28#63567—32no—
29#63566—32no—
30C23×H(3,𝔽3)D411yescharacteristic
31C22×(C3⋊D3)⋊C3—21no—
32C2×C4×H(3,𝔽3)—22no—
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35#638⋊C3—42no—
36C2×(C3⋊Dic3)⋊C3—24no—
37C2×(C3⋊Dic3)⋊C3—22no—
38C2×(C3⋊Dic3)⋊C3—21no—
39C4⋊((C3⋊D3)⋊C3)—22no—
40#634⋊C3—24no—
41#634⋊C3—22no—
42#634⋊C3D414yescomplemented
43#6180—22no—
44#5227—91no—
45#5040—91no—
46#4738—91no—
47C22×C32×C4—21no—
48C22×C32×C4C2×D311yescharacteristic
49C22×C32×C4C2×D311yescharacteristic
50C2×#638—21no—
51C22×C3⋊Dic3—21no—
52C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
53C2×#634—21no—
54C2×#634—24no—
55C2×#634C2×D312yes—
56C2×C3×#67—61no—
57C22×C3×Dic3—61no—
58C2×C3×D12—31no—
59C2×C3×Dic6—32no—
60C2×C3×Dic6—64no—
61C2×C3×Dic6—61no—
62#3752—21no—
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C3×(C2×D3)⋊C4—61no—
65C3×C4⋊Dic3—31no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—62no—
69#3717—64no—
70#3716—64no—
71C22×H(3,𝔽3)—21no—
72C22×H(3,𝔽3)—21no—
73C22×H(3,𝔽3)SD1612yescomplemented
74C22×H(3,𝔽3)C2×D411yescharacteristic
75C2×(C3⋊D3)⋊C3—41no—
76C2×(C3⋊D3)⋊C3—41no—
77C2×(C3⋊D3)⋊C3—41no—
78C4×H(3,𝔽3)—21no—
79C4×H(3,𝔽3)C2×D412yes—
80(C3⋊Dic3)⋊C3—24no—
81(C3⋊Dic3)⋊C3—41no—
82C22×C3×Q8—91no—
83C22×Dic6—31no—
84C22×Dic6—91no—
85C3×#121—91no—
86C2×#215—92no—
87C2×#214—92no—
88#930—31no—
89#930—91no—
90C2×#204—92no—
91C3×C22⋊C8—91no—
92#835—92no—
93#830—31no—
94#830—31no—
95#830—61no—
96#816—92no—
97#810—92no—
98C23×C32—21no—
99C23×C32#6711yescharacteristic
100C23×C32#6711yescharacteristic
101C22×C3⋊D3—21no—
102C22×C3×D3—61no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—42no—
108C2×C32×C4#6711yescharacteristic
109C2×C32×C4#6711yescharacteristic
110C2×C32×C4C22×D311yescharacteristic
111C2×C32×C4C22×D311yescharacteristic
112#638—42no—
113C2×C3⋊Dic3—24no—
114C2×C3⋊Dic3—22no—
115C2×C3⋊Dic3—21no—
116C4⋊(C3⋊D3)—22no—
117#634—24no—
118#634—22no—
119#634#6714yescomplemented
120C3×#67—122no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—61no—
124C3×D12—62no—
125C3×Dic6—34no—
126C3×Dic6—64no—
127C3×Dic6—62no—
128C3×C3⋊C8—62no—
129C3×C3⋊C8—62no—
130C3×C3⋊C8—122no—
131#448—271noSylow
132C2×H(3,𝔽3)—22no—
133C2×H(3,𝔽3)#12011yescharacteristic
134C2×H(3,𝔽3)C2×SD1611yescharacteristic
135C2×H(3,𝔽3)#13711yescharacteristic
136(C3⋊D3)⋊C3—81no—
137C2×C3×Q8—92no—
138C2×C3×Q8—181no—
139C2×C3×Q8—184no—
140C2×C3×D4—91no—
141C2×C3×D4—181no—
142C22×C3×C4—31no—
143C22×C3×C4—31no—
144C22×C3×C4—61no—
145C22×C3×C4—181no—
146C22×C3×C4D3211yescharacteristic
147C2×#67—61no—
148C2×#67—181no—
149C22×Dic3—61no—
150C22×Dic3—181no—
151C2×D12—31no—
152C2×D12—91no—
153C2×Dic6—32no—
154C2×Dic6—64no—
155C2×Dic6—61no—
156C2×Dic6—92no—
157C2×Dic6—184no—
158C2×Dic6—181no—
159C2×C3×C8—92no—
160C3×C4⋊C4—91no—
161C3×C22⋊C4—181no—
162#215—184no—
163#214—184no—
164(C2×D3)⋊C4—61no—
165(C2×D3)⋊C4—181no—
166C4⋊Dic3—31no—
167C4⋊Dic3—91no—
168C2×C3⋊C8—32no—
169C2×C3⋊C8—32no—
170C2×C3⋊C8—62no—
171#204—184no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—22no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32—41no—
180C22×C32#21412yescomplemented
181C22×C32#21512yescomplemented
182C22×C32C2×#6711yescharacteristic
183C22×C32C2×#6711yescharacteristic
184C2×C3⋊D3—41no—
185C2×C3⋊D3—41no—
186C2×C3⋊D3—41no—
187C2×C3×D3—121no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—22no—
193C32×C4—41no—
194C32×C4D3×D412yes—
195C32×C4C2×#6712yes—
196C3⋊Dic3—24no—
197C3⋊Dic3—41no—
198C3×Dic3—64no—
199C3×Dic3—121no—
200C22×Q8—91no—
201C2×SD16—272no—
202#121—91no—
203#103—272no—
204C22⋊C8—91no—
205H(3,𝔽3)#44811yescharacteristic, complemented, Sylow
206C23×C3—31no—
207C23×C3—31no—
208C23×C3—61no—
209C23×C3—181no—
210C23×C3#62511yescharacteristic
211C22×D3—61no—
212C22×D3—181no—
213C3×Q8—94no—
214C3×Q8—182no—
215C3×Q8—184no—
216C3×D4—182no—
217C3×D4—362no—
218C2×C3×C4—22no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—62no—
224C2×C3×C4—62no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—122no—
228C2×C3×C4—182no—
229C2×C3×C4—184no—
230C2×C3×C4—181no—
231C2×C3×C4#62511yescharacteristic
232C2×C3×C4C2×D3211yescharacteristic
233#67—122no—
234#67—362no—
235C2×Dic3—64no—
236C2×Dic3—62no—
237C2×Dic3—61no—
238C2×Dic3—182no—
239C2×Dic3—184no—
240C2×Dic3—181no—
241D12—62no—
242D12—182no—
243Dic6—34no—
244Dic6—64no—
245Dic6—62no—
246Dic6—94no—
247Dic6—182no—
248Dic6—184no—
249C3×C8—182no—
250C3⋊C8—62no—
251C3⋊C8—62no—
252C3⋊C8—122no—
253C2×C32—22no—
254C2×C32—22no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—42no—
259C2×C32C2×#21411yescharacteristic
260C2×C32#93711yescharacteristic
261C2×C32C2×#21511yescharacteristic
262C2×C32#94211yescharacteristic
263C2×C32#95111yescharacteristic
264C2×C32#95311yescharacteristic
265C3⋊D3—81no—
266C3×D3—241no—
267C2×Q8—92no—
268C2×Q8—184no—
269C2×Q8—181no—
270C2×D4—91no—
271C2×D4—181no—
272C22×C4—181no—
273C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
274SD16—544no—
275C2×C8—92no—
276C4⋊C4—91no—
277C22⋊C4—181no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—31no—
281C22×C3—31no—
282C22×C3—32no—
283C22×C3—32no—
284C22×C3—61no—
285C22×C3—61no—
286C22×C3—61no—
287C22×C3—61no—
288C22×C3—61no—
289C22×C3—62no—
290C22×C3—121no—
291C22×C3—121no—
292C22×C3—361no—
293C22×C3—361no—
294C22×C3—361no—
295C22×C3#371512yes—
296C22×C3C2×#62511yescharacteristic
297C2×D3—121no—
298C2×D3—121no—
299C2×D3—121no—
300C2×D3—361no—
301C2×D3—361no—
302C2×D3—361no—
303C3×C4—21no—
304C3×C4—32no—
305C3×C4—32no—
306C3×C4—61no—
307C3×C4—61no—
308C3×C4—62no—
309C3×C4—121no—
310C3×C4—184no—
311C3×C4—361no—
312C3×C4D3×#6712yes—
313Dic3—64no—
314Dic3—121no—
315Dic3—184no—
316Dic3—361no—
317C32—21no—
318C32#504011yescharacteristic, complemented
319C32#522711yescharacteristic, complemented
320C23—181no—
321C23#619811yescharacteristic
322Q8—94no—
323Q8—184no—
324Q8—182no—
325D4—182no—
326D4—362no—
327C2×C4—22no—
328C2×C4—184no—
329C2×C4—182no—
330C2×C4—181no—
331C2×C4#619811yescharacteristic
332C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
333C8—182no—
334C2×C3—22no—
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—62no—
342C2×C3—62no—
343C2×C3—61no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—122no—
347C2×C3—721no—
348C2×C3C2×#371511yescharacteristic
349C2×C3#6382711yescharacteristic
350C2×C3#6397611yescharacteristic
351D3—241no—
352D3—721no—
353C22—21no—
354C22—21no—
355C22—361no—
356C22—361no—
357C22—361no—
358C22#8843112yescomplemented
359C22C2×#619811yescharacteristic
360C4—21no—
361C4—184no—
362C4—361no—
363C4#8867612yes—
364C3—31no—
365C3—31no—
366C3—61no—
367C3#1059117111yescharacteristic
368C2—22no—
369C2—721no—
370C2C2×#8843111yescharacteristic
371C2#1172105811yescharacteristic
372C2#1172131611yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radical#64139⋊C3C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#448
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#634⋊C3C2semidirect
3. C2×#88431C2semidirect
4. #11720573C2semidirect
5. #11720313C2semidirect
19. C22×#634D3semidirect
42. #634⋊C3D4semidirect
73. C22×H(3,𝔽3)SD16semidirect
119. #634#67semidirect
180. C22×C32#214semidirect
181. C22×C32#215semidirect
205. H(3,𝔽3)#448semidirect
318. C32#5040semidirect
319. C32#5227semidirect
358. C22#88431semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
#64139⋊C32
C2×#884312
#117205732
#117203132
#105911813
#105911743