Finite Group Databasebeta

← #13148475

Subgroups — #13148475

Order 1728.

836 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 133 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
1#11721737C211yesmaximal, characteristic
2#64088⋊C3C212yesmaximal, complemented
3#11721309C211yesmaximal, characteristic, complemented
4#11721298C211yesmaximal, characteristic, complemented
5#11721294C212yesmaximal
6#11721117C211yesmaximal, characteristic, complemented
7#11721097C211yesmaximal, characteristic, complemented
8#11721096C211yesmaximal, characteristic, complemented
9#11721088C212yesmaximal, complemented
10#11721078C211yesmaximal, characteristic, complemented
11#11721077C212yesmaximal, complemented
12#10594248—32nomaximal
13C2×#6235C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C2×C4×#272C2211yescharacteristic
16C2×#6196C2212yes—
17C2×#6196C2211yescharacteristic
18C22⋊C4×H(3,𝔽3)C2211yescharacteristic
19#88542C2211yescharacteristic
20#88540C2211yescharacteristic
21C4×#1084C2211yescharacteristic
22#3757⋊C3C2212yes—
23#3757⋊C3C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2212yes—
25#3750⋊C3C2212yes—
26#3750⋊C3C2212yes—
27C4×(C3⋊Dic3)⋊C3C2212yes—
28#88451C2212yes—
29#88451C2212yes—
30#88449C2211yescharacteristic
31#88449C2211yescharacteristic
32#88449C2211yescharacteristic
33#88448C2212yes—
34#88447C2212yes—
35#88447C2212yes—
36#64088D312yescomplemented
37C3×#67⋊C4—32no—
38C3×#67⋊C4—61no—
39C3×#879—32no—
40#63964—32no—
41#63959—32no—
42#63959—32no—
43#63955—32no—
44#63918—32no—
45#63861—32no—
46#63858—32no—
47#63855—32no—
48#63851—32no—
49#63838—32no—
50#63837—32no—
51#63837—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C22×#272C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56#6235—22no—
57C2×#1084—22no—
58C2×#1084C2311yescharacteristic
59C2×#1084C2311yescharacteristic
60C2×#1084C2311yescharacteristic
61C4×#272—22no—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66#6196—24no—
67#6196—22no—
68#5702—92no—
69#5542—91no—
70C2×C3×#67—32no—
71C2×C3×#67—61no—
72C22×C3×Dic3—32no—
73C22×C3×Dic3—61no—
74C2×C3×C4×D3—32no—
75C2×C3×C4×D3—61no—
76C2×#627—32no—
77C2×#627—32no—
78C2×#627—32no—
79C32×C22⋊C4—21no—
80C32×C22⋊C4C2×D312yes—
81#3757C2×D312yes—
82#3757C2×D312yes—
83C4⋊(C3⋊Dic3)C2×D312yes—
84#3750C2×D312yes—
85#3750C2×D312yes—
86C4×C3⋊Dic3C2×D312yes—
87C3×#217—32no—
88C3×#217—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—61no—
91C3×C4⋊Dic3—32no—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4—32no—
94C3×Dic3⋊C4—32no—
95C3×Dic3⋊C4—61no—
96C3×C4×Dic3—32no—
97C3×C4×Dic3—32no—
98C3×C4×Dic3—61no—
99#3724—32no—
100Dic3⋊Dic3—32no—
101Dic3⋊Dic3—32no—
102Dic3⋊Dic3—32no—
103Dic3⋊Dic3—32no—
104(C2×D3)⋊Dic3—32no—
105(C2×D3)⋊Dic3—32no—
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108Dic32—32no—
109Dic32—32no—
110Dic32—32no—
111Dic32—32no—
112C22×H(3,𝔽3)—21no—
113C22×H(3,𝔽3)—21no—
114C22×H(3,𝔽3)—21no—
115C22×H(3,𝔽3)C2411yescharacteristic
116C2×#272—21no—
117C2×#272—21no—
118C2×#272—21no—
119C4×H(3,𝔽3)—21no—
120C4×H(3,𝔽3)—21no—
121#1084—21no—
122#1084#4112yes—
123#1084#4112yes—
124(C3⋊Dic3)⋊C3—22no—
125(C3⋊Dic3)⋊C3—22no—
126(C3⋊Dic3)⋊C3—22no—
127(C3⋊Dic3)⋊C3—22no—
128C3×#126—92no—
129C3×C4×D4—91no—
130#936—92no—
131#926—91no—
132#926—91no—
133#924—91no—
134#924—91no—
135#898—92no—
136#898—92no—
137#896—92no—
138#888—92no—
139#887—92no—
140#887—92no—
141#887—92no—
142#885—92no—
143#885—92no—
144#67⋊C4—32no—
145#67⋊C4—61no—
146#879—32no—
147#879—92no—
148#879—92no—
149#878—92no—
150#877—92no—
151#872—92no—
152#869—91no—
153C4×Dic6—91no—
154C4×Dic6—92no—
155C23×C32—21no—
156C23×C32C22×D312yes—
157C22×C3×D3—32no—
158C22×C3×D3—61no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4C22×D312yes—
162C2×C32×C4C22×D312yes—
163C2×C3⋊Dic3C22×D312yes—
164C2×C3⋊Dic3C22×D312yes—
165C2×C3⋊Dic3C22×D312yes—
166C2×C3⋊Dic3C22×D312yes—
167C3×#67—64no—
168C3×#67—64no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—64no—
180C2×C3×Dic3—122no—
181C3×C4×D3—64no—
182C3×C4×D3—122no—
183#627—64no—
184#627—64no—
185#627—64no—
186#556—271noSylow
187C2×H(3,𝔽3)—41no—
188C2×H(3,𝔽3)C2×#4111yescharacteristic
189C2×H(3,𝔽3)C2×#4111yescharacteristic
190C2×H(3,𝔽3)2−1+411yescharacteristic
191#272—41no—
192C2×C3×D4—91no—
193C22×C3×C4—91no—
194C22×C3×C4—91no—
195C2×#67—32no—
196C2×#67—61no—
197C22×Dic3—32no—
198C22×Dic3—61no—
199C2×C4×D3—32no—
200C2×C4×D3—61no—
201C2×Dic6—91no—
202C2×Dic6—92no—
203C2×Dic6—92no—
204C2×Dic6—92no—
205C2×Dic6—92no—
206C3×C4⋊C4—92no—
207C3×C4⋊C4—92no—
208C3×C4⋊C4—92no—
209C3×C4⋊C4—91no—
210C3×C22⋊C4—32no—
211C3×C22⋊C4—61no—
212C3×C22⋊C4—92no—
213C3×C22⋊C4—92no—
214C3×C22⋊C4—91no—
215C3×C22⋊C4D3211yescharacteristic
216C3×C42—92no—
217C3×C42—91no—
218#217—32no—
219#217—32no—
220#217—92no—
221#217—92no—
222#217—92no—
223#217—92no—
224(C2×D3)⋊C4—32no—
225(C2×D3)⋊C4—61no—
226(C2×D3)⋊C4—92no—
227(C2×D3)⋊C4—92no—
228(C2×D3)⋊C4—92no—
229(C2×D3)⋊C4—92no—
230C4⋊Dic3—32no—
231C4⋊Dic3—92no—
232C4⋊Dic3—91no—
233C4⋊Dic3—91no—
234C4⋊Dic3—91no—
235C4⋊Dic3—92no—
236Dic3⋊C4—32no—
237Dic3⋊C4—32no—
238Dic3⋊C4—32no—
239Dic3⋊C4—61no—
240Dic3⋊C4—92no—
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247C4×Dic3—32no—
248C4×Dic3—32no—
249C4×Dic3—61no—
250C4×Dic3—92no—
251C4×Dic3—92no—
252C4×Dic3—92no—
253C4×Dic3—92no—
254C4×Dic3—92no—
255C4×Dic3—92no—
256C4×Dic3—92no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—22no—
261C22×C32—21no—
262C22×C32—41no—
263C22×C32—41no—
264C22×C32C23×D312yes—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—121no—
270C2×C3×D3—121no—
271C32×C4—22no—
272C32×C4—22no—
273C32×C4—41no—
274C32×C4—41no—
275C3⋊Dic3—22no—
276C3⋊Dic3—22no—
277C3⋊Dic3—22no—
278C3⋊Dic3—22no—
279C3×Dic3—34no—
280C3×Dic3—34no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C3×Dic3—62no—
286C3×Dic3—62no—
287C3×Dic3—62no—
288C3×Dic3—121no—
289C4⋊Q8—271no—
290#126—92no—
291#126—272no—
292#124—272no—
293C22⋊Q8—271no—
294C22⋊Q8—271no—
295C4×Q8—272no—
296C4×Q8—271no—
297C4×D4—91no—
298#117—271no—
299#117—271no—
300H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
301C23×C3—32no—
302C23×C3—61no—
303C23×C3—91no—
304C23×C3C2×D3211yescharacteristic
305C22×D3—32no—
306C22×D3—61no—
307C3×D4—182no—
308C2×C3×C4—32no—
309C2×C3×C4—32no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—91no—
313C2×C3×C4—91no—
314C2×C3×C4—92no—
315C2×C3×C4—92no—
316C2×C3×C4—92no—
317C2×C3×C4—92no—
318C2×C3×C4—91no—
319C2×C3×C4—182no—
320C2×C3×C4—182no—
321C2×C3×C4C2×D3211yescharacteristic
322C2×C3×C4C2×D3211yescharacteristic
323#67—64no—
324#67—64no—
325C2×Dic3—32no—
326C2×Dic3—32no—
327C2×Dic3—32no—
328C2×Dic3—32no—
329C2×Dic3—32no—
330C2×Dic3—32no—
331C2×Dic3—32no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—64no—
336C2×Dic3—92no—
337C2×Dic3—92no—
338C2×Dic3—92no—
339C2×Dic3—92no—
340C2×Dic3—122no—
341C4×D3—64no—
342C4×D3—122no—
343Dic6—184no—
344Dic6—184no—
345Dic6—184no—
346Dic6—184no—
347Dic6—182no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—42no—
352C2×C32—42no—
353C2×C32D3×#4112yes—
354C2×C32D3×#4112yes—
355C2×C32#101012yes—
356C3×D3—122no—
357C3×D3—122no—
358C2×Q8—271no—
359C2×Q8—272no—
360C2×D4—91no—
361C22×C4—91no—
362C22×C4—91no—
363C4⋊C4—91no—
364C4⋊C4—92no—
365C4⋊C4—92no—
366C4⋊C4—92no—
367C4⋊C4—271no—
368C4⋊C4—271no—
369C4⋊C4—271no—
370C4⋊C4—272no—
371C22⋊C4—92no—
372C22⋊C4—92no—
373C22⋊C4—91no—
374C22⋊C4—272no—
375C22⋊C4—272no—
376C22⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
377C42—92no—
378C42—91no—
379C42—272no—
380C42—272no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—32no—
385C22×C3—62no—
386C22×C3—62no—
387C22×C3—62no—
388C22×C3—62no—
389C22×C3—61no—
390C22×C3—121no—
391C22×C3—121no—
392C22×C3—181no—
393C22×C3—181no—
394C22×C3—181no—
395C22×C3C22×D3211yescharacteristic
396C2×D3—62no—
397C2×D3—62no—
398C2×D3—62no—
399C2×D3—62no—
400C2×D3—121no—
401C2×D3—121no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—62no—
405C3×C4—62no—
406C3×C4—92no—
407C3×C4—92no—
408C3×C4—121no—
409C3×C4—121no—
410C3×C4—182no—
411C3×C4—182no—
412C3×C4—182no—
413C3×C4—182no—
414C3×C4—181no—
415Dic3—34no—
416Dic3—34no—
417Dic3—62no—
418Dic3—62no—
419Dic3—62no—
420Dic3—62no—
421Dic3—62no—
422Dic3—62no—
423Dic3—62no—
424Dic3—121no—
425Dic3—182no—
426Dic3—182no—
427Dic3—182no—
428Dic3—182no—
429C32—21no—
430C32#570212yescomplemented
431C23—91no—
432C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
433Q8—542no—
434Q8—544no—
435D4—182no—
436C2×C4—91no—
437C2×C4—91no—
438C2×C4—92no—
439C2×C4—92no—
440C2×C4—91no—
441C2×C4—92no—
442C2×C4—92no—
443C2×C4—182no—
444C2×C4—182no—
445C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
446C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
447C2×C3—32no—
448C2×C3—32no—
449C2×C3—32no—
450C2×C3—41no—
451C2×C3—61no—
452C2×C3—61no—
453C2×C3—61no—
454C2×C3—122no—
455C2×C3—122no—
456C2×C3—361no—
457C2×C3#6430611yescharacteristic
458C2×C3#6430911yescharacteristic
459C2×C3#6431211yescharacteristic
460D3—122no—
461D3—122no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—181no—
466C22—181no—
467C22—181no—
468C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
469C4—21no—
470C4—21no—
471C4—92no—
472C4—92no—
473C4—182no—
474C4—181no—
475C4—182no—
476C4—182no—
477C4—182no—
478C3—32no—
479C3—61no—
480C3#1059425011yescharacteristic
481C2—41no—
482C2—361no—
483C2#1172249111yescharacteristic
484C2#1172250211yescharacteristic
485C2#1172250511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22⋊C4×H(3,𝔽3)C22
prime radicalC2×#6235C22
hypercenterC22⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#556
LCSG ▷ C22×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. #64088⋊C3C2semidirect
3. #11721309C2semidirect
4. #11721298C2semidirect
6. #11721117C2semidirect
7. #11721097C2semidirect
8. #11721096C2semidirect
9. #11721088C2semidirect
10. #11721078C2semidirect
11. #11721077C2semidirect
36. #64088D3semidirect
300. H(3,𝔽3)#556semidirect
430. C32#5702semidirect

Maximal subgroups

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

subgroupindex
#117217372
#64088⋊C32
#117213092
#117212982
#117212942
#117211172
#117210972
#117210962
#117210882
#117210782
#117210772
#105942483