Finite Group Databasebeta

← #13150676

Subgroups — #13150676

Order 1728.

1121 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1121 conjugacy classes of subgroups — too many to draw. Only its 203 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×#88672C211yesmaximal, characteristic
2C2×#88540C211yesmaximal, characteristic, complemented
3C4×#638⋊C3C212yesmaximal, complemented
4#11721315C212yesmaximal, complemented
5#11721298C211yesmaximal, characteristic, complemented
6#11721122C212yesmaximal, complemented
7#11721120C212yesmaximal, complemented
8#11721119C212yesmaximal, complemented
9#11721077C212yesmaximal, complemented
10#10594874—32nomaximal
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C2×#638⋊C3C2212yes—
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×#6198C2211yescharacteristic
17#88672C418yescomplemented
18C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
19C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
20C2×#6196C2211yescharacteristic
21#88540C2212yes—
22#88540C2212yes—
23#3757⋊C3C2212yes—
24#3752⋊C3C2212yes—
25#3750⋊C3C2212yes—
26C4×(C3⋊Dic3)⋊C3C2212yes—
27#88450C2212yes—
28#88449C2211yescharacteristic
29#88449C2211yescharacteristic
30#88448C2212yes—
31#88447C2212yes—
32#88447C2212yes—
33C2×#3804—32no—
34C4×#638D312yescomplemented
35C3×C4×#67—32no—
36C2×C3×Dic3⋊C4—32no—
37C2×C3×Dic3⋊C4—61no—
38Dic3×#67—32no—
39#63955—32no—
40#63952—32no—
41#63904—32no—
42#63883—32no—
43#63881—32no—
44#63878—32no—
45D3×Dic3⋊C4—32no—
46#63838—32no—
47#63837—32no—
48#63836—32no—
49C23×H(3,𝔽3)C2311yescharacteristic
50C22×(C3⋊D3)⋊C3C2312yes—
51C2×C4×H(3,𝔽3)—41no—
52C2×C4×H(3,𝔽3)C2312yes—
53C2×#1084—41no—
54C2×#1084C2×C414yescomplemented
55C2×#1084C2312yes—
56C2×#1084C2311yescharacteristic
57C2×#1084C2311yescharacteristic
58#638⋊C3C2×C418yescomplemented
59C2×(C3⋊Dic3)⋊C3C2312yes—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C4×(C3⋊D3)⋊C3—42no—
63#6198C2×C414yescomplemented
64((C3⋊D3)⋊C3)⋊C4—42no—
65((C3⋊D3)⋊C3)⋊C4C2×C418yescomplemented
66#6196C2×C414yescomplemented
67#5546—92no—
68#5504—91no—
69C22×C32×C4—21no—
70C22×C32×C4C2×D312yes—
71C2×#638C2×D312yes—
72C2×C4×C3⋊D3C2×D312yes—
73C2×C3×#67—32no—
74C22×C3×Dic3—32no—
75C22×C3×Dic3—32no—
76C22×C3×Dic3—61no—
77C22×C3×Dic3—61no—
78C2×C3×C4×D3—32no—
79C2×#627—32no—
80C2×#626—32no—
81C2×#624—32no—
82C2×#624—32no—
83#3804—316no—
84C2×D3×Dic3—32no—
85C2×D3×Dic3—32no—
86#3757C2×D312yes—
87#3752C2×D312yes—
88#3750C2×D312yes—
89C4×C3⋊Dic3C2×D312yes—
90C3×#217—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4—32no—
94C3×Dic3⋊C4—32no—
95C3×Dic3⋊C4—32no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—62no—
98C3×Dic3⋊C4—62no—
99C3×C4×Dic3—32no—
100#3724—32no—
101Dic3⋊Dic3—32no—
102Dic3⋊Dic3—32no—
103Dic3⋊Dic3—32no—
104#3722—32no—
105(C2×D3)⋊Dic3—32no—
106Dic32—32no—
107Dic32—32no—
108Dic32—32no—
109Dic32—32no—
110C22×H(3,𝔽3)—21no—
111C22×H(3,𝔽3)—21no—
112C22×H(3,𝔽3)C22×C412yes—
113C22×H(3,𝔽3)C2411yescharacteristic
114C2×(C3⋊D3)⋊C3—22no—
115C2×(C3⋊D3)⋊C3—22no—
116C2×(C3⋊D3)⋊C3C22×C414yes—
117C4×H(3,𝔽3)—22no—
118#1084—22no—
119#1084C22×C412yes—
120#1084C22×C412yes—
121(C3⋊Dic3)⋊C3—22no—
122(C3⋊Dic3)⋊C3—22no—
123(C3⋊Dic3)⋊C3C22×C414yes—
124C2×#237—92no—
125C2×#235—91no—
126C3×C4×D4—92no—
127C2×C3×C4⋊C4—91no—
128D4×Dic3—92no—
129C4×#67—32no—
130C4×#67—92no—
131C4×#67—92no—
132C4×#67—92no—
133#926—91no—
134C2×Dic3⋊C4—32no—
135C2×Dic3⋊C4—61no—
136C2×Dic3⋊C4—92no—
137#893—92no—
138#892—92no—
139D3×C4⋊C4—92no—
140D3×C4⋊C4—92no—
141#887—92no—
142#887—92no—
143#67⋊C4—92no—
144#67⋊C4—92no—
145#877—92no—
146#877—92no—
147#872—92no—
148C4×Dic6—92no—
149C23×C32—21no—
150C23×C32C22×D312yes—
151C22×C3⋊D3C22×D312yes—
152C22×C3×D3—32no—
153C2×C32×C4—22no—
154C2×C32×C4—42no—
155C2×C32×C4—42no—
156C2×C32×C4C22×D312yes—
157C2×C32×C4C22×D312yes—
158#638C4×D318yescomplemented
159C2×C3⋊Dic3C22×D312yes—
160C2×C3⋊Dic3C22×D312yes—
161C2×C3⋊Dic3C22×D312yes—
162C4×C3⋊D3—42no—
163C3×#67—38no—
164C2×C3×Dic3—38no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—64no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—122no—
177C2×C3×Dic3—122no—
178C3×C4×D3—122no—
179#627—38no—
180#626—38no—
181#624—38no—
182#624—122no—
183D3×Dic3—38no—
184D3×Dic3—122no—
185#520—271noSylow
186C2×H(3,𝔽3)—22no—
187C2×H(3,𝔽3)C23×C411yescharacteristic
188C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
189C2×H(3,𝔽3)2−1+411yescharacteristic
190(C3⋊D3)⋊C3—24no—
191C2×C3×D4—92no—
192C22×C3×C4—32no—
193C22×C3×C4—61no—
194C22×C3×C4—92no—
195C22×C3×C4—91no—
196C22×C3×C4—91no—
197C22×C3×C4D3211yescharacteristic
198C2×#67—32no—
199C2×#67—92no—
200C2×#67—92no—
201C22×Dic3—32no—
202C22×Dic3—32no—
203C22×Dic3—61no—
204C22×Dic3—61no—
205C22×Dic3—92no—
206C22×Dic3—92no—
207#237—916no—
208#235—98no—
209C2×D12—91no—
210C2×C4×D3—32no—
211C2×C4×D3—92no—
212C2×C4×D3—92no—
213C2×C4×D3—92no—
214C2×C4×D3—92no—
215C2×C4×D3—92no—
216C2×Dic6—91no—
217C2×Dic6—92no—
218C3×C4⋊C4—92no—
219C3×C4⋊C4—92no—
220C3×C4⋊C4—92no—
221C3×C22⋊C4—92no—
222C3×C22⋊C4—92no—
223C3×C42—92no—
224#217—32no—
225#217—92no—
226#217—92no—
227(C2×D3)⋊C4—32no—
228(C2×D3)⋊C4—92no—
229(C2×D3)⋊C4—92no—
230(C2×D3)⋊C4—92no—
231C4⋊Dic3—91no—
232C4⋊Dic3—91no—
233C4⋊Dic3—92no—
234Dic3⋊C4—32no—
235Dic3⋊C4—32no—
236Dic3⋊C4—32no—
237Dic3⋊C4—32no—
238Dic3⋊C4—32no—
239Dic3⋊C4—62no—
240Dic3⋊C4—62no—
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246C4×Dic3—32no—
247C4×Dic3—92no—
248C4×Dic3—92no—
249C4×Dic3—92no—
250C4×Dic3—92no—
251C4×Dic3—92no—
252C4×Dic3—92no—
253C4×Dic3—92no—
254C22×C32—22no—
255C22×C32—22no—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32—21no—
259C22×C32—22no—
260C22×C32C2×C4×D314yes—
261C22×C32C23×D312yes—
262C2×C3⋊D3—22no—
263C2×C3⋊D3—22no—
264C2×C3⋊D3C2×C4×D314yes—
265C2×C3×D3—34no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C32×C4—22no—
269C32×C4—22no—
270C32×C4—42no—
271C3⋊Dic3—22no—
272C3⋊Dic3—22no—
273C3⋊Dic3C2×C4×D314yes—
274C3×Dic3—34no—
275C3×Dic3—34no—
276C3×Dic3—34no—
277C3×Dic3—62no—
278C3×Dic3—62no—
279C3×Dic3—62no—
280C3×Dic3—62no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—122no—
284C2×#41—271no—
285C4×Q8—272no—
286C4×D4—92no—
287C4×D4—272no—
288C4×D4—272no—
289#117—272no—
290#117—271no—
291C2×C4⋊C4—91no—
292C2×C4⋊C4—272no—
293H(3,𝔽3)#52011yescharacteristic, complemented, Sylow
294C23×C3—32no—
295C23×C3—61no—
296C23×C3—92no—
297C23×C3C2×D3211yescharacteristic
298C22×D3—32no—
299C22×D3—92no—
300C3×D4—98no—
301C2×C3×C4—32no—
302C2×C3×C4—32no—
303C2×C3×C4—41no—
304C2×C3×C4—62no—
305C2×C3×C4—92no—
306C2×C3×C4—91no—
307C2×C3×C4—91no—
308C2×C3×C4—94no—
309C2×C3×C4—92no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4—122no—
313C2×C3×C4—122no—
314C2×C3×C4—362no—
315C2×C3×C4—361no—
316C2×C3×C4C2×D3212yes—
317#67—38no—
318#67—98no—
319#67—98no—
320C2×Dic3—32no—
321C2×Dic3—32no—
322C2×Dic3—38no—
323C2×Dic3—32no—
324C2×Dic3—32no—
325C2×Dic3—32no—
326C2×Dic3—32no—
327C2×Dic3—32no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—64no—
331C2×Dic3—62no—
332C2×Dic3—92no—
333C2×Dic3—92no—
334C2×Dic3—92no—
335C2×Dic3—98no—
336C2×Dic3—122no—
337C2×Dic3—122no—
338C2×Dic3—362no—
339D12—94no—
340C4×D3—98no—
341C4×D3—98no—
342C4×D3—122no—
343C4×D3—362no—
344C4×D3—362no—
345C4×D3—362no—
346Dic6—94no—
347Dic6—98no—
348C2×C32—24no—
349C2×C32—24no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32C22×C4×D312yes—
354C2×C32#100412yes—
355C2×C32#101012yes—
356C3⋊D3—24no—
357C3×D3—64no—
358#41—278no—
359C2×Q8—271no—
360C2×D4—92no—
361C2×D4—271no—
362C22×C4—91no—
363C22×C4—91no—
364C22×C4—92no—
365C22×C4—272no—
366C22×C4—272no—
367C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
368C4⋊C4—92no—
369C4⋊C4—92no—
370C4⋊C4—92no—
371C4⋊C4—271no—
372C4⋊C4—272no—
373C4⋊C4—271no—
374C22⋊C4—92no—
375C22⋊C4—92no—
376C22⋊C4—272no—
377C42—92no—
378C42—272no—
379C42—272no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—32no—
383C22×C3—34no—
384C22×C3—62no—
385C22×C3—62no—
386C22×C3—62no—
387C22×C3—62no—
388C22×C3—61no—
389C22×C3—62no—
390C22×C3—94no—
391C22×C3—182no—
392C22×C3—182no—
393C22×C3C4×D3212yes—
394C22×C3C22×D3211yescharacteristic
395C2×D3—34no—
396C2×D3—62no—
397C2×D3—62no—
398C2×D3—94no—
399C2×D3—182no—
400C2×D3—182no—
401C3×C4—22no—
402C3×C4—62no—
403C3×C4—62no—
404C3×C4—94no—
405C3×C4—92no—
406C3×C4—92no—
407C3×C4—122no—
408C3×C4—182no—
409C3×C4—182no—
410C3×C4—182no—
411Dic3—34no—
412Dic3—34no—
413Dic3—34no—
414Dic3—62no—
415Dic3—62no—
416Dic3—62no—
417Dic3—62no—
418Dic3—62no—
419Dic3—62no—
420Dic3—94no—
421Dic3—122no—
422Dic3—182no—
423Dic3—182no—
424C32—21no—
425C32#554612yescomplemented
426C23—92no—
427C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
428Q8—274no—
429D4—98no—
430D4—274no—
431C2×C4—41no—
432C2×C4—92no—
433C2×C4—91no—
434C2×C4—91no—
435C2×C4—94no—
436C2×C4—92no—
437C2×C4—92no—
438C2×C4—92no—
439C2×C4—278no—
440C2×C4—362no—
441C2×C4—361no—
442C2×C4—1082no—
443C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
444C2×C3—22no—
445C2×C3—32no—
446C2×C3—32no—
447C2×C3—32no—
448C2×C3—64no—
449C2×C3—64no—
450C2×C3—61no—
451C2×C3—61no—
452C2×C3—61no—
453C2×C3—184no—
454C2×C3#6429811yescharacteristic
455C2×C3C2×C4×D3211yescharacteristic
456C2×C3#6433011yescharacteristic
457D3—64no—
458D3—184no—
459C22—21no—
460C22—21no—
461C22—94no—
462C22—182no—
463C22—182no—
464C22C4×((C3⋊D3)⋊C3)⋊C212yes—
465C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
466C4—22no—
467C4—92no—
468C4—92no—
469C4—94no—
470C4—182no—
471C4—182no—
472C4—182no—
473C3—32no—
474C3—61no—
475C3#1059487611yescharacteristic
476C2—22no—
477C2—184no—
478C2#1172248311yescharacteristic
479C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
480C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC2×#88672C2
cosocleC22×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#520
LCSG ▷ 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
2. C2×#88540C2semidirect
3. C4×#638⋊C3C2semidirect
4. #11721315C2semidirect
5. #11721298C2semidirect
6. #11721122C2semidirect
7. #11721120C2semidirect
8. #11721119C2semidirect
9. #11721077C2semidirect
17. #88672C4semidirect
34. C4×#638D3semidirect
54. C2×#1084C2×C4semidirect
58. #638⋊C3C2×C4semidirect
63. #6198C2×C4semidirect
65. ((C3⋊D3)⋊C3)⋊C4C2×C4semidirect
66. #6196C2×C4semidirect
158. #638C4×D3semidirect
293. H(3,𝔽3)#520semidirect
425. C32#5546semidirect

Maximal subgroups

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

subgroupindex
C2×#886722
C2×#885402
C4×#638⋊C32
#117213152
#117212982
#117211222
#117211202
#117211192
#117210772
#105948743