Finite Group Databasebeta

← #13149526

Subgroups — #13149526

Order 1728.

868 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 145 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#11721755C211yesmaximal, characteristic, complemented
2#64105⋊C3C212yesmaximal, complemented
3#11721138C211yesmaximal, characteristic
4#11721138C211yesmaximal, characteristic
5C4×#6198C211yesmaximal, characteristic
6#11721135C212yesmaximal, complemented
7#11721128C212yesmaximal, complemented
8#11721119C212yesmaximal, complemented
9#11721099C211yesmaximal, characteristic, complemented
10#11721098C211yesmaximal, characteristic, complemented
11#11721098C211yesmaximal, characteristic, complemented
12#10594392—32nomaximal
13C2×#6229C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2212yes—
15C2×#634⋊C3C2212yes—
16C2×#6198C2211yescharacteristic
17C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
18C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20#88540C2211yescharacteristic
21#88540C2211yescharacteristic
22C4×#1084C2211yescharacteristic
23C4×#1084C2211yescharacteristic
24C4×#1084C2211yescharacteristic
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27#3750⋊C3C2212yes—
28#3750⋊C3C2212yes—
29#88450C2212yes—
30#88449C2211yescharacteristic
31#88449C2211yescharacteristic
32#88449C2211yescharacteristic
33#88449C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#64105D312yescomplemented
37C3×#896—32no—
38C3×#887—32no—
39C3×#887—61no—
40C4×#626—32no—
41#63896—32no—
42#63888—32no—
43#63888—32no—
44#63887—32no—
45#63883—32no—
46#63880—32no—
47#63880—32no—
48D3×Dic3⋊C4—32no—
49#63867—32no—
50#63867—32no—
51#63866—32no—
52C22×(C3⋊D3)⋊C3C2312yes—
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×C4×H(3,𝔽3)C2311yescharacteristic
56C2×#1084C2311yescharacteristic
57C2×#1084C2311yescharacteristic
58C2×#1084C2311yescharacteristic
59C2×#1084C2311yescharacteristic
60#6229—22no—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C4×(C3⋊D3)⋊C3—42no—
65#634⋊C3—42no—
66#6198Q814yescomplemented
67((C3⋊D3)⋊C3)⋊C4—24no—
68((C3⋊D3)⋊C3)⋊C4—42no—
69#5636—92no—
70#5579—91no—
71C2×C4×C3⋊D3C2×D312yes—
72C2×#634C2×D312yes—
73C2×C3×C4×D3—32no—
74C2×C3×Dic6—32no—
75C2×C3×Dic6—32no—
76C2×C3×Dic6—61no—
77C2×#626—32no—
78C2×#624—32no—
79C2×#624—32no—
80C2×D3×Dic3—32no—
81C2×D3×Dic3—32no—
82C32×C4⋊C4—21no—
83C32×C4⋊C4C2×D312yes—
84#3752C2×D312yes—
85#3752C2×D312yes—
86#3750C2×D312yes—
87#3750C2×D312yes—
88C3×(C2×D3)⋊C4—32no—
89C3×(C2×D3)⋊C4—32no—
90C3×Dic3⋊C4—32no—
91C3×Dic3⋊C4—32no—
92C3×Dic3⋊C4—32no—
93C3×Dic3⋊C4—32no—
94C3×Dic3⋊C4—61no—
95C3×Dic3⋊C4—61no—
96C3×C4×Dic3—32no—
97C3×C4×Dic3—32no—
98C3×C4×Dic3—32no—
99C3×C4×Dic3—61no—
100C3×C4×Dic3—61no—
101C3×C4×Dic3—61no—
102#3724—32no—
103#3724—32no—
104Dic3⋊Dic3—32no—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108Dic3⋊Dic3—32no—
109Dic3⋊Dic3—32no—
110Dic3⋊Dic3—32no—
111#3722—32no—
112(C2×D3)⋊Dic3—32no—
113C22×H(3,𝔽3)C2411yescharacteristic
114C2×(C3⋊D3)⋊C3—22no—
115C2×(C3⋊D3)⋊C3—22no—
116C2×(C3⋊D3)⋊C3C2×Q814yes—
117C4×H(3,𝔽3)—21no—
118C4×H(3,𝔽3)—21no—
119C4×H(3,𝔽3)—21no—
120#1084—21no—
121#1084—21no—
122#1084C2×Q812yes—
123#1084#4112yes—
124(C3⋊Dic3)⋊C3—22no—
125(C3⋊Dic3)⋊C3—22no—
126(C3⋊Dic3)⋊C3—22no—
127C3×C22⋊Q8—92no—
128C3×C4×Q8—91no—
129#946—92no—
130#946—92no—
131#944—92no—
132C4×#67—92no—
133C2×Dic3⋊C4—92no—
134#896—32no—
135#896—92no—
136#896—92no—
137#896—92no—
138D3×C4⋊C4—92no—
139D3×C4⋊C4—92no—
140#889—92no—
141#889—92no—
142#887—32no—
143#887—61no—
144#67⋊C4—92no—
145#67⋊C4—92no—
146#878—92no—
147#878—92no—
148C4×D12—91no—
149C4×D12—91no—
150C4×D12—91no—
151#870—91no—
152#870—91no—
153#869—91no—
154C22×C3⋊D3C22×D312yes—
155C22×C3×D3—32no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4C22×D312yes—
160C2×C32×C4C22×D312yes—
161C2×C32×C4C22×D312yes—
162C2×C3⋊Dic3C22×D312yes—
163C2×C3⋊Dic3C22×D312yes—
164C2×C3⋊Dic3C22×D312yes—
165C4×C3⋊D3—42no—
166#634—42no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C3×C4×D3—122no—
179C3×Dic6—64no—
180C3×Dic6—64no—
181C3×Dic6—122no—
182#626—38no—
183#624—64no—
184#624—122no—
185D3×Dic3—64no—
186D3×Dic3—122no—
187#554—271noSylow
188C2×H(3,𝔽3)C22×Q811yescharacteristic
189C2×H(3,𝔽3)C2×#4111yescharacteristic
190C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
191(C3⋊D3)⋊C3—24no—
192C2×C3×Q8—92no—
193C2×C3×Q8—91no—
194C22×C3×C4—92no—
195C2×#67—92no—
196C22×Dic3—92no—
197C22×Dic3—92no—
198C2×D12—91no—
199C2×C4×D3—32no—
200C2×C4×D3—92no—
201C2×C4×D3—92no—
202C2×C4×D3—92no—
203C2×C4×D3—92no—
204C2×C4×D3—92no—
205C2×Dic6—32no—
206C2×Dic6—32no—
207C2×Dic6—61no—
208C2×Dic6—92no—
209C3×C4⋊C4—32no—
210C3×C4⋊C4—61no—
211C3×C4⋊C4—92no—
212C3×C4⋊C4—92no—
213C3×C4⋊C4—91no—
214C3×C4⋊C4—91no—
215C3×C4⋊C4D3211yescharacteristic
216C3×C22⋊C4—92no—
217C3×C22⋊C4—92no—
218C3×C42—91no—
219C3×C42—91no—
220C3×C42—91no—
221#217—92no—
222(C2×D3)⋊C4—32no—
223(C2×D3)⋊C4—32no—
224(C2×D3)⋊C4—92no—
225(C2×D3)⋊C4—92no—
226(C2×D3)⋊C4—92no—
227(C2×D3)⋊C4—92no—
228C4⋊Dic3—91no—
229C4⋊Dic3—91no—
230C4⋊Dic3—92no—
231C4⋊Dic3—91no—
232C4⋊Dic3—92no—
233C4⋊Dic3—92no—
234Dic3⋊C4—32no—
235Dic3⋊C4—32no—
236Dic3⋊C4—32no—
237Dic3⋊C4—32no—
238Dic3⋊C4—61no—
239Dic3⋊C4—61no—
240Dic3⋊C4—92no—
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251C4×Dic3—32no—
252C4×Dic3—32no—
253C4×Dic3—32no—
254C4×Dic3—61no—
255C4×Dic3—61no—
256C4×Dic3—61no—
257C22×C32—21no—
258C22×C32C23×D312yes—
259C2×C3⋊D3—22no—
260C2×C3⋊D3—22no—
261C2×C3⋊D3D3×Q814yes—
262C2×C3×D3—34no—
263C2×C3×D3—62no—
264C2×C3×D3—62no—
265C32×C4—22no—
266C32×C4—22no—
267C32×C4—22no—
268C32×C4—22no—
269C32×C4—41no—
270C32×C4—41no—
271C3⋊Dic3—22no—
272C3⋊Dic3—22no—
273C3⋊Dic3—22no—
274C3×Dic3—34no—
275C3×Dic3—34no—
276C3×Dic3—62no—
277C3×Dic3—62no—
278C3×Dic3—62no—
279C3×Dic3—62no—
280C3×Dic3—62no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—62no—
284C3×Dic3—121no—
285C4⋊Q8—271no—
286#125—271no—
287#125—271no—
288C22⋊Q8—92no—
289C22⋊Q8—272no—
290C22⋊Q8—272no—
291C4×Q8—91no—
292C4×D4—271no—
293C4×D4—271no—
294C4×D4—271no—
295C2×C4⋊C4—272no—
296H(3,𝔽3)#55411yescharacteristic, complemented, Sylow
297C23×C3—92no—
298C22×D3—32no—
299C22×D3—92no—
300C3×Q8—182no—
301C3×Q8—362no—
302C2×C3×C4—32no—
303C2×C3×C4—32no—
304C2×C3×C4—32no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—91no—
309C2×C3×C4—91no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4—92no—
313C2×C3×C4—91no—
314C2×C3×C4—91no—
315C2×C3×C4—362no—
316C2×C3×C4C2×D3211yescharacteristic
317C2×C3×C4C2×D3211yescharacteristic
318C2×C3×C4C2×D3211yescharacteristic
319#67—98no—
320C2×Dic3—32no—
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—32no—
324C2×Dic3—32no—
325C2×Dic3—32no—
326C2×Dic3—32no—
327C2×Dic3—61no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—92no—
332C2×Dic3—92no—
333C2×Dic3—92no—
334C2×Dic3—184no—
335C2×Dic3—362no—
336D12—94no—
337C4×D3—122no—
338C4×D3—184no—
339C4×D3—184no—
340C4×D3—362no—
341C4×D3—362no—
342C4×D3—362no—
343Dic6—64no—
344Dic6—64no—
345Dic6—122no—
346Dic6—362no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32#100412yes—
351C2×C32C2×D3×Q812yes—
352C2×C32D3×#4112yes—
353C3⋊D3—24no—
354C3×D3—64no—
355C2×Q8—91no—
356C2×Q8—92no—
357C2×D4—271no—
358C22×C4—92no—
359C22×C4—272no—
360C22×C4—272no—
361C4⋊C4—91no—
362C4⋊C4—92no—
363C4⋊C4—91no—
364C4⋊C4—92no—
365C4⋊C4—272no—
366C4⋊C4—272no—
367C4⋊C4—271no—
368C4⋊C4—271no—
369C4⋊C4—271no—
370C4⋊C4—272no—
371C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
372C22⋊C4—92no—
373C22⋊C4—92no—
374C22⋊C4—272no—
375C42—91no—
376C42—91no—
377C42—91no—
378C22×C3—32no—
379C22×C3—61no—
380C22×C3—94no—
381C22×C3—182no—
382C22×C3—182no—
383C22×C3C22×D3211yescharacteristic
384C2×D3—34no—
385C2×D3—62no—
386C2×D3—62no—
387C2×D3—94no—
388C2×D3—182no—
389C2×D3—182no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—62no—
394C3×C4—62no—
395C3×C4—62no—
396C3×C4—62no—
397C3×C4—92no—
398C3×C4—92no—
399C3×C4—121no—
400C3×C4—121no—
401C3×C4—182no—
402C3×C4—182no—
403C3×C4—182no—
404C3×C4—181no—
405C3×C4—181no—
406Dic3—34no—
407Dic3—34no—
408Dic3—62no—
409Dic3—62no—
410Dic3—62no—
411Dic3—62no—
412Dic3—62no—
413Dic3—62no—
414Dic3—62no—
415Dic3—62no—
416Dic3—121no—
417Dic3—182no—
418Dic3—182no—
419Dic3—182no—
420C32—21no—
421C32#563612yescomplemented
422C23—92no—
423Q8—182no—
424Q8—362no—
425D4—274no—
426C2×C4—91no—
427C2×C4—91no—
428C2×C4—91no—
429C2×C4—91no—
430C2×C4—92no—
431C2×C4—92no—
432C2×C4—92no—
433C2×C4—362no—
434C2×C4—544no—
435C2×C4—1082no—
436C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
437C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
438C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
439C2×C3—32no—
440C2×C3—32no—
441C2×C3—32no—
442C2×C3—61no—
443C2×C3—61no—
444C2×C3—61no—
445C2×C3—184no—
446C2×C3#6430611yescharacteristic
447C2×C3#6431311yescharacteristic
448C2×C3D32×Q811yescharacteristic
449D3—64no—
450D3—184no—
451C22—94no—
452C22—182no—
453C22—182no—
454C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
455C4—21no—
456C4—21no—
457C4—21no—
458C4—92no—
459C4—92no—
460C4—182no—
461C4—181no—
462C4—182no—
463C4—182no—
464C4—181no—
465C3—32no—
466C3—61no—
467C3#1059453411yescharacteristic
468C2—184no—
469C2#1172249111yescharacteristic
470C2#1172250611yescharacteristic
471C2Q8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC2×#6198C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#554
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
1. #11721755C2semidirect
2. #64105⋊C3C2semidirect
6. #11721135C2semidirect
7. #11721128C2semidirect
8. #11721119C2semidirect
9. #11721099C2semidirect
10. #11721098C2semidirect
11. #11721098C2semidirect
36. #64105D3semidirect
66. #6198Q8semidirect
296. H(3,𝔽3)#554semidirect
421. C32#5636semidirect

Maximal subgroups

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

subgroupindex
#117217552
#64105⋊C32
#117211382
C4×#61982
#117211352
#117211282
#117211192
#117210992
#117210982
#105943923