Finite Group Databasebeta

← #13149605

Subgroups — #13149605

Order 1728.

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

normal subgroups only

This group has 857 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#11721763C211yesmaximal, characteristic
2#64098⋊C3C212yesmaximal, complemented
3#11721151C212yesmaximal, complemented
4#11721151C212yesmaximal, complemented
5#11721150C212yesmaximal, complemented
6#11721149C212yesmaximal, complemented
7#11721147C211yesmaximal, characteristic, complemented
8#11721096C211yesmaximal, characteristic, complemented
9#11721092C212yesmaximal, complemented
10#10594428—32nomaximal
11C2×C4⋊#272C2212yes—
12C2×C4⋊#272C2211yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C2×#6199C2211yescharacteristic
15C2×#6199C2211yescharacteristic
16C2×#6199C2212yes—
17C2×#6196C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19#88542C2212yes—
20C4⋊((C3⋊Dic3)⋊C3)C2212yes—
21#3750⋊C3C2212yes—
22#3750⋊C3C2212yes—
23#3750⋊C3C2212yes—
24#3750⋊C3C2212yes—
25C4×(C3⋊Dic3)⋊C3C2212yes—
26#88451C2212yes—
27#88451C2212yes—
28#88451C2212yes—
29#88451C2212yes—
30#88449C2211yescharacteristic
31#88448C2212yes—
32#64098D312yescomplemented
33C3×#895—32no—
34C3×#895—61no—
35C3×#889—32no—
36#63899—32no—
37#63898—32no—
38#63893—32no—
39Dic3×D12—32no—
40#63884—32no—
41#63884—32no—
42#63884—32no—
43#63884—32no—
44D3×Dic3⋊C4—32no—
45#63862—32no—
46#63859—32no—
47#63851—32no—
48C22×#272C2312yes—
49C22×#272C2311yescharacteristic
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×C4×H(3,𝔽3)C2312yes—
52C2×#1084C2311yescharacteristic
53C4⋊#272—24no—
54C4⋊#272—41no—
55C4×#272—41no—
56C2×(C3⋊Dic3)⋊C3C2312yes—
57C2×(C3⋊Dic3)⋊C3C2312yes—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C2×(C3⋊Dic3)⋊C3C2312yes—
60#6199—22no—
61#6199—42no—
62#6199—41no—
63#6196—41no—
64#5692—92no—
65#5602—91no—
66C2×C3×D12—32no—
67C2×C3×D12—32no—
68C2×C3×D12—32no—
69C2×C3×D12—62no—
70C2×C3×D12—61no—
71C2×C3×C4×D3—32no—
72C2×C3×C4×D3—61no—
73C2×#627—32no—
74C2×D3×Dic3—32no—
75C2×D3×Dic3—32no—
76C2×D3×Dic3—32no—
77C2×D3×Dic3—32no—
78C32×C4⋊C4—21no—
79C32×C4⋊C4C2×D312yes—
80C4⋊(C3⋊Dic3)C2×D312yes—
81#3750C2×D312yes—
82#3750C2×D312yes—
83#3750C2×D312yes—
84#3750C2×D312yes—
85C4×C3⋊Dic3C2×D312yes—
86C3×(C2×D3)⋊C4—32no—
87C3×(C2×D3)⋊C4—32no—
88C3×(C2×D3)⋊C4—62no—
89C3×C4⋊Dic3—32no—
90C3×Dic3⋊C4—32no—
91C3×Dic3⋊C4—32no—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4—32no—
94C3×C4×Dic3—32no—
95#3724—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103(C2×D3)⋊Dic3—32no—
104(C2×D3)⋊Dic3—32no—
105(C2×D3)⋊Dic3—32no—
106C22×H(3,𝔽3)C2411yescharacteristic
107C2×#272—21no—
108C2×#272—21no—
109C2×#272—22no—
110C2×#272—22no—
111C2×#272—22no—
112C2×#272#4112yes—
113C4×H(3,𝔽3)—22no—
114C4×H(3,𝔽3)—21no—
115#1084—21no—
116(C3⋊Dic3)⋊C3—22no—
117(C3⋊Dic3)⋊C3—22no—
118(C3⋊Dic3)⋊C3—22no—
119(C3⋊Dic3)⋊C3—22no—
120C3×#125—92no—
121C3×#121—91no—
122#935—92no—
123#935—92no—
124D4×Dic3—92no—
125#924—91no—
126C2×Dic3⋊C4—91no—
127#898—92no—
128#897—92no—
129#896—92no—
130#896—92no—
131#895—32no—
132#895—61no—
133#894—92no—
134#894—92no—
135#894—92no—
136#894—92no—
137#893—92no—
138D3×C4⋊C4—92no—
139#889—32no—
140#889—92no—
141#879—92no—
142#878—92no—
143#876—92no—
144C4×D12—92no—
145C22×C3×D3—32no—
146C22×C3×D3—32no—
147C22×C3×D3—32no—
148C22×C3×D3—61no—
149C22×C3×D3—62no—
150C2×C32×C4—21no—
151C2×C32×C4—22no—
152C2×C32×C4C22×D312yes—
153C2×C32×C4C22×D312yes—
154C2×C32×C4C22×D312yes—
155C2×C3⋊Dic3C22×D312yes—
156C2×C3⋊Dic3C22×D312yes—
157C2×C3⋊Dic3C22×D312yes—
158C2×C3⋊Dic3C22×D312yes—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—32no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—61no—
165C3×D12—64no—
166C3×D12—64no—
167C3×D12—124no—
168C3×D12—122no—
169C3×D12—122no—
170C3×C4×D3—122no—
171C3×C4×D3—122no—
172#627—122no—
173D3×Dic3—64no—
174D3×Dic3—122no—
175D3×Dic3—122no—
176D3×Dic3—122no—
177#539—271noSylow
178C2×H(3,𝔽3)C2×#4111yescharacteristic
179C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
180C2×H(3,𝔽3)2−1+411yescharacteristic
181#272—22no—
182#272—42no—
183C2×C3×D4—92no—
184C2×C3×D4—91no—
185C22×C3×C4—91no—
186C22×Dic3—91no—
187C22×Dic3—92no—
188C22×Dic3—91no—
189C2×D12—32no—
190C2×D12—32no—
191C2×D12—32no—
192C2×D12—62no—
193C2×D12—61no—
194C2×C4×D3—32no—
195C2×C4×D3—61no—
196C2×C4×D3—92no—
197C2×C4×D3—92no—
198C2×C4×D3—92no—
199C2×C4×D3—92no—
200C2×Dic6—91no—
201C2×Dic6—92no—
202C3×C4⋊C4—32no—
203C3×C4⋊C4—61no—
204C3×C4⋊C4—92no—
205C3×C4⋊C4—92no—
206C3×C4⋊C4—92no—
207C3×C4⋊C4—92no—
208C3×C4⋊C4—92no—
209C3×C4⋊C4D3211yescharacteristic
210C3×C22⋊C4—92no—
211C3×C42—92no—
212#217—92no—
213#217—92no—
214#217—92no—
215#217—92no—
216(C2×D3)⋊C4—32no—
217(C2×D3)⋊C4—32no—
218(C2×D3)⋊C4—62no—
219(C2×D3)⋊C4—92no—
220(C2×D3)⋊C4—92no—
221(C2×D3)⋊C4—92no—
222(C2×D3)⋊C4—92no—
223(C2×D3)⋊C4—92no—
224(C2×D3)⋊C4—92no—
225(C2×D3)⋊C4—92no—
226(C2×D3)⋊C4—92no—
227C4⋊Dic3—32no—
228C4⋊Dic3—92no—
229C4⋊Dic3—91no—
230C4⋊Dic3—92no—
231Dic3⋊C4—32no—
232Dic3⋊C4—32no—
233Dic3⋊C4—32no—
234Dic3⋊C4—32no—
235Dic3⋊C4—92no—
236Dic3⋊C4—92no—
237Dic3⋊C4—92no—
238Dic3⋊C4—92no—
239Dic3⋊C4—92no—
240Dic3⋊C4—92no—
241Dic3⋊C4—92no—
242C4×Dic3—32no—
243C4×Dic3—92no—
244C22×C32—21no—
245C22×C32C23×D312yes—
246C2×C3×D3—34no—
247C2×C3×D3—62no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C2×C3×D3—62no—
251C2×C3×D3—62no—
252C2×C3×D3—62no—
253C2×C3×D3—62no—
254C2×C3×D3—62no—
255C2×C3×D3—62no—
256C2×C3×D3—121no—
257C2×C3×D3—121no—
258C2×C3×D3—122no—
259C2×C3×D3—122no—
260C2×C3×D3—122no—
261C32×C4—22no—
262C32×C4—22no—
263C32×C4—22no—
264C32×C4—22no—
265C32×C4—42no—
266C3⋊Dic3—22no—
267C3⋊Dic3—22no—
268C3⋊Dic3—22no—
269C3⋊Dic3—22no—
270C3×Dic3—62no—
271C3×Dic3—62no—
272C3×Dic3—62no—
273C3×Dic3—62no—
274C3×Dic3—62no—
275C3×Dic3—121no—
276#126—272no—
277#125—92no—
278#123—272no—
279#123—272no—
280C22⋊Q8—272no—
281C22⋊Q8—271no—
282#121—91no—
283C4×D4—272no—
284C2×C4⋊C4—271no—
285H(3,𝔽3)#53911yescharacteristic, complemented, Sylow
286C23×C3—91no—
287C23×C3—92no—
288C22×D3—32no—
289C22×D3—32no—
290C22×D3—32no—
291C22×D3—61no—
292C22×D3—62no—
293C3×D4—184no—
294C3×D4—361no—
295C2×C3×C4—32no—
296C2×C3×C4—32no—
297C2×C3×C4—32no—
298C2×C3×C4—61no—
299C2×C3×C4—62no—
300C2×C3×C4—92no—
301C2×C3×C4—92no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4—91no—
305C2×C3×C4—361no—
306C2×C3×C4C2×D3212yes—
307C2×C3×C4C2×D3211yescharacteristic
308C2×Dic3—32no—
309C2×Dic3—32no—
310C2×Dic3—32no—
311C2×Dic3—32no—
312C2×Dic3—32no—
313C2×Dic3—61no—
314C2×Dic3—92no—
315C2×Dic3—92no—
316C2×Dic3—92no—
317C2×Dic3—92no—
318C2×Dic3—182no—
319C2×Dic3—362no—
320C2×Dic3—361no—
321D12—64no—
322D12—64no—
323D12—124no—
324D12—122no—
325D12—122no—
326C4×D3—122no—
327C4×D3—122no—
328C4×D3—184no—
329C4×D3—362no—
330C4×D3—362no—
331C4×D3—362no—
332Dic6—361no—
333Dic6—362no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32#100712yes—
338C2×C32D3×#4112yes—
339C2×C32#100912yes—
340C3×D3—64no—
341C3×D3—122no—
342C3×D3—122no—
343C3×D3—122no—
344C3×D3—242no—
345C2×Q8—271no—
346C2×D4—92no—
347C2×D4—91no—
348C22×C4—91no—
349C22×C4—271no—
350C22×C4—271no—
351C22×C4—272no—
352C4⋊C4—92no—
353C4⋊C4—92no—
354C4⋊C4—92no—
355C4⋊C4—92no—
356C4⋊C4—92no—
357C4⋊C4—272no—
358C4⋊C4—271no—
359C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
360C22⋊C4—92no—
361C22⋊C4—272no—
362C22⋊C4—272no—
363C22⋊C4—272no—
364C22⋊C4—272no—
365C42—92no—
366C22×C3—32no—
367C22×C3—61no—
368C22×C3—92no—
369C22×C3—181no—
370C22×C3—181no—
371C22×C3—182no—
372C22×C3—182no—
373C22×C3—182no—
374C22×C3C22×D3211yescharacteristic
375C2×D3—34no—
376C2×D3—62no—
377C2×D3—62no—
378C2×D3—62no—
379C2×D3—62no—
380C2×D3—62no—
381C2×D3—62no—
382C2×D3—62no—
383C2×D3—62no—
384C2×D3—62no—
385C2×D3—121no—
386C2×D3—121no—
387C2×D3—122no—
388C2×D3—122no—
389C2×D3—122no—
390C3×C4—21no—
391C3×C4—22no—
392C3×C4—62no—
393C3×C4—62no—
394C3×C4—62no—
395C3×C4—62no—
396C3×C4—122no—
397C3×C4—182no—
398C3×C4—182no—
399C3×C4—182no—
400C3×C4—182no—
401C3×C4—181no—
402Dic3—62no—
403Dic3—62no—
404Dic3—62no—
405Dic3—62no—
406Dic3—62no—
407Dic3—121no—
408Dic3—182no—
409Dic3—182no—
410Dic3—182no—
411Dic3—182no—
412C32—21no—
413C32#569212yescomplemented
414C23—91no—
415C23—92no—
416Q8—1081no—
417D4—184no—
418D4—361no—
419C2×C4—92no—
420C2×C4—92no—
421C2×C4—91no—
422C2×C4—92no—
423C2×C4—92no—
424C2×C4—361no—
425C2×C4—542no—
426C2×C4—1081no—
427C2×C4—1082no—
428C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
429C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
430C2×C3—32no—
431C2×C3—32no—
432C2×C3—32no—
433C2×C3—61no—
434C2×C3—61no—
435C2×C3—61no—
436C2×C3—182no—
437C2×C3—362no—
438C2×C3#6430811yescharacteristic
439C2×C3#6431211yescharacteristic
440C2×C3#6431611yescharacteristic
441D3—64no—
442D3—122no—
443D3—122no—
444D3—122no—
445D3—242no—
446C22—92no—
447C22—181no—
448C22—181no—
449C22—182no—
450C22—182no—
451C22—182no—
452C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
453C4—22no—
454C4—21no—
455C4—182no—
456C4—182no—
457C4—181no—
458C4—182no—
459C4—182no—
460C3—32no—
461C3—61no—
462C3#1059448311yescharacteristic
463C2—182no—
464C2—362no—
465C2#1172249311yescharacteristic
466C2#1172250511yescharacteristic
467C2#1172251711yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radical#11721763C2
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#539
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #64098⋊C3C2semidirect
3. #11721151C2semidirect
4. #11721151C2semidirect
5. #11721150C2semidirect
6. #11721149C2semidirect
7. #11721147C2semidirect
8. #11721096C2semidirect
9. #11721092C2semidirect
32. #64098D3semidirect
285. H(3,𝔽3)#539semidirect
413. C32#5692semidirect

Maximal subgroups

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

subgroupindex
#117217632
#64098⋊C32
#117211512
#117211502
#117211492
#117211472
#117210962
#117210922
#105944283