Finite Group Databasebeta

← #10592746⋊C3 #13142885

Subgroups — #10592746⋊C3 #13142885

Order 1728.

606 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 100 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
2H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#3755⋊C3C211yesmaximal, characteristic, complemented
4C2×#3756⋊C3C211yesmaximal, characteristic, complemented
5#63657⋊C3C211yesmaximal, characteristic, complemented
6#63636⋊C3C211yesmaximal, characteristic, complemented
7#63637⋊C3C211yesmaximal, characteristic, complemented
8#10592746C311yesmaximal, characteristic, complemented
9C3×#5047—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—21no—
13#3828⋊C3—22no—
14#3828⋊C3—22no—
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3—21no—
17C2×#634⋊C3C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)—21no—
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20C22⋊C4×H(3,𝔽3)—21no—
21#3755⋊C3—22no—
22#3756⋊C3—22no—
23C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
24C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
25C2×#3828C2×C311yescharacteristic
26C2×C3×#235—31no—
27C32×C22⋊Q8—21no—
28C32×C22⋊Q8D311yescharacteristic, complemented
29C32×C22⋊Q8C2×C311yescharacteristic, complemented
30C2×#3756C2×C311yescharacteristic, complemented
31C2×#3755C2×C311yescharacteristic, complemented
32C2×C3×#216—31no—
33C2×C3×#215—31no—
34#63657C2×C311yescharacteristic, complemented
35#63637C2×C311yescharacteristic, complemented
36#63636C2×C311yescharacteristic, complemented
37C3×#830—31no—
38C3×#810—31no—
39C3×#809—31no—
40C23×H(3,𝔽3)D411yescharacteristic
41C22×(C3⋊D3)⋊C3—21no—
42Q8×H(3,𝔽3)—21no—
43Q8×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)—21no—
48C2×C4×H(3,𝔽3)D411yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50#638⋊C3—24no—
51C2×(C3⋊Dic3)⋊C3—21no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53C4⋊((C3⋊D3)⋊C3)D412yes—
54C4×(C3⋊D3)⋊C3—24no—
55#634⋊C3—21no—
56#634⋊C3D412yes—
57(C3⋊(C3⋊C8))⋊C3—21no—
58(C3⋊(C3⋊C8))⋊C3—21no—
59C3×#452—91no—
60#5047—31no—
61#5047—91no—
62C2×C32×Q8—21no—
63C2×C32×Q8C2×D311yescharacteristic
64C2×C32×Q8C22×C311yescharacteristic
65C22×C32×C4—21no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C22×C311yescharacteristic
68C2×#638—21no—
69#3828—22no—
70#3828—22no—
71C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
72C2×C4×C3⋊D3—21no—
73C2×#634C22×C311yescharacteristic
74C2×C3×#67—61no—
75C3×#235—62no—
76C3×#235—62no—
77C2×C3×D12—31no—
78C2×C3×C4×D3—61no—
79C2×C3×Dic6—31no—
80C32×C4⋊C4—22no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4—21no—
83C32×C4⋊C4—21no—
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C4⋊C4C22×C311yescharacteristic
86C32×C22⋊C4—22no—
87C32×C22⋊C4—21no—
88C32×C22⋊C4—21no—
89#3756—22no—
90#3755—22no—
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
93C3×#216—62no—
94C3×#215—62no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—31no—
97C22×H(3,𝔽3)—21no—
98C22×H(3,𝔽3)—21no—
99C22×H(3,𝔽3)—21no—
100C22×H(3,𝔽3)C2×D411yescharacteristic
101C2×(C3⋊D3)⋊C3—22no—
102C2×(C3⋊D3)⋊C3—41no—
103C2×(C3⋊D3)⋊C3—41no—
104C4×H(3,𝔽3)—21no—
105C4×H(3,𝔽3)—41no—
106C4×H(3,𝔽3)—41no—
107C4×H(3,𝔽3)C2×D411yescharacteristic
108C4×H(3,𝔽3)C2×D411yescharacteristic
109(C3⋊Dic3)⋊C3—22no—
110C2×C3×#41—91no—
111C2×#235—31no—
112C2×#235—91no—
113C2×C3×Q16—91no—
114C2×C3×SD16—91no—
115C3×C22⋊Q8—31no—
116C3×C22⋊Q8—31no—
117C3×C22⋊Q8—61no—
118C3×C22⋊Q8C3×D311yescharacteristic
119C2×#216—31no—
120C2×#216—91no—
121C2×#215—31no—
122C2×#215—91no—
123C3×#103—91no—
124C3×#102—91no—
125C3×C22⋊C8—91no—
126#830—31no—
127#830—91no—
128#810—31no—
129#810—91no—
130#809—31no—
131#809—91no—
132C23×C32—21no—
133C23×C32#6711yescharacteristic
134C23×C32C3×D411yescharacteristic
135C22×C3⋊D3—21no—
136C22×C3×D3—61no—
137C32×Q8—21no—
138C32×Q8—21no—
139C32×Q8—21no—
140C32×Q8—21no—
141C32×Q8—41no—
142C32×Q8—41no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—22no—
148C2×C32×C4—22no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—41no—
156C2×C32×C4—41no—
157C2×C32×C4#6711yescharacteristic
158C2×C32×C4C3×D411yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C23×C311yescharacteristic
161#638—24no—
162C2×C3⋊Dic3—21no—
163C4⋊(C3⋊D3)—21no—
164C4⋊(C3⋊D3)C3×D412yes—
165C4×C3⋊D3—24no—
166#634—21no—
167#634C3×D412yes—
168C3×#67—64no—
169C2×C3×Dic3—61no—
170C3×D12—32no—
171C3×D12—61no—
172C3×C4×D3—64no—
173C3×Dic6—32no—
174C3×Dic6—61no—
175C3⋊(C3⋊C8)—21no—
176C3⋊(C3⋊C8)—21no—
177C3×C3⋊C8—61no—
178C3×C3⋊C8—61no—
179#452—91noSylow
180C2×H(3,𝔽3)—41no—
181C2×H(3,𝔽3)#12011yescharacteristic
182C2×H(3,𝔽3)#13511yescharacteristic
183C2×H(3,𝔽3)#13711yescharacteristic
184(C3⋊D3)⋊C3—42no—
185C3×#41—182no—
186C3×#41—182no—
187C2×C3×Q8—31no—
188C2×C3×Q8—31no—
189C2×C3×Q8—61no—
190C2×C3×Q8—91no—
191C2×C3×Q8C2×C3×D311yescharacteristic
192C2×C3×D4—91no—
193C2×C3×D4—181no—
194C22×C3×C4—31no—
195C22×C3×C4—31no—
196C22×C3×C4—61no—
197C22×C3×C4—181no—
198C22×C3×C4C2×C3×D311yescharacteristic
199C2×#67—61no—
200C2×#67—181no—
201#235—62no—
202#235—62no—
203#235—182no—
204#235—182no—
205C2×D12—31no—
206C2×D12—91no—
207C2×C4×D3—61no—
208C2×C4×D3—181no—
209C2×Dic6—31no—
210C2×Dic6—91no—
211C3×Q16—182no—
212C3×SD16—182no—
213C2×C3×C8—91no—
214C2×C3×C8—91no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—31no—
217C3×C4⋊C4—31no—
218C3×C4⋊C4—61no—
219C3×C4⋊C4—61no—
220C3×C4⋊C4—62no—
221C3×C4⋊C4—61no—
222C3×C4⋊C4C2×C3×D311yescharacteristic
223C3×C22⋊C4—21no—
224C3×C22⋊C4—61no—
225C3×C22⋊C4—61no—
226C3×C22⋊C4—62no—
227#216—62no—
228#216—182no—
229#215—62no—
230#215—182no—
231C2×C3⋊C8—31no—
232C2×C3⋊C8—31no—
233C2×C3⋊C8—91no—
234C2×C3⋊C8—91no—
235C22×C32—22no—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—41no—
244C22×C32—41no—
245C22×C32C2×#6711yescharacteristic
246C22×C32C2×C3×D411yescharacteristic
247C2×C3⋊D3—22no—
248C2×C3⋊D3—41no—
249C2×C3⋊D3—41no—
250C2×C3×D3—62no—
251C2×C3×D3—121no—
252C2×C3×D3—121no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—41no—
258C32×C4—42no—
259C32×C4—42no—
260C32×C4—41no—
261C32×C4—41no—
262C32×C4—41no—
263C32×C4—41no—
264C32×C4D3×D411yescharacteristic
265C32×C4D3×D411yescharacteristic
266C32×C4C2×C3×D411yescharacteristic
267C32×C4C2×C3×D411yescharacteristic
268C3⋊Dic3—22no—
269C3×Dic3—62no—
270C2×#41—91no—
271C2×Q16—91no—
272C2×SD16—91no—
273C22⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
274#103—91no—
275#102—91no—
276C22⋊C8—91no—
277H(3,𝔽3)#45211yescharacteristic, complemented, Sylow
278C23×C3—31no—
279C23×C3—31no—
280C23×C3—61no—
281C23×C3—181no—
282C23×C3C3×#6711yescharacteristic
283C22×D3—61no—
284C22×D3—181no—
285C3×Q8—21no—
286C3×Q8—21no—
287C3×Q8—61no—
288C3×Q8—61no—
289C3×Q8—61no—
290C3×Q8—61no—
291C3×Q8—92no—
292C3×Q8—121no—
293C3×Q8—121no—
294C3×Q8—181no—
295C3×D4—92no—
296C3×D4—181no—
297C3×D4—184no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—31no—
303C2×C3×C4—31no—
304C2×C3×C4—31no—
305C2×C3×C4—31no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—121no—
319C2×C3×C4—121no—
320C2×C3×C4—184no—
321C2×C3×C4—181no—
322C2×C3×C4C3×#6711yescharacteristic
323C2×C3×C4C22×C3×D311yescharacteristic
324#67—64no—
325#67—184no—
326C2×Dic3—61no—
327C2×Dic3—181no—
328D12—32no—
329D12—61no—
330D12—92no—
331D12—181no—
332C4×D3—64no—
333C4×D3—184no—
334Dic6—32no—
335Dic6—61no—
336Dic6—92no—
337Dic6—181no—
338C3×C8—181no—
339C3×C8—181no—
340C3⋊C8—61no—
341C3⋊C8—61no—
342C3⋊C8—181no—
343C3⋊C8—181no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—42no—
348C2×C32—41no—
349C2×C32—41no—
350C2×C32#93711yescharacteristic
351C2×C32#94211yescharacteristic
352C2×C32#95011yescharacteristic
353C2×C32C3×#12011yescharacteristic
354C2×C32C3×#13511yescharacteristic
355C2×C32C3×#13711yescharacteristic
356C3⋊D3—42no—
357C3×D3—122no—
358#41—182no—
359#41—182no—
360C2×Q8—91no—
361C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
362C2×D4—91no—
363C2×D4—181no—
364C22×C4—181no—
365C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
366Q16—182no—
367SD16—182no—
368C2×C8—91no—
369C2×C8—91no—
370C4⋊C4—21no—
371C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
372C22⋊C4—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—31no—
377C22×C3—31no—
378C22×C3—61no—
379C22×C3—61no—
380C22×C3—61no—
381C22×C3—61no—
382C22×C3—61no—
383C22×C3—61no—
384C22×C3—62no—
385C22×C3—61no—
386C22×C3—121no—
387C22×C3—121no—
388C22×C3—182no—
389C22×C3—361no—
390C22×C3—361no—
391C22×C3C2×C3×#6711yescharacteristic
392C2×D3—62no—
393C2×D3—121no—
394C2×D3—121no—
395C2×D3—182no—
396C2×D3—361no—
397C2×D3—361no—
398C3×C4—21no—
399C3×C4—31no—
400C3×C4—31no—
401C3×C4—31no—
402C3×C4—31no—
403C3×C4—41no—
404C3×C4—41no—
405C3×C4—61no—
406C3×C4—61no—
407C3×C4—61no—
408C3×C4—61no—
409C3×C4—121no—
410C3×C4—121no—
411C3×C4—121no—
412C3×C4—121no—
413C3×C4—121no—
414C3×C4—122no—
415C3×C4—122no—
416C3×C4—182no—
417C3×C4C3×D3×D411yescharacteristic
418C3×C4C3×D3×D411yescharacteristic
419Dic3—62no—
420Dic3—182no—
421C32—21no—
422C32#504711yescharacteristic, complemented
423C32C3×#45211yescharacteristic, complemented
424C23—181no—
425C23#638⋊C311yescharacteristic
426Q8—21no—
427Q8—21no—
428Q8—92no—
429Q8—181no—
430D4—92no—
431D4—184no—
432D4—181no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—184no—
438C2×C4—181no—
439C2×C4#638⋊C311yescharacteristic
440C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
441C8—181no—
442C8—181no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—31no—
447C2×C3—31no—
448C2×C3—31no—
449C2×C3—41no—
450C2×C3—61no—
451C2×C3—61no—
452C2×C3—61no—
453C2×C3—121no—
454C2×C3—121no—
455C2×C3—122no—
456C2×C3—362no—
457C2×C3C3×#93711yescharacteristic
458C2×C3C3×#94211yescharacteristic
459C2×C3C3×#95011yescharacteristic
460D3—122no—
461D3—362no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—182no—
466C22—361no—
467C22—361no—
468C22C2×#638⋊C311yescharacteristic
469C4—21no—
470C4—41no—
471C4—41no—
472C4—182no—
473C4D4×(C3⋊D3)⋊C311yescharacteristic
474C4D4×(C3⋊D3)⋊C311yescharacteristic
475C3—31no—
476C3—31no—
477C3—61no—
478C3C3×#504711yescharacteristic
479C2—41no—
480C2—362no—
481C2#64146⋊C311yescharacteristic
482C2#64151⋊C311yescharacteristic
483C2#64159⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC2×#3828⋊C3C2
hypercenterC22⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#452
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×C22⋊Q8C2semidirect
3. C2×#3755⋊C3C2semidirect
4. C2×#3756⋊C3C2semidirect
5. #63657⋊C3C2semidirect
6. #63636⋊C3C2semidirect
7. #63637⋊C3C2semidirect
8. #10592746C3semidirect
28. C32×C22⋊Q8D3semidirect
29. C32×C22⋊Q8C2×C3semidirect
30. C2×#3756C2×C3semidirect
31. C2×#3755C2×C3semidirect
34. #63657C2×C3semidirect
35. #63637C2×C3semidirect
36. #63636C2×C3semidirect
273. C22⋊Q8(C3⋊D3)⋊C3semidirect
277. H(3,𝔽3)#452semidirect
422. C32#5047semidirect
423. C32C3×#452semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
H(3,𝔽3)×C22⋊Q82
C2×#3755⋊C32
C2×#3756⋊C32
#63657⋊C32
#63636⋊C32
#63637⋊C32
#105927463
C3×#50473