Finite Group Databasebeta

← #13151460

Subgroups — #13151460

Order 1728.

1193 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1193 conjugacy classes of subgroups — too many to draw. Only its 219 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×#88744C211yesmaximal, characteristic
2D4×(C3⋊Dic3)⋊C3C212yesmaximal, complemented
3C2×#88449C212yesmaximal, complemented
4#11721298C212yesmaximal, complemented
5#11721293C214yesmaximal, complemented
6#11721152C211yesmaximal, characteristic, complemented
7#11721097C211yesmaximal, characteristic, complemented
8#11721083C212yesmaximal, complemented
9#10595222—32nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235C2212yes—
12C22×#1084C2212yes—
13#88744C418yescomplemented
14C2×C4×#272C2211yescharacteristic
15C2×#6229C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2214yes—
17C2×#6199C2212yes—
18#3757⋊C3C2214yes—
19C4⋊((C3⋊Dic3)⋊C3)C2212yes—
20C4×(C3⋊Dic3)⋊C3C2212yes—
21#88451C2212yes—
22#88449C2214yes—
23#88449C2211yescharacteristic
24#88449C2212yes—
25#88449C2211yescharacteristic
26#88447C2214yes—
27C2×C3×#237—32no—
28C2×C3×#237—61no—
29D4×C3⋊Dic3D312yescomplemented
30C3×D4×Dic3—32no—
31#67⋊Dic3—34no—
32Dic3×#67—34no—
33C2×Dic3⋊Dic3—34no—
34#63955—34no—
35D3×C4⋊Dic3—32no—
36#63855—32no—
37#63843—32no—
38Dic3×Dic6—32no—
39C23×H(3,𝔽3)C2312yes—
40C22×#272C2311yescharacteristic
41D4×H(3,𝔽3)C2×C414yescomplemented
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43#6235C2×C418yescomplemented
44C2×#1084C2×C418yescomplemented
45C2×#1084C2312yes—
46C2×#1084C2311yescharacteristic
47C4×#272C2×C414yescomplemented
48#6229C2×C414yescomplemented
49C2×(C3⋊Dic3)⋊C3—44no—
50C2×(C3⋊Dic3)⋊C3C2314yes—
51C2×(C3⋊Dic3)⋊C3C2312yes—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53#6199—42no—
54#5827—91no—
55#5546—92no—
56C2×C32×D4—21no—
57C2×C32×D4C2×D312yes—
58C22×C3⋊Dic3C2×D314yes—
59C2×C3×#67—34no—
60C2×C3×#67—62no—
61C22×C3×Dic3—34no—
62C22×C3×Dic3—34no—
63C22×C3×Dic3—62no—
64C3×#237—316no—
65C3×#237—68no—
66C2×C3×C4×D3—32no—
67C2×C3×C4×D3—61no—
68C2×C3×Dic6—32no—
69C2×C3×Dic6—61no—
70C2×D3×Dic3—34no—
71#3757C2×D314yes—
72C4⋊(C3⋊Dic3)C2×D312yes—
73C4×C3⋊Dic3C2×D312yes—
74C3×#217—34no—
75C3×C4⋊Dic3—32no—
76C3×C4×Dic3—32no—
77Dic3⋊Dic3—34no—
78Dic3⋊Dic3—34no—
79Dic3⋊Dic3—32no—
80Dic3⋊Dic3—32no—
81Dic3⋊Dic3—34no—
82(C2×D3)⋊Dic3—32no—
83(C2×D3)⋊Dic3—32no—
84Dic32—34no—
85Dic32—34no—
86C22×H(3,𝔽3)—22no—
87C22×H(3,𝔽3)—22no—
88C22×H(3,𝔽3)C22×C414yes—
89C22×H(3,𝔽3)C2411yescharacteristic
90C2×#272—21no—
91C2×#272—21no—
92C2×#272C22×C412yes—
93C4×H(3,𝔽3)C22×C412yes—
94#1084C22×C414yes—
95#1084C22×C412yes—
96(C3⋊Dic3)⋊C3—24no—
97(C3⋊Dic3)⋊C3—22no—
98(C3⋊Dic3)⋊C3—22no—
99C2×C3×#41—91no—
100C2×#237—32no—
101C2×#237—61no—
102C3×C4×D4—92no—
103Q8×Dic3—92no—
104D4×Dic3—32no—
105D4×Dic3—92no—
106D4×Dic3—94no—
107C4×#67—94no—
108#926—91no—
109#926—92no—
110C2×C4⋊Dic3—91no—
111C2×C4⋊Dic3—92no—
112C2×Dic3⋊C4—94no—
113D3×C4⋊C4—92no—
114#887—92no—
115#67⋊C4—94no—
116#877—94no—
117#872—92no—
118C4×Dic6—92no—
119C23×C32—22no—
120C23×C32C22×D314yes—
121C22×C3×D3—32no—
122C22×C3×D3—61no—
123C32×D4—24no—
124C32×D4C4×D318yescomplemented
125C2×C32×C4—21no—
126C2×C32×C4C22×D312yes—
127C2×C3⋊Dic3—44no—
128C2×C3⋊Dic3C22×D312yes—
129C2×C3⋊Dic3C22×D314yes—
130C2×C3⋊Dic3C22×D312yes—
131C3×#67—316no—
132C3×#67—68no—
133C2×C3×Dic3—316no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—32no—
139C2×C3×Dic3—68no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—124no—
143C3×C4×D3—38no—
144C3×C4×D3—64no—
145C3×Dic6—38no—
146C3×Dic6—64no—
147D3×Dic3—124no—
148#520—271noSylow
149C2×H(3,𝔽3)—24no—
150C2×H(3,𝔽3)C23×C411yescharacteristic
151C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
152C2×H(3,𝔽3)2−1+411yescharacteristic
153#272—22no—
154C3×#41—98no—
155C2×C3×Q8—91no—
156C2×C3×D4—32no—
157C2×C3×D4—61no—
158C2×C3×D4—92no—
159C2×C3×D4D3211yescharacteristic
160C22×C3×C4—92no—
161C22×C3×C4—91no—
162C22×C3×C4—94no—
163C2×#67—34no—
164C2×#67—62no—
165C22×Dic3—34no—
166C22×Dic3—34no—
167C22×Dic3—62no—
168C22×Dic3—94no—
169C22×Dic3—92no—
170#237—316no—
171#237—68no—
172C2×C4×D3—32no—
173C2×C4×D3—61no—
174C2×C4×D3—94no—
175C2×Dic6—32no—
176C2×Dic6—61no—
177C3×C4⋊C4—92no—
178C3×C22⋊C4—94no—
179C3×C42—92no—
180#217—34no—
181#217—94no—
182#217—92no—
183(C2×D3)⋊C4—92no—
184(C2×D3)⋊C4—92no—
185C4⋊Dic3—32no—
186C4⋊Dic3—92no—
187C4⋊Dic3—91no—
188C4⋊Dic3—92no—
189C4⋊Dic3—94no—
190C4⋊Dic3—91no—
191Dic3⋊C4—94no—
192Dic3⋊C4—92no—
193Dic3⋊C4—92no—
194Dic3⋊C4—94no—
195Dic3⋊C4—94no—
196C4×Dic3—32no—
197C4×Dic3—92no—
198C4×Dic3—94no—
199C4×Dic3—94no—
200C4×Dic3—94no—
201C22×C32—24no—
202C22×C32—24no—
203C22×C32—21no—
204C22×C32—24no—
205C22×C32—42no—
206C22×C32—42no—
207C22×C32C2×C4×D318yes—
208C22×C32C23×D312yes—
209C2×C3×D3—34no—
210C2×C3×D3—62no—
211C2×C3×D3—62no—
212C2×C3×D3—62no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215C32×C4—22no—
216C32×C4C2×C4×D314yes—
217C3⋊Dic3—24no—
218C3⋊Dic3—22no—
219C3⋊Dic3—22no—
220C3×Dic3—38no—
221C3×Dic3—34no—
222C3×Dic3—64no—
223C3×Dic3—62no—
224C3×Dic3—64no—
225C3×Dic3—62no—
226C3×Dic3—62no—
227C2×#41—91no—
228C4×Q8—272no—
229C4×D4—92no—
230C4×D4—274no—
231#117—271no—
232#117—272no—
233C2×C4⋊C4—271no—
234C2×C4⋊C4—272no—
235H(3,𝔽3)#52011yescharacteristic, complemented, Sylow
236C23×C3—34no—
237C23×C3—62no—
238C23×C3—91no—
239C23×C3C2×D3212yes—
240C22×D3—32no—
241C22×D3—61no—
242C3×Q8—94no—
243C3×D4—38no—
244C3×D4—64no—
245C3×D4—98no—
246C3×D4#62414yes—
247C2×C3×C4—32no—
248C2×C3×C4—61no—
249C2×C3×C4—92no—
250C2×C3×C4—91no—
251C2×C3×C4—98no—
252C2×C3×C4—94no—
253C2×C3×C4—92no—
254C2×C3×C4—94no—
255C2×C3×C4—92no—
256C2×C3×C4—364no—
257C2×C3×C4C2×D3211yescharacteristic
258#67—316no—
259#67—68no—
260C2×Dic3—34no—
261C2×Dic3—32no—
262C2×Dic3—316no—
263C2×Dic3—32no—
264C2×Dic3—32no—
265C2×Dic3—34no—
266C2×Dic3—62no—
267C2×Dic3—61no—
268C2×Dic3—68no—
269C2×Dic3—92no—
270C2×Dic3—92no—
271C2×Dic3—94no—
272C2×Dic3—124no—
273C2×Dic3—364no—
274C2×Dic3—362no—
275C4×D3—38no—
276C4×D3—64no—
277C4×D3—364no—
278Dic6—38no—
279Dic6—64no—
280C2×C32—28no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—44no—
285C2×C32C22×C4×D312yes—
286C2×C32#100412yes—
287C2×C32#101012yes—
288C3×D3—64no—
289C3×D3—122no—
290#41—98no—
291C2×Q8—91no—
292C2×D4—92no—
293C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
294C22×C4—92no—
295C22×C4—91no—
296C22×C4—94no—
297C22×C4—272no—
298C4⋊C4—92no—
299C4⋊C4—271no—
300C4⋊C4—272no—
301C4⋊C4—274no—
302C4⋊C4—271no—
303C22⋊C4—94no—
304C22⋊C4—272no—
305C42—92no—
306C42—274no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—32no—
310C22×C3—38no—
311C22×C3—64no—
312C22×C3—64no—
313C22×C3—61no—
314C22×C3—64no—
315C22×C3—92no—
316C22×C3—122no—
317C22×C3—122no—
318C22×C3—181no—
319C22×C3—181no—
320C22×C3C2×#62414yes—
321C22×C3C22×D3211yescharacteristic
322C2×D3—34no—
323C2×D3—62no—
324C2×D3—62no—
325C2×D3—62no—
326C2×D3—121no—
327C2×D3—121no—
328C3×C4—34no—
329C3×C4—62no—
330C3×C4—92no—
331C3×C4—94no—
332C3×C4—184no—
333C3×C4—182no—
334C3×C4—182no—
335C3×C4C2×#62412yes—
336Dic3—38no—
337Dic3—34no—
338Dic3—64no—
339Dic3—62no—
340Dic3—62no—
341Dic3—62no—
342Dic3—64no—
343Dic3—184no—
344Dic3—182no—
345Dic3—182no—
346C32—21no—
347C32#554612yescomplemented
348C23—91no—
349C23C2×((C3⋊D3)⋊C3)⋊C212yes—
350Q8—94no—
351D4—98no—
352D4#619914yescomplemented
353C2×C4—92no—
354C2×C4—91no—
355C2×C4—98no—
356C2×C4—94no—
357C2×C4—92no—
358C2×C4—92no—
359C2×C4—94no—
360C2×C4—364no—
361C2×C4—1082no—
362C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
363C2×C3—24no—
364C2×C3—32no—
365C2×C3—32no—
366C2×C3—32no—
367C2×C3—68no—
368C2×C3—61no—
369C2×C3—61no—
370C2×C3—61no—
371C2×C3—124no—
372C2×C3—182no—
373C2×C3#6430911yescharacteristic
374C2×C3#6431311yescharacteristic
375C2×C3C22×#62411yescharacteristic
376D3—64no—
377D3—122no—
378C22—22no—
379C22—22no—
380C22—92no—
381C22—181no—
382C22—181no—
383C22C2×#619914yes—
384C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
385C4—92no—
386C4—94no—
387C4—182no—
388C4—182no—
389C4—184no—
390C4C2×#619912yes—
391C3—32no—
392C3—61no—
393C3#1059521911yescharacteristic
394C2—24no—
395C2—182no—
396C2#1172250211yescharacteristic
397C2#1172250611yescharacteristic
398C2C22×#619911yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC2×D4×H(3,𝔽3)C22
prime radicalC2×#88744C2
cosocleC22×H(3,𝔽3)C24
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#520
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D4×(C3⋊Dic3)⋊C3C2semidirect
3. C2×#88449C2semidirect
4. #11721298C2semidirect
5. #11721293C2semidirect
6. #11721152C2semidirect
7. #11721097C2semidirect
8. #11721083C2semidirect
13. #88744C4semidirect
29. D4×C3⋊Dic3D3semidirect
41. D4×H(3,𝔽3)C2×C4semidirect
43. #6235C2×C4semidirect
44. C2×#1084C2×C4semidirect
47. C4×#272C2×C4semidirect
48. #6229C2×C4semidirect
124. C32×D4C4×D3semidirect
235. H(3,𝔽3)#520semidirect
347. C32#5546semidirect
352. D4#6199semidirect

Maximal subgroups

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

subgroupindex
C2×#887442
D4×(C3⋊Dic3)⋊C32
C2×#884492
#117212982
#117212932
#117211522
#117210972
#117210832
#105952223