Finite Group Databasebeta

← #13150865

Subgroups — #13150865

Order 1728.

1405 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1405 conjugacy classes of subgroups — too many to draw. Only its 179 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
1C22×C4⋊#272C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C211yesmaximal, characteristic, complemented
3#64135⋊C3C211yesmaximal, characteristic, complemented
4#11721323C214yesmaximal, complemented
5#11721158C212yesmaximal, complemented
6#11721156C214yesmaximal, complemented
7#11721114C212yesmaximal, complemented
8#10595042—31nomaximal
9#10595040—31nomaximal
10C23×#272C2212yes—
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×C4⋊#272—24no—
13C2×C4⋊#272C2212yes—
14C2×C4⋊#272C2212yes—
15C2×#638⋊C3C2212yes—
16#3828⋊C3—24no—
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
18C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
19C2×#634⋊C3C2211yescharacteristic, complemented
20C2×#6200C2218yescomplemented
21#3757⋊C3C2212yes—
22C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
23C4×(C3⋊Dic3)⋊C3C2212yescomplemented
24#88451C2218yescomplemented
25C2×#3828D311yescharacteristic, complemented
26C2×C3×#235—31no—
27C22×C3×D12—31no—
28C22×C3×D12—31no—
29C22×C3×D12—61no—
30#64135D311yescharacteristic, complemented
31C3×#926—31no—
32#63977—34no—
33#63976—34no—
34#63916—34no—
35#63914—32no—
36#63911—32no—
37#63908—34no—
38#63863—32no—
39#63860—32no—
40C23×H(3,𝔽3)C2311yescharacteristic
41C22×#272—24no—
42C22×#272—44no—
43C22×#272C2314yes—
44C22×(C3⋊D3)⋊C3C2312yes—
45C2×C4×H(3,𝔽3)D414yescomplemented
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C4⋊#272—24no—
49C4⋊#272—24no—
50#638⋊C3—24no—
51C2×(C3⋊Dic3)⋊C3C2314yes—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53C4⋊((C3⋊D3)⋊C3)—22no—
54C4×(C3⋊D3)⋊C3—24no—
55#634⋊C3—22no—
56#6200—48no—
57#5830—91no—
58#5805—91no—
59#5525—91no—
60C23×C3×D3—32no—
61C23×C3×D3—32no—
62C23×C3×D3—62no—
63C22×C32×C4—21no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C2×D311yescharacteristic
66C2×#638C2×D312yes—
67#3828—24no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
69C2×C4×C3⋊D3C2×D312yescomplemented
70C2×#634C2×D311yescharacteristic, complemented
71C2×C3×#67—32no—
72C3×#235—64no—
73C2×C3×D12—32no—
74C2×C3×D12—32no—
75C2×C3×D12—32no—
76C2×C3×D12—32no—
77C2×C3×D12—31no—
78C2×C3×D12—62no—
79C2×C3×D12—62no—
80C2×C3×D12—68no—
81C2×C3×D12—64no—
82C2×C3×D12—64no—
83C2×C3×C4×D3—32no—
84C2×C3×Dic6—31no—
85C2×#626—38no—
86C2×#625—38no—
87#3757C2×D312yes—
88C4⋊(C3⋊Dic3)C2×D312yescomplemented
89C4×C3⋊Dic3C2×D312yescomplemented
90C3×#217—32no—
91C3×C4⋊Dic3—32no—
92C3×C4×Dic3—32no—
93(C2×D3)⋊Dic3—38no—
94(C2×D3)⋊Dic3—38no—
95C22×H(3,𝔽3)—22no—
96C22×H(3,𝔽3)C2×D412yes—
97C22×H(3,𝔽3)C2411yescharacteristic
98C2×#272—24no—
99C2×#272—28no—
100C2×#272—48no—
101C2×(C3⋊D3)⋊C3—22no—
102C2×(C3⋊D3)⋊C3—24no—
103C4×H(3,𝔽3)C2×D412yes—
104C4×H(3,𝔽3)C2×D412yes—
105(C3⋊Dic3)⋊C3—22no—
106(C3⋊Dic3)⋊C3—24no—
107C2×C3×#41—91no—
108C22×C3×D4—91no—
109C2×#235—31no—
110C2×#235—91no—
111C22×D12—31no—
112C22×D12—31no—
113C22×D12—61no—
114C3×#117—91no—
115#953—94no—
116#947—92no—
117#940—92no—
118#940—92no—
119#938—94no—
120#937—94no—
121#936—92no—
122#930—94no—
123#926—31no—
124#926—91no—
125#895—94no—
126#882—94no—
127#875—92no—
128#874—92no—
129C23×C32—21no—
130C23×C32C22×D311yescharacteristic
131C23×C32C22×D311yescharacteristic
132C22×C3⋊D3C22×D312yes—
133C22×C3×D3—34no—
134C22×C3×D3—34no—
135C22×C3×D3—32no—
136C22×C3×D3—64no—
137C22×C3×D3—68no—
138C22×C3×D3—64no—
139C22×C3×D3—64no—
140C22×C3×D3—124no—
141C22×C3×D3—124no—
142C22×C3×D3—128no—
143C2×C32×C4—24no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4D1214yescomplemented
147C2×C32×C4#6714yescomplemented
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152#638—24no—
153C2×C3⋊Dic3C22×D312yes—
154C2×C3⋊Dic3C22×D314yes—
155C4⋊(C3⋊D3)—22no—
156C4×C3⋊D3—24no—
157#634—22no—
158C3×#67—64no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—34no—
161C3×D12—68no—
162C3×D12—68no—
163C3×D12—62no—
164C3×D12—64no—
165C3×D12—64no—
166C3×D12—64no—
167C3×D12—64no—
168C3×C4×D3—64no—
169C3×Dic6—62no—
170#626—128no—
171#625—128no—
172#535—271noSylow
173C2×H(3,𝔽3)—22no—
174C2×H(3,𝔽3)C22×D411yescharacteristic
175C2×H(3,𝔽3)H(5,𝔽2)12yes—
176#272—44no—
177(C3⋊D3)⋊C3—42no—
178C24×C3—92no—
179C23×D3—32no—
180C23×D3—32no—
181C23×D3—62no—
182C3×#41—184no—
183C2×C3×Q8—91no—
184C2×C3×D4—91no—
185C2×C3×D4—92no—
186C2×C3×D4—92no—
187C2×C3×D4—92no—
188C2×C3×D4—184no—
189C22×C3×C4—31no—
190C22×C3×C4—31no—
191C22×C3×C4—61no—
192C22×C3×C4—92no—
193C22×C3×C4D3211yescharacteristic
194C2×#67—32no—
195C2×#67—92no—
196C2×#67—98no—
197C2×#67—98no—
198#235—64no—
199#235—184no—
200C2×D12—32no—
201C2×D12—32no—
202C2×D12—32no—
203C2×D12—32no—
204C2×D12—31no—
205C2×D12—62no—
206C2×D12—62no—
207C2×D12—64no—
208C2×D12—64no—
209C2×D12—68no—
210C2×D12—91no—
211C2×D12—98no—
212C2×C4×D3—32no—
213C2×C4×D3—92no—
214C2×Dic6—31no—
215C2×Dic6—91no—
216C3×C4⋊C4—92no—
217C3×C22⋊C4—92no—
218C3×C42—92no—
219#217—32no—
220#217—92no—
221#217—98no—
222(C2×D3)⋊C4—98no—
223(C2×D3)⋊C4—98no—
224C4⋊Dic3—32no—
225C4⋊Dic3—92no—
226C4×Dic3—32no—
227C4×Dic3—92no—
228C22×C32—24no—
229C22×C32—22no—
230C22×C32—22no—
231C22×C32—21no—
232C22×C32—22no—
233C22×C32C2×D1212yes—
234C22×C32C2×#6712yes—
235C22×C32C23×D311yescharacteristic
236C22×C32C23×D311yescharacteristic
237C2×C3⋊D3—24no—
238C2×C3⋊D3—22no—
239C2×C3×D3—68no—
240C2×C3×D3—62no—
241C2×C3×D3—64no—
242C2×C3×D3—64no—
243C2×C3×D3—68no—
244C2×C3×D3—68no—
245C2×C3×D3—64no—
246C2×C3×D3—128no—
247C2×C3×D3—1216no—
248C2×C3×D3—128no—
249C2×C3×D3—128no—
250C32×C4—22no—
251C32×C4—22no—
252C32×C4C2×D1212yes—
253C32×C4C2×D1212yes—
254C32×C4C2×#6712yes—
255C32×C4C2×#6712yes—
256C3⋊Dic3—24no—
257C3⋊Dic3—22no—
258C3×Dic3—62no—
259C3×Dic3—64no—
260C2×#41—91no—
261C22×D4—91no—
262#127—272no—
263#124—272no—
264#121—274no—
265#120—274no—
266#117—91no—
267H(3,𝔽3)#53511yescharacteristic, complemented, Sylow
268C23×C3—31no—
269C23×C3—31no—
270C23×C3—61no—
271C23×C3—92no—
272C23×C3—94no—
273C23×C3—184no—
274C23×C3—364no—
275C23×C3C2×D3211yescharacteristic
276C22×D3—34no—
277C22×D3—34no—
278C22×D3—32no—
279C22×D3—64no—
280C22×D3—64no—
281C22×D3—64no—
282C22×D3—68no—
283C22×D3—92no—
284C22×D3—124no—
285C22×D3—124no—
286C22×D3—128no—
287C3×Q8—182no—
288C3×D4—182no—
289C3×D4—184no—
290C3×D4—184no—
291C3×D4—184no—
292C2×C3×C4—34no—
293C2×C3×C4—34no—
294C2×C3×C4—31no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—64no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—92no—
302C2×C3×C4—94no—
303C2×C3×C4—184no—
304C2×C3×C4#62614yes—
305C2×C3×C4C2×D3211yescharacteristic
306C2×C3×C4C2×D3211yescharacteristic
307#67—64no—
308#67—184no—
309#67—368no—
310#67—368no—
311C2×Dic3—34no—
312C2×Dic3—32no—
313C2×Dic3—94no—
314C2×Dic3—92no—
315D12—64no—
316D12—64no—
317D12—64no—
318D12—64no—
319D12—68no—
320D12—68no—
321D12—62no—
322D12—182no—
323D12—368no—
324C4×D3—64no—
325C4×D3—184no—
326Dic6—62no—
327Dic6—182no—
328C2×C32—24no—
329C2×C32—22no—
330C2×C32—22no—
331C2×C32—22no—
332C2×C32—21no—
333C2×C32C22×D1211yescharacteristic
334C2×C32#100912yes—
335C2×C32#100912yes—
336C2×C32C22×#6711yescharacteristic
337C3⋊D3—42no—
338C3×D3—128no—
339C3×D3—122no—
340C3×D3—124no—
341C3×D3—124no—
342C24—92no—
343#41—184no—
344C2×Q8—91no—
345C2×D4—92no—
346C2×D4—92no—
347C2×D4—92no—
348C2×D4—91no—
349C2×D4—184no—
350C2×D4—278no—
351C22×C4—92no—
352C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
353C4⋊C4—92no—
354C22⋊C4—92no—
355C22⋊C4—278no—
356C42—92no—
357C22×C3—22no—
358C22×C3—31no—
359C22×C3—31no—
360C22×C3—32no—
361C22×C3—32no—
362C22×C3—62no—
363C22×C3—62no—
364C22×C3—64no—
365C22×C3—61no—
366C22×C3—62no—
367C22×C3—184no—
368C22×C3—182no—
369C22×C3—188no—
370C22×C3—184no—
371C22×C3—368no—
372C22×C3C2×#62612yes—
373C22×C3C22×D3211yescharacteristic
374C2×D3—64no—
375C2×D3—64no—
376C2×D3—68no—
377C2×D3—68no—
378C2×D3—68no—
379C2×D3—64no—
380C2×D3—62no—
381C2×D3—128no—
382C2×D3—128no—
383C2×D3—128no—
384C2×D3—1216no—
385C2×D3—182no—
386C2×D3—184no—
387C3×C4—32no—
388C3×C4—32no—
389C3×C4—32no—
390C3×C4—32no—
391C3×C4—62no—
392C3×C4—62no—
393C3×C4—184no—
394C3×C4—182no—
395C3×C4C2×#62612yes—
396C3×C4C2×#62612yes—
397Dic3—64no—
398Dic3—62no—
399Dic3—182no—
400Dic3—184no—
401C32—21no—
402C32#552511yescharacteristic, complemented
403C32#583011yescharacteristic, complemented
404C23—94no—
405C23—92no—
406C23—184no—
407C23—364no—
408C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
409Q8—182no—
410D4—184no—
411D4—184no—
412D4—184no—
413D4—182no—
414D4—1088no—
415C2×C4—94no—
416C2×C4—92no—
417C2×C4—184no—
418C2×C4#620014yescomplemented
419C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
420C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
421C2×C3—22no—
422C2×C3—31no—
423C2×C3—31no—
424C2×C3—32no—
425C2×C3—32no—
426C2×C3—62no—
427C2×C3—62no—
428C2×C3—64no—
429C2×C3—61no—
430C2×C3—62no—
431C2×C3—362no—
432C2×C3—364no—
433C2×C3#6430812yes—
434C2×C3C22×#62611yescharacteristic
435D3—124no—
436D3—124no—
437D3—128no—
438D3—122no—
439D3—362no—
440C22—22no—
441C22—182no—
442C22—184no—
443C22—184no—
444C22—188no—
445C22—368no—
446C22C2×#620012yes—
447C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
448C4—182no—
449C4—184no—
450C4C2×#620012yes—
451C4C2×#620012yes—
452C3—31no—
453C3—31no—
454C3—61no—
455C3#1059504511yescharacteristic
456C2—22no—
457C2—364no—
458C2—362no—
459C2#1172249312yes—
460C2C22×#620011yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#535
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
1. C22×C4⋊#272C2semidirect
2. C2×#3828⋊C3C2semidirect
3. #64135⋊C3C2semidirect
4. #11721323C2semidirect
5. #11721158C2semidirect
6. #11721156C2semidirect
7. #11721114C2semidirect
17. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
18. C2×C4×(C3⋊D3)⋊C3C22semidirect
19. C2×#634⋊C3C22semidirect
20. C2×#6200C22semidirect
22. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
23. C4×(C3⋊Dic3)⋊C3C22semidirect
24. #88451C22semidirect
25. C2×#3828D3semidirect
30. #64135D3semidirect
45. C2×C4×H(3,𝔽3)D4semidirect
68. C2×C4⋊(C3⋊D3)C2×D3semidirect
69. C2×C4×C3⋊D3C2×D3semidirect
70. C2×#634C2×D3semidirect
88. C4⋊(C3⋊Dic3)C2×D3semidirect
89. C4×C3⋊Dic3C2×D3semidirect
146. C2×C32×C4D12semidirect
147. C2×C32×C4#67semidirect
267. H(3,𝔽3)#535semidirect
402. C32#5525semidirect
403. C32#5830semidirect
418. C2×C4#6200semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
C2×#3828⋊C32
#64135⋊C32
#117213232
#117211582
#117211562
#117211142
#105950423
#105950403