Finite Group Databasebeta

← #13149523

Subgroups — #13149523

Order 1728.

1592 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 1592 conjugacy classes of subgroups — too many to draw. Only its 161 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×#88655C212yesmaximal, complemented
2C2×#88654C212yesmaximal, complemented
3#11721761C211yesmaximal, characteristic, complemented
4#64104⋊C3C212yesmaximal, complemented
5#11721162C214yesmaximal, complemented
6#11721144C211yesmaximal, characteristic, complemented
7#11721142C212yesmaximal, complemented
8C4×#6198C211yesmaximal, characteristic, complemented
9#10594621—32nomaximal
10C22×((C3⋊D3)⋊C3)⋊C2C2214yescomplemented
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272C2212yescomplemented
13C2×C4⋊((C3⋊D3)⋊C3)C2212yescomplemented
14C2×C4⋊((C3⋊D3)⋊C3)C2214yescomplemented
15C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
16C2×#6200C2214yescomplemented
17C2×#6198C2211yescharacteristic
18C2×#6198C2212yescomplemented
19C2×#6198C2212yescomplemented
20#88655—28no—
21#88654—28no—
22C4⋊C4×H(3,𝔽3)C2211yescharacteristic, complemented
23#88542C2212yescomplemented
24C4×#1084C2211yescharacteristic, complemented
25#3752⋊C3C2214yescomplemented
26#88450C2212yescomplemented
27#88449C2211yescharacteristic, complemented
28C2×#3802—32no—
29C2×D3×D12—32no—
30C2×D3×D12—34no—
31#64104D312yescomplemented
32C3×#895—32no—
33C3×#893—32no—
34C3×#893—61no—
35#63923—34no—
36#63922—34no—
37#63911—32no—
38#63906—34no—
39C4×#626—32no—
40C22×#272C2312yes—
41C22×(C3⋊D3)⋊C3C2314yes—
42C22×(C3⋊D3)⋊C3C2312yes—
43C2×((C3⋊D3)⋊C3)⋊C2—28no—
44C2×((C3⋊D3)⋊C3)⋊C2—28no—
45C2×((C3⋊D3)⋊C3)⋊C2—44no—
46C2×C4×H(3,𝔽3)C2312yes—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×#1084C2311yescharacteristic
49C2×#1084C2311yescharacteristic
50C4⋊#272D414yescomplemented
51C4×#272—24no—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53C4⋊((C3⋊D3)⋊C3)—28no—
54C4⋊((C3⋊D3)⋊C3)—24no—
55C4×(C3⋊D3)⋊C3—24no—
56#6200—28no—
57#6198—24no—
58#6198—42no—
59#6198D414yescomplemented
60#5609—92no—
61D4×D12—91no—
62C22×D32—34no—
63C22×D32—34no—
64C2×C4⋊(C3⋊D3)C2×D312yescomplemented
65C2×C4⋊(C3⋊D3)C2×D314yescomplemented
66C2×C4×C3⋊D3C2×D312yescomplemented
67C2×C3×D12—32no—
68C2×C3×D12—34no—
69C2×C3×D12—32no—
70C2×C3×D12—61no—
71C2×C3×C4×D3—34no—
72C2×C3×C4×D3—32no—
73C2×C3×C4×D3—62no—
74C2×#626—32no—
75C2×#626—34no—
76C2×#626—34no—
77C2×#626—34no—
78C2×#625—34no—
79#3802—68no—
80D3×D12—616no—
81D3×D12—68no—
82C32×C4⋊C4—21no—
83C32×C4⋊C4C2×D312yescomplemented
84#3752C2×D314yescomplemented
85C3×(C2×D3)⋊C4—34no—
86C3×(C2×D3)⋊C4—34no—
87C3×(C2×D3)⋊C4—62no—
88C3×C4×Dic3—32no—
89C3×C4×Dic3—61no—
90Dic3⋊Dic3—32no—
91#3722—32no—
92(C2×D3)⋊Dic3—32no—
93C22×H(3,𝔽3)C2411yescharacteristic
94C2×#272—22no—
95C2×#272—22no—
96C2×#272C2×D414yescomplemented
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3—24no—
100C2×(C3⋊D3)⋊C3—24no—
101C2×(C3⋊D3)⋊C3—24no—
102C2×(C3⋊D3)⋊C3C2×D414yescomplemented
103((C3⋊D3)⋊C3)⋊C2—416no—
104C4×H(3,𝔽3)—22no—
105C4×H(3,𝔽3)C2×D412yescomplemented
106#1084—21no—
107#1084C2×D412yescomplemented
108(C3⋊Dic3)⋊C3—22no—
109C2×D3×D4—92no—
110C2×D3×D4—94no—
111C2×D3×D4—92no—
112C22×D12—92no—
113C22×D12—92no—
114C3×#121—92no—
115C3×C4×D4—91no—
116#940—92no—
117#937—94no—
118#930—92no—
119C4×#67—92no—
120#895—32no—
121#895—92no—
122#893—32no—
123#893—61no—
124#884—94no—
125#882—94no—
126#882—94no—
127#874—91no—
128C4×D12—91no—
129C22×C3⋊D3C22×D314yes—
130C22×C3⋊D3C22×D312yes—
131C22×C3×D3—32no—
132C22×C3×D3—34no—
133C22×C3×D3—34no—
134C22×C3×D3—62no—
135C2×D32—68no—
136C2×D32—68no—
137C2×D32—68no—
138C2×D32—68no—
139C2×D32—124no—
140C2×D32—124no—
141C2×C32×C4—21no—
142C2×C32×C4—22no—
143C2×C32×C4C22×D314yes—
144C2×C32×C4C22×D312yes—
145C2×C3⋊Dic3C22×D312yes—
146C4⋊(C3⋊D3)—28no—
147C4⋊(C3⋊D3)—24no—
148C4×C3⋊D3—24no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C3×D12—38no—
155C3×D12—64no—
156C3×D12—68no—
157C3×D12—64no—
158C3×C4×D3—68no—
159C3×C4×D3—64no—
160C3×C4×D3—124no—
161#626—38no—
162#626—68no—
163#626—68no—
164#626—124no—
165#625—68no—
166D42—271noSylow
167C2×H(3,𝔽3)C22×D411yescharacteristic
168C2×H(3,𝔽3)C22×D411yescharacteristic
169C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
170#272—24no—
171(C3⋊D3)⋊C3—24no—
172(C3⋊D3)⋊C3—44no—
173C23×D3—94no—
174C23×D3—94no—
175C23×D3—94no—
176C2×C3×D4—94no—
177C2×C3×D4—92no—
178C2×C3×D4—91no—
179C22×C3×C4—92no—
180C22×C3×C4—92no—
181C2×#67—92no—
182C2×#67—94no—
183C2×#67—94no—
184C2×#67—94no—
185C2×#67—94no—
186D3×D4—188no—
187D3×D4—1816no—
188D3×D4—188no—
189C2×D12—32no—
190C2×D12—32no—
191C2×D12—34no—
192C2×D12—61no—
193C2×D12—94no—
194C2×D12—92no—
195C2×D12—91no—
196C2×D12—92no—
197C2×D12—94no—
198C2×D12—92no—
199C2×D12—188no—
200C2×D12—188no—
201C2×C4×D3—34no—
202C2×C4×D3—32no—
203C2×C4×D3—62no—
204C2×C4×D3—92no—
205C3×C4⋊C4—32no—
206C3×C4⋊C4—61no—
207C3×C4⋊C4D3211yescharacteristic
208C3×C22⋊C4—94no—
209C3×C22⋊C4—92no—
210C3×C42—91no—
211#217—92no—
212(C2×D3)⋊C4—34no—
213(C2×D3)⋊C4—34no—
214(C2×D3)⋊C4—62no—
215(C2×D3)⋊C4—94no—
216(C2×D3)⋊C4—92no—
217(C2×D3)⋊C4—92no—
218C4⋊Dic3—91no—
219Dic3⋊C4—92no—
220C4×Dic3—32no—
221C4×Dic3—61no—
222C22×C32—21no—
223C22×C32C23×D312yes—
224C2×C3⋊D3—22no—
225C2×C3⋊D3—22no—
226C2×C3⋊D3—24no—
227C2×C3⋊D3—24no—
228C2×C3⋊D3—24no—
229C2×C3⋊D3D3×D414yescomplemented
230C2×C3×D3—34no—
231C2×C3×D3—38no—
232C2×C3×D3—64no—
233C2×C3×D3—62no—
234C2×C3×D3—64no—
235C2×C3×D3—62no—
236C2×C3×D3—64no—
237C2×C3×D3—64no—
238C2×C3×D3—64no—
239C2×C3×D3—64no—
240C2×C3×D3—122no—
241C2×C3×D3—122no—
242D32—1216no—
243D32—1216no—
244C32×C4—24no—
245C32×C4—22no—
246C32×C4—42no—
247C32×C4D3×D414yescomplemented
248C3⋊Dic3—22no—
249C3×Dic3—34no—
250C3×Dic3—62no—
251C3×Dic3—62no—
252C3×Dic3—62no—
253C3×Dic3—121no—
254C22×D4—272no—
255C22×D4—272no—
256#127—271no—
257#121—92no—
258#121—272no—
259#120—274no—
260C4×D4—91no—
261C4×D4—271no—
262H(3,𝔽3)D4211yescharacteristic, complemented, Sylow
263C23×C3—92no—
264C23×C3—94no—
265C23×C3—92no—
266C22×D3—34no—
267C22×D3—32no—
268C22×D3—34no—
269C22×D3—62no—
270C22×D3—92no—
271C22×D3—94no—
272C22×D3—188no—
273C22×D3—188no—
274C22×D3—188no—
275C22×D3—188no—
276C22×D3—188no—
277C22×D3—188no—
278C22×D3—364no—
279C22×D3—364no—
280C22×D3—364no—
281C3×D4—94no—
282C3×D4—184no—
283C3×D4—188no—
284C2×C3×C4—32no—
285C2×C3×C4—34no—
286C2×C3×C4—61no—
287C2×C3×C4—62no—
288C2×C3×C4—91no—
289C2×C3×C4—92no—
290C2×C3×C4—91no—
291C2×C3×C4—184no—
292C2×C3×C4—184no—
293C2×C3×C4C2×D3212yes—
294C2×C3×C4C2×D3211yescharacteristic
295#67—98no—
296#67—188no—
297#67—188no—
298#67—188no—
299#67—364no—
300C2×Dic3—32no—
301C2×Dic3—32no—
302C2×Dic3—32no—
303C2×Dic3—61no—
304C2×Dic3—61no—
305C2×Dic3—92no—
306D12—38no—
307D12—64no—
308D12—68no—
309D12—64no—
310D12—94no—
311D12—184no—
312D12—188no—
313D12—184no—
314D12—188no—
315D12—362no—
316C4×D3—68no—
317C4×D3—64no—
318C4×D3—124no—
319C4×D3—184no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32C2×D3×D412yes—
324C2×C32C2×D3×D412yes—
325C2×C32#100912yes—
326C3⋊D3—24no—
327C3⋊D3—44no—
328C3×D3—64no—
329C3×D3—68no—
330C3×D3—124no—
331C3×D3—124no—
332C24—274no—
333C2×D4—91no—
334C2×D4—94no—
335C2×D4—92no—
336C2×D4—271no—
337C2×D4—274no—
338C2×D4—272no—
339C2×D4—272no—
340C2×D4—548no—
341C2×D4—548no—
342C22×C4—92no—
343C22×C4—92no—
344C4⋊C4—271no—
345C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
346C22⋊C4—92no—
347C22⋊C4—94no—
348C22⋊C4—272no—
349C42—91no—
350C22×C3—32no—
351C22×C3—61no—
352C22×C3—94no—
353C22×C3—94no—
354C22×C3—182no—
355C22×C3—182no—
356C22×C3—184no—
357C22×C3—184no—
358C22×C3—184no—
359C22×C3—182no—
360C22×C3—182no—
361C22×C3C22×D3211yescharacteristic
362C2×D3—38no—
363C2×D3—34no—
364C2×D3—64no—
365C2×D3—64no—
366C2×D3—64no—
367C2×D3—62no—
368C2×D3—62no—
369C2×D3—64no—
370C2×D3—64no—
371C2×D3—64no—
372C2×D3—94no—
373C2×D3—122no—
374C2×D3—122no—
375C2×D3—182no—
376C2×D3—182no—
377C2×D3—184no—
378C2×D3—184no—
379C2×D3—184no—
380C2×D3—3616no—
381C2×D3—3616no—
382C2×D3—3616no—
383C3×C4—22no—
384C3×C4—34no—
385C3×C4—64no—
386C3×C4—62no—
387C3×C4—92no—
388C3×C4—122no—
389C3×C4—182no—
390C3×C4—181no—
391C3×C4#380212yes—
392Dic3—34no—
393Dic3—62no—
394Dic3—62no—
395Dic3—62no—
396Dic3—121no—
397Dic3—182no—
398C32—21no—
399C32#560912yescomplemented
400C23—92no—
401C23—92no—
402C23—94no—
403C23—548no—
404C23—548no—
405C23—1084no—
406D4—94no—
407D4—188no—
408D4—184no—
409D4—274no—
410D4—544no—
411D4—548no—
412D4—1082no—
413C2×C4—91no—
414C2×C4—91no—
415C2×C4—92no—
416C2×C4—184no—
417C2×C4—184no—
418C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
419C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
420C2×C3—32no—
421C2×C3—32no—
422C2×C3—32no—
423C2×C3—61no—
424C2×C3—61no—
425C2×C3—61no—
426C2×C3—184no—
427C2×C3—184no—
428C2×C3—364no—
429C2×C3C2×#380211yescharacteristic
430C2×C3D32×D411yescharacteristic
431C2×C3#6432011yescharacteristic
432D3—68no—
433D3—64no—
434D3—124no—
435D3—124no—
436D3—184no—
437D3—364no—
438C22—94no—
439C22—94no—
440C22—182no—
441C22—182no—
442C22—182no—
443C22—182no—
444C22—184no—
445C22—184no—
446C22—184no—
447C22—10816no—
448C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
449C4—22no—
450C4—92no—
451C4—181no—
452C4—182no—
453C4#8865512yescomplemented
454C3—32no—
455C3—61no—
456C3#1059461811yescharacteristic
457C2—184no—
458C2—184no—
459C2—364no—
460C2C2×#8865511yescharacteristic
461C2D4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
462C2#1172252111yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)D42
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. C2×#88655C2semidirect
2. C2×#88654C2semidirect
3. #11721761C2semidirect
4. #64104⋊C3C2semidirect
5. #11721162C2semidirect
6. #11721144C2semidirect
7. #11721142C2semidirect
8. C4×#6198C2semidirect
10. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
12. C2×C4×#272C22semidirect
13. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
14. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
15. C2×C4×(C3⋊D3)⋊C3C22semidirect
16. C2×#6200C22semidirect
18. C2×#6198C22semidirect
19. C2×#6198C22semidirect
22. C4⋊C4×H(3,𝔽3)C22semidirect
23. #88542C22semidirect
24. C4×#1084C22semidirect
25. #3752⋊C3C22semidirect
26. #88450C22semidirect
27. #88449C22semidirect
31. #64104D3semidirect
50. C4⋊#272D4semidirect
59. #6198D4semidirect
64. C2×C4⋊(C3⋊D3)C2×D3semidirect
65. C2×C4⋊(C3⋊D3)C2×D3semidirect
66. C2×C4×C3⋊D3C2×D3semidirect
83. C32×C4⋊C4C2×D3semidirect
84. #3752C2×D3semidirect
96. C2×#272C2×D4semidirect
102. C2×(C3⋊D3)⋊C3C2×D4semidirect
105. C4×H(3,𝔽3)C2×D4semidirect
107. #1084C2×D4semidirect
229. C2×C3⋊D3D3×D4semidirect
247. C32×C4D3×D4semidirect
262. H(3,𝔽3)D42semidirect
345. C4⋊C4((C3⋊D3)⋊C3)⋊C2semidirect
399. C32#5609semidirect
453. C4#88655semidirect

Maximal subgroups

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

subgroupindex
C2×#886552
C2×#886542
#117217612
#64104⋊C32
#117211622
#117211442
#117211422
C4×#61982
#105946213