Finite Group Databasebeta

← #13148474

Subgroups — #13148474

Order 1728.

748 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 131 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#11721733C211yesmaximal, characteristic
2#64088⋊C3C212yesmaximal
3#11721309C211yesmaximal, characteristic
4#11721309C211yesmaximal, characteristic
5#11721297C212yesmaximal
6#11721117C211yesmaximal, characteristic, complemented
7C4×#6196C211yesmaximal, characteristic, complemented
8#11721098C211yesmaximal, characteristic, complemented
9#11721091C212yesmaximal, complemented
10#11721087C212yesmaximal, complemented
11#11721078C211yesmaximal, characteristic, complemented
12#10594247—32nomaximal
13C22×#1084C2211yescharacteristic
14C2×#6196C2211yescharacteristic
15C2×#6196C2211yescharacteristic
16C22⋊C4×H(3,𝔽3)C2211yescharacteristic
17#88552C2211yescharacteristic
18#88540C2211yescharacteristic
19#88540C2211yescharacteristic
20C4×#1084C2211yescharacteristic
21C4×#1084C2211yescharacteristic
22#3757⋊C3C2212yes—
23#3757⋊C3C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2212yes—
25#3750⋊C3C2212yes—
26#3750⋊C3C2212yes—
27C4×(C3⋊Dic3)⋊C3C2212yes—
28#88449C2212yes—
29#88449C2211yescharacteristic
30#88449C2211yescharacteristic
31#88448C2212yes—
32#88448C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88447C2212yes—
36#64088D312yes—
37C3×#879—32no—
38C3×#877—32no—
39C3×#877—61no—
40#63964—32no—
41#63964—32no—
42#63956—32no—
43#63956—32no—
44#63918—32no—
45C4×#627—32no—
46#63870—32no—
47#63867—32no—
48#63867—32no—
49#63847—32no—
50#63845—32no—
51#63837—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×#1084—22no—
56C2×#1084C2311yescharacteristic
57C2×#1084C2311yescharacteristic
58C2×#1084C2311yescharacteristic
59C2×#1084C2311yescharacteristic
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64#6196—22no—
65#6196—22no—
66#5701—92no—
67#5518—91no—
68C22×C3×Dic3—32no—
69C22×C3×Dic3—61no—
70C2×#627—32no—
71C2×#627—32no—
72C32×C22⋊C4—21no—
73C32×C22⋊C4C2×D312yes—
74#3757C2×D312yes—
75#3757C2×D312yes—
76C4⋊(C3⋊Dic3)C2×D312yes—
77#3750C2×D312yes—
78#3750C2×D312yes—
79C4×C3⋊Dic3C2×D312yes—
80C3×#217—32no—
81C3×#217—32no—
82C3×#217—32no—
83C3×#217—61no—
84C3×C4⋊Dic3—32no—
85C3×Dic3⋊C4—32no—
86C3×Dic3⋊C4—32no—
87C3×Dic3⋊C4—32no—
88C3×Dic3⋊C4—32no—
89C3×Dic3⋊C4—61no—
90C3×Dic3⋊C4—61no—
91C3×C4×Dic3—32no—
92C3×C4×Dic3—32no—
93C3×C4×Dic3—32no—
94C3×C4×Dic3—61no—
95C3×C4×Dic3—61no—
96#3724—32no—
97#3724—32no—
98#3724—32no—
99#3724—32no—
100Dic3⋊Dic3—32no—
101Dic3⋊Dic3—32no—
102Dic3⋊Dic3—32no—
103Dic3⋊Dic3—32no—
104Dic3⋊Dic3—32no—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108Dic32—32no—
109Dic32—32no—
110C22×H(3,𝔽3)—21no—
111C22×H(3,𝔽3)—21no—
112C22×H(3,𝔽3)—21no—
113C22×H(3,𝔽3)C2411yescharacteristic
114C4×H(3,𝔽3)—21no—
115C4×H(3,𝔽3)—21no—
116#1084—21no—
117#1084—21no—
118#1084—21no—
119#1084#4112yes—
120(C3⋊Dic3)⋊C3—22no—
121(C3⋊Dic3)⋊C3—22no—
122(C3⋊Dic3)⋊C3—22no—
123(C3⋊Dic3)⋊C3—22no—
124C3×#126—92no—
125C3×#117—91no—
126#924—91no—
127#924—91no—
128#890—92no—
129#889—92no—
130#889—92no—
131#889—92no—
132#889—92no—
133#889—92no—
134#888—92no—
135#887—92no—
136#879—32no—
137#879—92no—
138#879—92no—
139#879—92no—
140#879—92no—
141#878—92no—
142#878—92no—
143#877—32no—
144#877—61no—
145#870—91no—
146#870—92no—
147#869—91no—
148C4×Dic6—91no—
149C4×Dic6—92no—
150C4×Dic6—91no—
151C23×C32—21no—
152C23×C32C22×D312yes—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4C22×D312yes—
156C2×C32×C4C22×D312yes—
157C2×C3⋊Dic3C22×D312yes—
158C2×C3⋊Dic3C22×D312yes—
159C2×C3⋊Dic3C22×D312yes—
160C2×C3⋊Dic3C22×D312yes—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—64no—
174C2×C3×Dic3—64no—
175#627—64no—
176#627—64no—
177#541—271noSylow
178C2×H(3,𝔽3)—41no—
179C2×H(3,𝔽3)C2×#4111yescharacteristic
180C2×H(3,𝔽3)2−1+411yescharacteristic
181C2×H(3,𝔽3)2−1+411yescharacteristic
182C22×C3×C4—91no—
183C22×Dic3—32no—
184C22×Dic3—61no—
185C2×Dic6—91no—
186C2×Dic6—91no—
187C2×Dic6—92no—
188C2×Dic6—92no—
189C3×C4⋊C4—92no—
190C3×C4⋊C4—92no—
191C3×C4⋊C4—92no—
192C3×C4⋊C4—91no—
193C3×C4⋊C4—91no—
194C3×C22⋊C4—32no—
195C3×C22⋊C4—61no—
196C3×C22⋊C4—92no—
197C3×C22⋊C4—92no—
198C3×C22⋊C4—91no—
199C3×C22⋊C4D3211yescharacteristic
200C3×C42—92no—
201C3×C42—91no—
202C3×C42—91no—
203#217—32no—
204#217—32no—
205#217—32no—
206#217—61no—
207#217—92no—
208#217—92no—
209C4⋊Dic3—32no—
210C4⋊Dic3—92no—
211C4⋊Dic3—91no—
212C4⋊Dic3—91no—
213C4⋊Dic3—92no—
214C4⋊Dic3—92no—
215C4⋊Dic3—92no—
216C4⋊Dic3—92no—
217C4⋊Dic3—92no—
218Dic3⋊C4—32no—
219Dic3⋊C4—32no—
220Dic3⋊C4—32no—
221Dic3⋊C4—32no—
222Dic3⋊C4—61no—
223Dic3⋊C4—61no—
224Dic3⋊C4—92no—
225Dic3⋊C4—92no—
226Dic3⋊C4—92no—
227Dic3⋊C4—92no—
228Dic3⋊C4—92no—
229Dic3⋊C4—92no—
230Dic3⋊C4—92no—
231Dic3⋊C4—92no—
232Dic3⋊C4—92no—
233Dic3⋊C4—92no—
234Dic3⋊C4—92no—
235Dic3⋊C4—92no—
236Dic3⋊C4—92no—
237Dic3⋊C4—92no—
238C4×Dic3—32no—
239C4×Dic3—32no—
240C4×Dic3—32no—
241C4×Dic3—61no—
242C4×Dic3—61no—
243C4×Dic3—92no—
244C4×Dic3—92no—
245C4×Dic3—92no—
246C4×Dic3—92no—
247C22×C32—22no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—21no—
252C22×C32—41no—
253C22×C32—41no—
254C22×C32C23×D312yes—
255C32×C4—22no—
256C32×C4—22no—
257C32×C4—41no—
258C32×C4—41no—
259C3⋊Dic3—22no—
260C3⋊Dic3—22no—
261C3⋊Dic3—22no—
262C3⋊Dic3—22no—
263C3×Dic3—34no—
264C3×Dic3—62no—
265C3×Dic3—62no—
266C3×Dic3—62no—
267C3×Dic3—62no—
268C3×Dic3—62no—
269C3×Dic3—62no—
270C3×Dic3—62no—
271C3×Dic3—62no—
272C3×Dic3—62no—
273C3×Dic3—121no—
274C3×Dic3—121no—
275C4⋊Q8—271no—
276#126—92no—
277#126—272no—
278#125—271no—
279#125—272no—
280#125—272no—
281C22⋊Q8—271no—
282C22⋊Q8—271no—
283C4×Q8—271no—
284C4×Q8—271no—
285#117—91no—
286H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
287C23×C3—32no—
288C23×C3—61no—
289C23×C3C2×D3211yescharacteristic
290C2×C3×C4—32no—
291C2×C3×C4—32no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—91no—
295C2×C3×C4—92no—
296C2×C3×C4—92no—
297C2×C3×C4—92no—
298C2×C3×C4—92no—
299C2×C3×C4—91no—
300C2×C3×C4—91no—
301C2×C3×C4—91no—
302C2×C3×C4—182no—
303C2×C3×C4C2×D3211yescharacteristic
304C2×C3×C4C2×D3211yescharacteristic
305C2×Dic3—32no—
306C2×Dic3—32no—
307C2×Dic3—32no—
308C2×Dic3—32no—
309C2×Dic3—32no—
310C2×Dic3—32no—
311C2×Dic3—32no—
312C2×Dic3—32no—
313C2×Dic3—61no—
314C2×Dic3—61no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—64no—
318C2×Dic3—64no—
319C2×Dic3—92no—
320C2×Dic3—92no—
321C2×Dic3—92no—
322C2×Dic3—92no—
323Dic6—184no—
324Dic6—184no—
325Dic6—182no—
326Dic6—182no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—42no—
331C2×C32—42no—
332C2×C32D3×#4112yes—
333C2×C32#101012yes—
334C2×C32#101012yes—
335C2×Q8—271no—
336C2×Q8—271no—
337C22×C4—91no—
338C4⋊C4—91no—
339C4⋊C4—91no—
340C4⋊C4—92no—
341C4⋊C4—92no—
342C4⋊C4—92no—
343C4⋊C4—271no—
344C4⋊C4—271no—
345C4⋊C4—272no—
346C4⋊C4—272no—
347C4⋊C4—272no—
348C4⋊C4—272no—
349C4⋊C4—272no—
350C22⋊C4—91no—
351C22⋊C4—92no—
352C22⋊C4—92no—
353C22⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
354C42—91no—
355C42—91no—
356C42—92no—
357C42—272no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—32no—
362C22×C3—62no—
363C22×C3—62no—
364C22×C3—62no—
365C22×C3—62no—
366C22×C3—61no—
367C22×C3—121no—
368C22×C3—121no—
369C22×C3C22×D3211yescharacteristic
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—62no—
373C3×C4—62no—
374C3×C4—92no—
375C3×C4—121no—
376C3×C4—121no—
377C3×C4—182no—
378C3×C4—182no—
379C3×C4—182no—
380C3×C4—182no—
381C3×C4—181no—
382C3×C4—181no—
383C3×C4—181no—
384Dic3—34no—
385Dic3—62no—
386Dic3—62no—
387Dic3—62no—
388Dic3—62no—
389Dic3—62no—
390Dic3—62no—
391Dic3—62no—
392Dic3—62no—
393Dic3—62no—
394Dic3—121no—
395Dic3—121no—
396Dic3—182no—
397Dic3—182no—
398Dic3—182no—
399Dic3—182no—
400C32—21no—
401C32#570112yescomplemented
402C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
403Q8—542no—
404Q8—542no—
405C2×C4—91no—
406C2×C4—91no—
407C2×C4—91no—
408C2×C4—92no—
409C2×C4—92no—
410C2×C4—92no—
411C2×C4—92no—
412C2×C4—91no—
413C2×C4—182no—
414C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
415C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
416C2×C3—32no—
417C2×C3—32no—
418C2×C3—32no—
419C2×C3—41no—
420C2×C3—61no—
421C2×C3—61no—
422C2×C3—61no—
423C2×C3—122no—
424C2×C3—122no—
425C2×C3#6430611yescharacteristic
426C2×C3#6430911yescharacteristic
427C2×C3#6430911yescharacteristic
428C22—21no—
429C22—21no—
430C22—21no—
431C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
432C4—21no—
433C4—21no—
434C4—92no—
435C4—181no—
436C4—181no—
437C4—182no—
438C4—182no—
439C4—181no—
440C4—182no—
441C4—182no—
442C3—32no—
443C3—61no—
444C3#1059424911yescharacteristic
445C2—41no—
446C2#1172249111yescharacteristic
447C2#1172250211yescharacteristic
448C2#1172250211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22⋊C4×H(3,𝔽3)C22
prime radicalC23×H(3,𝔽3)C23
hypercenterC22⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#541
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
6. #11721117C2semidirect
7. C4×#6196C2semidirect
8. #11721098C2semidirect
9. #11721091C2semidirect
10. #11721087C2semidirect
11. #11721078C2semidirect
286. H(3,𝔽3)#541semidirect
401. C32#5701semidirect

Maximal subgroups

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

subgroupindex
#117217332
#64088⋊C32
#117213092
#117212972
#117211172
C4×#61962
#117210982
#117210912
#117210872
#117210782
#105942473