Finite Group Databasebeta

← #10592833⋊C3 #13143217

Subgroups — #10592833⋊C3 #13143217

Order 1728.

488 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 102 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×M16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic
3#64133⋊C3C211yesmaximal, characteristic, complemented
4C2×#3744⋊C3C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic, complemented
6#63645⋊C3C211yesmaximal, characteristic, complemented
7#63643⋊C3C211yesmaximal, characteristic, complemented
8#10592833C311yesmaximal, characteristic, complemented
9C3×#5134—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14M16×H(3,𝔽3)—22no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16C2×C8×H(3,𝔽3)C2211yescharacteristic
17#3757⋊C3—21no—
18#3752⋊C3—21no—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21#3750⋊C3—21no—
22#3744⋊C3—22no—
23C2×C32×M16—21no—
24C2×C32×M16D311yescharacteristic, complemented
25C2×C32×M16C2×C311yescharacteristic, complemented
26#64139C2×C311yescharacteristic
27#64133C2×C311yescharacteristic, complemented
28C2×#3744C2×C311yescharacteristic, complemented
29C3×#930—31no—
30C3×#924—31no—
31C2×C3×#204—31no—
32#63648C2×C311yescharacteristic, complemented
33#63645C2×C311yescharacteristic, complemented
34#63643C2×C311yescharacteristic, complemented
35C3×#821—31no—
36C3×#818—31no—
37C3×#816—31no—
38C23×H(3,𝔽3)D411yescharacteristic
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)D411yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#638⋊C3—42no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52C8×H(3,𝔽3)—21no—
53C8×H(3,𝔽3)D412yescomplemented
54C3×#468—91no—
55#5134—31no—
56#5134—91no—
57C22×C32×C4—21no—
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C22×C311yescharacteristic
60C2×#638—21no—
61C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
62C2×#634C22×C311yescharacteristic
63C2×C3×#67—61no—
64C2×C3×D12—31no—
65C2×C3×Dic6—31no—
66C32×M16—24no—
67C32×M16—22no—
68C32×M16—22no—
69C2×C32×C8—21no—
70C2×C32×C8—21no—
71C2×C32×C8C2×D311yescharacteristic
72C2×C32×C8C2×D311yescharacteristic
73C2×C32×C8C22×C311yescharacteristic
74C2×C32×C8C22×C311yescharacteristic
75#3757—21no—
76#3752—21no—
77C4⋊(C3⋊Dic3)C22×C311yescharacteristic
78C4⋊(C3⋊Dic3)C22×C311yescharacteristic
79#3750—21no—
80#3744—22no—
81C3×#217—61no—
82C3×(C2×D3)⋊C4—61no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×Dic3⋊C4—61no—
86C3×#204—62no—
87C22×H(3,𝔽3)—21no—
88C22×H(3,𝔽3)—21no—
89C22×H(3,𝔽3)—21no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C4×H(3,𝔽3)—21no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97(C3⋊Dic3)⋊C3—41no—
98(C3⋊Dic3)⋊C3—41no—
99(C3⋊Dic3)⋊C3—41no—
100C2×C3×SD16—91no—
101C2×C3×M16—31no—
102C2×C3×M16—31no—
103C2×C3×M16—61no—
104C2×C3×M16C3×D311yescharacteristic
105C3×C22⋊Q8—91no—
106C3×#121—91no—
107#930—31no—
108#930—91no—
109#924—31no—
110#924—91no—
111C2×#204—31no—
112C2×#204—91no—
113C3×#107—91no—
114C3×#103—91no—
115C3×#102—91no—
116#821—31no—
117#821—91no—
118#818—31no—
119#818—91no—
120#816—31no—
121#816—91no—
122C23×C32—21no—
123C23×C32D1211yescharacteristic
124C23×C32C3×D411yescharacteristic
125C22×C3⋊D3—21no—
126C22×C3×D3—61no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4D1211yescharacteristic
136C2×C32×C4C3×D411yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C23×C311yescharacteristic
139#638—42no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—21no—
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—21no—
145#634—21no—
146#634—21no—
147C3×#67—122no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155C32×C8—22no—
156C32×C8—21no—
157C32×C8—21no—
158C32×C8—22no—
159C32×C8#6712yescomplemented
160C32×C8C3×D412yescomplemented
161#468—91noSylow
162C2×H(3,𝔽3)—41no—
163C2×H(3,𝔽3)#12111yescharacteristic
164C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
165C2×H(3,𝔽3)#13711yescharacteristic
166(C3⋊D3)⋊C3—81no—
167C2×C3×Q8—91no—
168C2×C3×D4—91no—
169C2×C3×D4—181no—
170C22×C3×C4—31no—
171C22×C3×C4—31no—
172C22×C3×C4—61no—
173C22×C3×C4C2×C3×D311yescharacteristic
174C2×#67—61no—
175C2×#67—181no—
176C2×D12—31no—
177C2×D12—91no—
178C2×Dic6—31no—
179C2×Dic6—91no—
180C3×SD16—182no—
181C3×M16—22no—
182C3×M16—62no—
183C3×M16—62no—
184C3×M16—64no—
185C2×C3×C8—31no—
186C2×C3×C8—31no—
187C2×C3×C8—31no—
188C2×C3×C8—31no—
189C2×C3×C8—61no—
190C2×C3×C8—61no—
191C2×C3×C8C2×C3×D311yescharacteristic
192C2×C3×C8C2×C3×D311yescharacteristic
193C3×C4⋊C4—91no—
194C3×C4⋊C4—91no—
195C3×C4⋊C4—181no—
196C3×C22⋊C4—181no—
197C3×C22⋊C4—181no—
198#217—61no—
199#217—181no—
200(C2×D3)⋊C4—61no—
201(C2×D3)⋊C4—181no—
202C4⋊Dic3—31no—
203C4⋊Dic3—31no—
204C4⋊Dic3—91no—
205C4⋊Dic3—91no—
206Dic3⋊C4—61no—
207Dic3⋊C4—181no—
208#204—62no—
209#204—182no—
210C22×C32—22no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—21no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32C2×D1211yescharacteristic
221C22×C32C2×C3×D411yescharacteristic
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3—41no—
225C2×C3×D3—121no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C32×C4—22no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4#23511yescharacteristic
234C32×C4C2×#6711yescharacteristic
235C32×C4C2×C3×D411yescharacteristic
236C32×C4C3×#4111yescharacteristic
237C3⋊Dic3—41no—
238C3⋊Dic3—41no—
239C3⋊Dic3—41no—
240C3×Dic3—121no—
241C3×Dic3—121no—
242C3×Dic3—121no—
243C2×SD16—91no—
244C2×M16(C3⋊D3)⋊C311yescharacteristic, complemented
245C22⋊Q8—91no—
246#121—91no—
247#107—91no—
248#103—91no—
249#102—91no—
250H(3,𝔽3)#46811yescharacteristic, complemented, Sylow
251C23×C3—31no—
252C23×C3—31no—
253C23×C3—61no—
254C23×C3—181no—
255C23×C3C3×D1211yescharacteristic
256C22×D3—61no—
257C22×D3—181no—
258C3×Q8—181no—
259C3×Q8—181no—
260C3×D4—181no—
261C3×D4—181no—
262C3×D4—362no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—62no—
274C2×C3×C4—62no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×C3×C4C3×D1211yescharacteristic
281C2×C3×C4C22×C3×D311yescharacteristic
282#67—122no—
283#67—362no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—181no—
288C2×Dic3—181no—
289C2×Dic3—181no—
290D12—61no—
291D12—61no—
292D12—181no—
293D12—181no—
294Dic6—61no—
295Dic6—61no—
296Dic6—181no—
297Dic6—181no—
298C3×C8—21no—
299C3×C8—32no—
300C3×C8—32no—
301C3×C8—61no—
302C3×C8—61no—
303C3×C8—62no—
304C3×C8—62no—
305C3×C8C3×#6712yes—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32—42no—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32#90811yescharacteristic
313C2×C32#90911yescharacteristic
314C2×C32#93011yescharacteristic
315C2×C32C3×#12111yescharacteristic
316C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
317C2×C32C3×#13711yescharacteristic
318C3⋊D3—81no—
319C3×D3—241no—
320C2×Q8—91no—
321C2×D4—91no—
322C2×D4—181no—
323C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
324SD16—182no—
325M16—22no—
326C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
327C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
328C4⋊C4—91no—
329C4⋊C4—91no—
330C4⋊C4—181no—
331C22⋊C4—181no—
332C22⋊C4—181no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—61no—
339C22×C3—61no—
340C22×C3—61no—
341C22×C3—61no—
342C22×C3—61no—
343C22×C3—61no—
344C22×C3—62no—
345C22×C3—61no—
346C22×C3—121no—
347C22×C3—121no—
348C22×C3—361no—
349C22×C3—361no—
350C22×C3—361no—
351C22×C3C2×C3×D1211yescharacteristic
352C2×D3—121no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—361no—
356C2×D3—361no—
357C2×D3—361no—
358C3×C4—21no—
359C3×C4—31no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—61no—
364C3×C4—61no—
365C3×C4—62no—
366C3×C4—61no—
367C3×C4—61no—
368C3×C4—361no—
369C3×C4—361no—
370C3×C4—361no—
371C3×C4C3×#23511yescharacteristic
372C3×C4C2×C3×#6711yescharacteristic
373Dic3—121no—
374Dic3—121no—
375Dic3—121no—
376Dic3—361no—
377Dic3—361no—
378Dic3—361no—
379C32—21no—
380C32#513411yescharacteristic, complemented
381C32C3×#46811yescharacteristic, complemented
382C23—181no—
383C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
384Q8—181no—
385Q8—181no—
386D4—181no—
387D4—181no—
388D4—362no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—181no—
392C2×C4—181no—
393C2×C4—181no—
394C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
395C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
396C8—21no—
397C8#638⋊C312yescomplemented
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—41no—
405C2×C3—61no—
406C2×C3—61no—
407C2×C3—61no—
408C2×C3—121no—
409C2×C3—121no—
410C2×C3—122no—
411C2×C3—721no—
412C2×C3C3×#90811yescharacteristic
413C2×C3C3×#90911yescharacteristic
414C2×C3C3×#93011yescharacteristic
415D3—241no—
416D3—721no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—361no—
421C22—361no—
422C22—361no—
423C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
424C4—21no—
425C4—361no—
426C4—361no—
427C4—361no—
428C4#3828⋊C311yescharacteristic
429C4C2×#638⋊C311yescharacteristic
430C3—31no—
431C3—31no—
432C3—61no—
433C3C3×#513411yescharacteristic
434C2—41no—
435C2—721no—
436C2#64117⋊C311yescharacteristic
437C2#64118⋊C311yescharacteristic
438C2#64139⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×M16×H(3,𝔽3)C2
socleC22×C3C2×C3×D12
prime radical#64139⋊C3C2
hypercenterC2×M16(C3⋊D3)⋊C3
hypocenterC32C3×#468
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×M16×H(3,𝔽3)C2semidirect
3. #64133⋊C3C2semidirect
4. C2×#3744⋊C3C2semidirect
5. #63648⋊C3C2semidirect
6. #63645⋊C3C2semidirect
7. #63643⋊C3C2semidirect
8. #10592833C3semidirect
24. C2×C32×M16D3semidirect
25. C2×C32×M16C2×C3semidirect
27. #64133C2×C3semidirect
28. C2×#3744C2×C3semidirect
32. #63648C2×C3semidirect
33. #63645C2×C3semidirect
34. #63643C2×C3semidirect
53. C8×H(3,𝔽3)D4semidirect
159. C32×C8#67semidirect
160. C32×C8C3×D4semidirect
244. C2×M16(C3⋊D3)⋊C3semidirect
250. H(3,𝔽3)#468semidirect
380. C32#5134semidirect
381. C32C3×#468semidirect
397. C8#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×M16×H(3,𝔽3)2
#64139⋊C32
#64133⋊C32
C2×#3744⋊C32
#63648⋊C32
#63645⋊C32
#63643⋊C32
#105928333
C3×#51343