Finite Group Databasebeta

← #13140036

Subgroups — #13140036

Order 1728.

582 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 83 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×#3828⋊C3C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic, complemented
3C2×#88433C211yesmaximal, characteristic, complemented
4C2×#88431C211yesmaximal, characteristic, complemented
5#11720573C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10591184—31nomaximal
9#10591173—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12#3828⋊C3—22no—
13#3828⋊C3—22no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3—21no—
16C2×#634⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×#6180C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20#3757⋊C3—21no—
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
22#3750⋊C3—21no—
23#88433—22no—
24#88431—22no—
25C2×#3828D311yescharacteristic
26C2×C3×#235—31no—
27#64133D311yescharacteristic, complemented
28C3×#924—31no—
29C2×#3719—31no—
30C2×#3719—31no—
31C2×#3717—31no—
32C2×#3716—31no—
33C3×#830—31no—
34C3×#830—31no—
35C3×#830—61no—
36#63567—31no—
37#63566—31no—
38#63564—31no—
39#63563—31no—
40C23×H(3,𝔽3)D411yescharacteristic
41C22×(C3⋊D3)⋊C3—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)D411yescharacteristic
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46#638⋊C3—24no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)D412yes—
52C4×(C3⋊D3)⋊C3—24no—
53#634⋊C3—21no—
54#634⋊C3—21no—
55#634⋊C3—21no—
56#634⋊C3D412yes—
57#6180—21no—
58#6180—21no—
59#5239—91no—
60#5047—91no—
61#4733—91no—
62C22×C32×C4—21no—
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C2×D311yescharacteristic
65C2×#638—21no—
66#3828—22no—
67#3828—22no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×C4×C3⋊D3—21no—
70C2×#634C2×D311yescharacteristic
71C2×#634C2×D311yescharacteristic
72C2×C3×#67—61no—
73C3×#235—62no—
74C3×#235—62no—
75C2×C3×D12—31no—
76C2×C3×C4×D3—61no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—31no—
79#3757—21no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81#3750—21no—
82C3×#217—61no—
83C3×C4⋊Dic3—31no—
84C3×Dic3⋊C4—61no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—61no—
90C2×C3×C3⋊C8—61no—
91#3719—62no—
92#3719—62no—
93#3717—62no—
94#3716—62no—
95C22×H(3,𝔽3)—21no—
96C22×H(3,𝔽3)—21no—
97C22×H(3,𝔽3)—21no—
98C22×H(3,𝔽3)C2×D411yescharacteristic
99C2×(C3⋊D3)⋊C3—22no—
100C2×(C3⋊D3)⋊C3—41no—
101C2×(C3⋊D3)⋊C3—41no—
102C4×H(3,𝔽3)—21no—
103C4×H(3,𝔽3)C2×D411yescharacteristic
104C4×H(3,𝔽3)C2×D411yescharacteristic
105(C3⋊Dic3)⋊C3—22no—
106(C3⋊Dic3)⋊C3—41no—
107(C3⋊Dic3)⋊C3—41no—
108C2×C3×#41—91no—
109C2×#235—31no—
110C2×#235—91no—
111C3×C22⋊Q8—91no—
112C2×#216—91no—
113C2×#216—91no—
114C2×#215—91no—
115C2×#214—91no—
116#924—31no—
117#924—91no—
118C2×Dic12—91no—
119C2×#204—91no—
120C3×C22⋊C8—91no—
121#835—91no—
122#832—91no—
123#830—31no—
124#830—31no—
125#830—61no—
126#821—91no—
127#816—91no—
128#810—91no—
129#809—91no—
130C23×C32—21no—
131C23×C32#6711yescharacteristic
132C23×C32#6711yescharacteristic
133C22×C3⋊D3—21no—
134C22×C3×D3—61no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—41no—
142C2×C32×C4—41no—
143C2×C32×C4#6711yescharacteristic
144C2×C32×C4#6711yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147#638—24no—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—21no—
151C4⋊(C3⋊D3)—21no—
152C4⋊(C3⋊D3)#6712yes—
153C4×C3⋊D3—24no—
154#634—21no—
155#634—21no—
156#634—21no—
157#634#6712yes—
158C3×#67—64no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—61no—
162C3×D12—32no—
163C3×D12—61no—
164C3×C4×D3—64no—
165C3×Dic6—32no—
166C3×Dic6—61no—
167C3×Dic6—61no—
168C3×Dic6—61no—
169C3×C3⋊C8—61no—
170C3×C3⋊C8—61no—
171C3×C3⋊C8—61no—
172C3×C3⋊C8—61no—
173C3×C3⋊C8—121no—
174C3×C3⋊C8—121no—
175#452—271noSylow
176C2×H(3,𝔽3)—41no—
177C2×H(3,𝔽3)#12011yescharacteristic
178C2×H(3,𝔽3)#13511yescharacteristic
179C2×H(3,𝔽3)#13711yescharacteristic
180(C3⋊D3)⋊C3—42no—
181C3×#41—182no—
182C3×#41—182no—
183C2×C3×Q8—91no—
184C2×C3×Q8—91no—
185C2×C3×D4—91no—
186C2×C3×D4—181no—
187C22×C3×C4—31no—
188C22×C3×C4—31no—
189C22×C3×C4—61no—
190C22×C3×C4—181no—
191C22×C3×C4D3211yescharacteristic
192C2×#67—61no—
193C2×#67—181no—
194#235—62no—
195#235—62no—
196#235—182no—
197#235—182no—
198C2×D12—31no—
199C2×D12—91no—
200C2×C4×D3—61no—
201C2×C4×D3—181no—
202C2×Dic6—31no—
203C2×Dic6—31no—
204C2×Dic6—91no—
205C2×Dic6—91no—
206C2×C3×C8—91no—
207C2×C3×C8—91no—
208C3×C4⋊C4—91no—
209C3×C4⋊C4—181no—
210C3×C22⋊C4—181no—
211#217—61no—
212#217—181no—
213#216—182no—
214#216—182no—
215#215—182no—
216#214—182no—
217C4⋊Dic3—31no—
218C4⋊Dic3—91no—
219Dic3⋊C4—61no—
220Dic3⋊C4—181no—
221C2×C3⋊C8—31no—
222C2×C3⋊C8—31no—
223C2×C3⋊C8—31no—
224C2×C3⋊C8—31no—
225C2×C3⋊C8—61no—
226C2×C3⋊C8—61no—
227Dic12—182no—
228#204—182no—
229C22×C32—22no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—21no—
233C22×C32—21no—
234C22×C32—21no—
235C22×C32—21no—
236C22×C32—21no—
237C22×C32—41no—
238C22×C32—41no—
239C22×C32C2×#6711yescharacteristic
240C22×C32C2×#6711yescharacteristic
241C2×C3⋊D3—22no—
242C2×C3⋊D3—41no—
243C2×C3⋊D3—41no—
244C2×C3×D3—62no—
245C2×C3×D3—121no—
246C2×C3×D3—121no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—41no—
252C32×C4D3×D411yescharacteristic
253C32×C4D3×D411yescharacteristic
254C32×C4C2×#6711yescharacteristic
255C32×C4C2×#6711yescharacteristic
256C3⋊Dic3—22no—
257C3⋊Dic3—41no—
258C3⋊Dic3—41no—
259C3×Dic3—62no—
260C3×Dic3—121no—
261C3×Dic3—121no—
262C2×#41—91no—
263C2×Q16—271no—
264C2×SD16—271no—
265C22⋊Q8—91no—
266#103—271no—
267#102—271no—
268C22⋊C8—91no—
269H(3,𝔽3)#45211yescharacteristic, complemented, Sylow
270C23×C3—31no—
271C23×C3—31no—
272C23×C3—61no—
273C23×C3—181no—
274C23×C3#62511yescharacteristic
275C22×D3—61no—
276C22×D3—181no—
277C3×Q8—92no—
278C3×Q8—181no—
279C3×Q8—181no—
280C3×Q8—181no—
281C3×D4—92no—
282C3×D4—181no—
283C3×D4—184no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—31no—
290C2×C3×C4—61no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—121no—
297C2×C3×C4—121no—
298C2×C3×C4—184no—
299C2×C3×C4—181no—
300C2×C3×C4—181no—
301C2×C3×C4—181no—
302C2×C3×C4#62511yescharacteristic
303C2×C3×C4C2×D3211yescharacteristic
304#67—64no—
305#67—184no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—61no—
309C2×Dic3—181no—
310C2×Dic3—181no—
311C2×Dic3—181no—
312D12—32no—
313D12—61no—
314D12—92no—
315D12—181no—
316C4×D3—64no—
317C4×D3—184no—
318Dic6—32no—
319Dic6—61no—
320Dic6—61no—
321Dic6—61no—
322Dic6—92no—
323Dic6—181no—
324Dic6—181no—
325Dic6—181no—
326C3×C8—181no—
327C3×C8—181no—
328C3⋊C8—61no—
329C3⋊C8—61no—
330C3⋊C8—61no—
331C3⋊C8—61no—
332C3⋊C8—121no—
333C3⋊C8—121no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32—42no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32#93711yescharacteristic
341C2×C32#94211yescharacteristic
342C2×C32#95011yescharacteristic
343C2×C32#95011yescharacteristic
344C2×C32#95111yescharacteristic
345C2×C32#95311yescharacteristic
346C3⋊D3—42no—
347C3×D3—122no—
348#41—182no—
349#41—182no—
350C2×Q8—91no—
351C2×Q8—91no—
352C2×D4—91no—
353C2×D4—181no—
354C22×C4—181no—
355C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
356Q16—542no—
357SD16—542no—
358C2×C8—91no—
359C2×C8—91no—
360C4⋊C4—91no—
361C4⋊C4—181no—
362C22⋊C4—181no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—31no—
367C22×C3—31no—
368C22×C3—61no—
369C22×C3—61no—
370C22×C3—61no—
371C22×C3—61no—
372C22×C3—61no—
373C22×C3—61no—
374C22×C3—62no—
375C22×C3—61no—
376C22×C3—121no—
377C22×C3—121no—
378C22×C3—182no—
379C22×C3—361no—
380C22×C3—361no—
381C22×C3C2×#62511yescharacteristic
382C2×D3—62no—
383C2×D3—121no—
384C2×D3—121no—
385C2×D3—182no—
386C2×D3—361no—
387C2×D3—361no—
388C3×C4—21no—
389C3×C4—31no—
390C3×C4—31no—
391C3×C4—31no—
392C3×C4—31no—
393C3×C4—61no—
394C3×C4—61no—
395C3×C4—61no—
396C3×C4—61no—
397C3×C4—121no—
398C3×C4—182no—
399C3×C4—361no—
400C3×C4—361no—
401C3×C4D3×#6711yescharacteristic
402C3×C4D3×#6711yescharacteristic
403Dic3—62no—
404Dic3—121no—
405Dic3—121no—
406Dic3—182no—
407Dic3—361no—
408Dic3—361no—
409C32—21no—
410C32#504711yescharacteristic, complemented
411C32#523911yescharacteristic, complemented
412C23—181no—
413C23#619811yescharacteristic
414Q8—92no—
415Q8—181no—
416Q8—181no—
417Q8—181no—
418D4—92no—
419D4—184no—
420D4—181no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—184no—
424C2×C4—181no—
425C2×C4—181no—
426C2×C4—181no—
427C2×C4#619811yescharacteristic
428C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
429C8—181no—
430C8—181no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—41no—
438C2×C3—61no—
439C2×C3—61no—
440C2×C3—61no—
441C2×C3—121no—
442C2×C3—121no—
443C2×C3—122no—
444C2×C3—362no—
445C2×C3#6382211yescharacteristic
446C2×C3#6382711yescharacteristic
447C2×C3#6397611yescharacteristic
448D3—122no—
449D3—362no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—182no—
454C22—361no—
455C22—361no—
456C22C2×#619811yescharacteristic
457C4—21no—
458C4—182no—
459C4—361no—
460C4—361no—
461C4#8867611yescharacteristic
462C4#8867611yescharacteristic
463C3—31no—
464C3—31no—
465C3—61no—
466C3#1059117011yescharacteristic
467C2—41no—
468C2—362no—
469C2#1172105811yescharacteristic
470C2#1172106011yescharacteristic
471C2#1172131611yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC2×#3828⋊C3C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#452
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #64133⋊C3C2semidirect
3. C2×#88433C2semidirect
4. C2×#88431C2semidirect
5. #11720573C2semidirect
6. #11720318C2semidirect
7. #11720313C2semidirect
27. #64133D3semidirect
269. H(3,𝔽3)#452semidirect
410. C32#5047semidirect
411. C32#5239semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
#64133⋊C32
C2×#884332
C2×#884312
#117205732
#117203182
#117203132
#105911843
#105911733