Finite Group Databasebeta

← #13140192

Subgroups — #13140192

Order 1728.

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

normal subgroups only

This group has 606 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#11721824C211yesmaximal, characteristic, complemented
3C2×#88439C211yesmaximal, characteristic, complemented
4C2×#88438C211yesmaximal, characteristic, complemented
5#63657⋊C3C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10591184—31nomaximal
9#10591170—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
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
18#88552—21no—
19C4⋊#1084C2211yescharacteristic
20#88540—21no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88439—22no—
24#88438—22no—
25C2×#3828D311yescharacteristic
26C2×C3×#235—31no—
27C3×#924—31no—
28C3×#924—31no—
29C3×#924—61no—
30C2×#3719—31no—
31C2×#3718—31no—
32C2×#3717—31no—
33C2×#3715—31no—
34#63657D311yescharacteristic, complemented
35C3×#830—31no—
36#63567—31no—
37#63565—31no—
38#63564—31no—
39#63562—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
46C2×#1084—21no—
47C2×#1084—21no—
48#6229—21no—
49#6229—21no—
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—
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×C3×#67—61no—
72C3×#235—62no—
73C3×#235—62no—
74C2×C3×D12—31no—
75C2×C3×C4×D3—61no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—61no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×#217—62no—
83C3×#217—61no—
84C3×#217—61no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—61no—
88C3×Dic3⋊C4—62no—
89C3×Dic3⋊C4—61no—
90C3×Dic3⋊C4—61no—
91C2×C3×C3⋊C8—31no—
92C2×C3×C3⋊C8—31no—
93#3719—62no—
94#3718—62no—
95#3717—62no—
96#3715—62no—
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)C2×D411yescharacteristic
106C4×H(3,𝔽3)C2×D411yescharacteristic
107#1084—41no—
108#1084—41no—
109(C3⋊Dic3)⋊C3—22no—
110C2×C3×#41—91no—
111C2×#235—31no—
112C2×#235—91no—
113C3×C22⋊Q8—91no—
114C2×#216—91no—
115C2×#216—91no—
116C2×#215—91no—
117C2×#214—91no—
118#924—31no—
119#924—31no—
120#924—61no—
121C2×Dic12—91no—
122C2×#204—91no—
123C3×C22⋊C8—91no—
124#835—91no—
125#832—91no—
126#830—31no—
127#830—91no—
128#821—91no—
129#816—91no—
130#810—91no—
131#809—91no—
132C23×C32—21no—
133C23×C32D1211yescharacteristic
134C23×C32#6711yescharacteristic
135C22×C3⋊D3—21no—
136C22×C3×D3—61no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—41no—
144C2×C32×C4—41no—
145C2×C32×C4D1211yescharacteristic
146C2×C32×C4#6711yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149#638—24no—
150C2×C3⋊Dic3—21no—
151C4⋊(C3⋊D3)—21no—
152C4⋊(C3⋊D3)#6712yes—
153C4×C3⋊D3—24no—
154#634—21no—
155#634#6712yes—
156C3×#67—64no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C3×D12—32no—
165C3×D12—61no—
166C3×C4×D3—64no—
167C3×Dic6—32no—
168C3×Dic6—61no—
169C3×Dic6—61no—
170C3×Dic6—61no—
171C3×Dic6—61no—
172C3×Dic6—61no—
173C3×Dic6—121no—
174C3×Dic6—121no—
175C3⋊(C3⋊C8)—21no—
176C3⋊(C3⋊C8)—21no—
177C3×C3⋊C8—61no—
178C3×C3⋊C8—61no—
179#452—271noSylow
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—91no—
188C2×C3×Q8—91no—
189C2×C3×D4—91no—
190C2×C3×D4—181no—
191C22×C3×C4—31no—
192C22×C3×C4—31no—
193C22×C3×C4—61no—
194C22×C3×C4—181no—
195C22×C3×C4D3211yescharacteristic
196C2×#67—61no—
197C2×#67—181no—
198#235—62no—
199#235—62no—
200#235—182no—
201#235—182no—
202C2×D12—31no—
203C2×D12—91no—
204C2×C4×D3—61no—
205C2×C4×D3—181no—
206C2×Dic6—31no—
207C2×Dic6—31no—
208C2×Dic6—31no—
209C2×Dic6—61no—
210C2×Dic6—91no—
211C2×C3×C8—91no—
212C2×C3×C8—91no—
213C3×C4⋊C4—91no—
214C3×C4⋊C4—181no—
215C3×C22⋊C4—181no—
216#217—61no—
217#217—61no—
218#217—62no—
219#216—182no—
220#216—182no—
221#215—182no—
222#214—182no—
223C4⋊Dic3—31no—
224C4⋊Dic3—31no—
225C4⋊Dic3—61no—
226Dic3⋊C4—61no—
227Dic3⋊C4—61no—
228Dic3⋊C4—62no—
229C2×C3⋊C8—31no—
230C2×C3⋊C8—31no—
231C2×C3⋊C8—91no—
232C2×C3⋊C8—91no—
233Dic12—182no—
234#204—182no—
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×D1211yescharacteristic
246C22×C32C2×#6711yescharacteristic
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×C4D3×D411yescharacteristic
259C32×C4D3×D411yescharacteristic
260C32×C4C2×#6711yescharacteristic
261C32×C4C2×#6711yescharacteristic
262C3⋊Dic3—22no—
263C3×Dic3—62no—
264C3×Dic3—122no—
265C3×Dic3—122no—
266C3×Dic3—121no—
267C3×Dic3—121no—
268C3×Dic3—121no—
269C3×Dic3—121no—
270C2×#41—91no—
271C2×Q16—271no—
272C2×SD16—271no—
273C22⋊Q8—91no—
274#103—271no—
275#102—271no—
276C22⋊C8—91no—
277H(3,𝔽3)#45211yescharacteristic, complemented, Sylow
278C23×C3—31no—
279C23×C3—31no—
280C23×C3—61no—
281C23×C3—181no—
282C23×C3#62611yescharacteristic
283C22×D3—61no—
284C22×D3—181no—
285C3×Q8—92no—
286C3×Q8—181no—
287C3×Q8—181no—
288C3×Q8—181no—
289C3×D4—92no—
290C3×D4—181no—
291C3×D4—184no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—31no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—31no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—121no—
305C2×C3×C4—121no—
306C2×C3×C4—184no—
307C2×C3×C4—181no—
308C2×C3×C4—181no—
309C2×C3×C4—181no—
310C2×C3×C4#62611yescharacteristic
311C2×C3×C4C2×D3211yescharacteristic
312#67—64no—
313#67—184no—
314C2×Dic3—61no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—62no—
320C2×Dic3—62no—
321C2×Dic3—181no—
322D12—32no—
323D12—61no—
324D12—92no—
325D12—181no—
326C4×D3—64no—
327C4×D3—184no—
328Dic6—32no—
329Dic6—61no—
330Dic6—61no—
331Dic6—61no—
332Dic6—61no—
333Dic6—61no—
334Dic6—92no—
335Dic6—121no—
336Dic6—121no—
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#88211yescharacteristic
351C2×C32#90411yescharacteristic
352C2×C32#90911yescharacteristic
353C2×C32#95011yescharacteristic
354C2×C32#95111yescharacteristic
355C2×C32#95311yescharacteristic
356C3⋊D3—42no—
357C3×D3—122no—
358#41—182no—
359#41—182no—
360C2×Q8—91no—
361C2×Q8—91no—
362C2×D4—91no—
363C2×D4—181no—
364C22×C4—181no—
365C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
366Q16—542no—
367SD16—542no—
368C2×C8—91no—
369C2×C8—91no—
370C4⋊C4—91no—
371C4⋊C4—181no—
372C22⋊C4—181no—
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×#62611yescharacteristic
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—61no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—61no—
407C3×C4—121no—
408C3×C4—182no—
409C3×C4—361no—
410C3×C4—361no—
411C3×C4D3×#6711yescharacteristic
412C3×C4D3×#6711yescharacteristic
413Dic3—62no—
414Dic3—121no—
415Dic3—121no—
416Dic3—121no—
417Dic3—121no—
418Dic3—122no—
419Dic3—122no—
420Dic3—182no—
421C32—21no—
422C32#473311yescharacteristic, complemented
423C32#523911yescharacteristic, complemented
424C23—181no—
425C23#620011yescharacteristic
426Q8—92no—
427Q8—181no—
428Q8—181no—
429Q8—181no—
430D4—92no—
431D4—184no—
432D4—181no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—184no—
436C2×C4—181no—
437C2×C4—181no—
438C2×C4—181no—
439C2×C4#620011yescharacteristic
440C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
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×C3#6382911yescharacteristic
458C2×C3#6383111yescharacteristic
459C2×C3#6397711yescharacteristic
460D3—122no—
461D3—362no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—182no—
466C22—361no—
467C22—361no—
468C22C2×#620011yescharacteristic
469C4—21no—
470C4—182no—
471C4—361no—
472C4—361no—
473C4#8867611yescharacteristic
474C4#8867611yescharacteristic
475C3—31no—
476C3—31no—
477C3—61no—
478C3#1059117311yescharacteristic
479C2—41no—
480C2—362no—
481C2#1172107311yescharacteristic
482C2#1172107411yescharacteristic
483C2#1172132111yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
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. #11721824C2semidirect
3. C2×#88439C2semidirect
4. C2×#88438C2semidirect
5. #63657⋊C3C2semidirect
6. #11720309C2semidirect
7. #11720308C2semidirect
34. #63657D3semidirect
277. H(3,𝔽3)#452semidirect
422. C32#4733semidirect
423. 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
#117218242
C2×#884392
C2×#884382
#63657⋊C32
#117203092
#117203082
#105911843
#105911703