Finite Group Databasebeta

← #10592870⋊C3 #13143374

Subgroups — #10592870⋊C3 #13143374

Order 1728.

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

normal subgroups only

This group has 502 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×SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64155⋊C3C211yesmaximal, characteristic, complemented
3#64147⋊C3C211yesmaximal, characteristic
4C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#63659⋊C3C211yesmaximal, characteristic, complemented
7#63644⋊C3C211yesmaximal, characteristic, complemented
8#10592870C311yesmaximal, characteristic, complemented
9C3×#5171—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—21no—
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14SD16×H(3,𝔽3)—22no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16#3757⋊C3—21no—
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20#3750⋊C3—21no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C8×(C3⋊D3)⋊C3—22no—
23C2×C32×SD16—21no—
24C2×C32×SD16D311yescharacteristic, complemented
25C2×C32×SD16C2×C311yescharacteristic, complemented
26#64155C2×C311yescharacteristic, complemented
27#64147C2×C311yescharacteristic
28C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
29C3×#946—31no—
30C3×#938—31no—
31C2×C3×D3×C8—31no—
32#63662C2×C311yescharacteristic, complemented
33#63659C2×C311yescharacteristic, complemented
34#63644C2×C311yescharacteristic, complemented
35C3×#835—31no—
36C3×#832—31no—
37C3×#817—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43D4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46#638⋊C3—42no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3D411yescharacteristic
50C4×(C3⋊D3)⋊C3—21no—
51C4×(C3⋊D3)⋊C3—21no—
52C8×H(3,𝔽3)D412yescomplemented
53(C3⋊(C3⋊C8))⋊C3—21no—
54C3×#465—91no—
55#5171—31no—
56#5171—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C22×C311yescharacteristic
60C2×C32×D4—21no—
61C2×C32×D4C2×D311yescharacteristic
62C2×C32×D4C22×C311yescharacteristic
63C2×#638—21no—
64C2×C4×C3⋊D3C22×C311yescharacteristic
65C2×C3×#67—61no—
66C2×C3×C4×D3—31no—
67C32×SD16—24no—
68C32×SD16—22no—
69C32×SD16—22no—
70C2×C32×C8—21no—
71C2×C32×C8C2×D311yescharacteristic
72C2×C32×C8C22×C311yescharacteristic
73#3757—21no—
74#3752—21no—
75C4⋊(C3⋊Dic3)C22×C311yescharacteristic
76C4⋊(C3⋊Dic3)C22×C311yescharacteristic
77#3750—21no—
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C8×C3⋊D3—22no—
80C3×#217—61no—
81C3×(C2×D3)⋊C4—61no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×Dic3⋊C4—61no—
85C2×C3×C3⋊C8—31no—
86C3×D3×C8—62no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3SD1612yescomplemented
94C4×H(3,𝔽3)—41no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99(C3⋊Dic3)⋊C3—41no—
100C2×C3×SD16—31no—
101C2×C3×SD16—31no—
102C2×C3×SD16—61no—
103C2×C3×SD16C3×D311yescharacteristic
104C22×C3×C8—91no—
105C3×C22⋊Q8—91no—
106C3×#121—91no—
107#946—31no—
108#946—91no—
109#938—31no—
110#938—91no—
111C2×D3×C8—31no—
112C2×D3×C8—91no—
113C3×#106—91no—
114C3×#103—91no—
115C3×#102—91no—
116#835—31no—
117#835—91no—
118#832—31no—
119#832—91no—
120#817—31no—
121#817—91no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—41no—
125C22×C3⋊D3C3×D411yescharacteristic
126C22×C3×D3—31no—
127C32×Q8—22no—
128C32×Q8—21no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—41no—
133C32×D4—22no—
134C32×D4—21no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—21no—
138C32×D4—41no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—41no—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C23×C311yescharacteristic
145#638—42no—
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3C3×D411yescharacteristic
149C4×C3⋊D3—21no—
150C4×C3⋊D3—21no—
151C3×#67—122no—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C3×C4×D3—61no—
156C3×C4×D3—61no—
157C32×C8—22no—
158C32×C8#6712yescomplemented
159C32×C8C3×D412yescomplemented
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—61no—
162#465—91noSylow
163C2×H(3,𝔽3)—81no—
164C2×H(3,𝔽3)#12111yescharacteristic
165C2×H(3,𝔽3)C2×SD1611yescharacteristic
166C2×H(3,𝔽3)#13511yescharacteristic
167(C3⋊D3)⋊C3—22no—
168C2×C3×Q8—31no—
169C2×C3×Q8—31no—
170C2×C3×Q8—61no—
171C2×C3×Q8C2×C3×D311yescharacteristic
172C2×C3×D4—31no—
173C2×C3×D4—31no—
174C2×C3×D4—61no—
175C2×C3×D4—181no—
176C2×C3×D4C2×C3×D311yescharacteristic
177C22×C3×C4—91no—
178C2×#67—61no—
179C2×#67—181no—
180C2×C4×D3—31no—
181C2×C4×D3—91no—
182C3×SD16—22no—
183C3×SD16—62no—
184C3×SD16—62no—
185C3×SD16—64no—
186C2×C3×C8—31no—
187C2×C3×C8—31no—
188C2×C3×C8—61no—
189C2×C3×C8—91no—
190C2×C3×C8—182no—
191C2×C3×C8C2×C3×D311yescharacteristic
192C3×C4⋊C4—91no—
193C3×C4⋊C4—91no—
194C3×C4⋊C4—181no—
195C3×C22⋊C4—181no—
196C3×C22⋊C4—181no—
197#217—61no—
198#217—181no—
199(C2×D3)⋊C4—61no—
200(C2×D3)⋊C4—181no—
201C4⋊Dic3—31no—
202C4⋊Dic3—31no—
203C4⋊Dic3—91no—
204C4⋊Dic3—91no—
205Dic3⋊C4—61no—
206Dic3⋊C4—181no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—91no—
209D3×C8—62no—
210D3×C8—182no—
211C22×C32—21no—
212C22×C32—42no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—81no—
220C22×C32—81no—
221C22×C32D3×D411yescharacteristic
222C22×C32C2×C3×D411yescharacteristic
223C2×C3⋊D3—21no—
224C2×C3⋊D3—21no—
225C2×C3⋊D3C3×SD1612yescomplemented
226C2×C3×D3—32no—
227C2×C3×D3—61no—
228C2×C3×D3—61no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—42no—
232C32×C4—41no—
233C32×C4—41no—
234C32×C4#23711yescharacteristic
235C32×C4C2×#6711yescharacteristic
236C32×C4C2×C3×D411yescharacteristic
237C32×C4C3×#4111yescharacteristic
238C3⋊Dic3—21no—
239C3⋊Dic3—41no—
240C3⋊Dic3—41no—
241C3×Dic3—61no—
242C3×Dic3—121no—
243C3×Dic3—121no—
244C2×SD16(C3⋊D3)⋊C311yescharacteristic, complemented
245C22×C8—91no—
246C22⋊Q8—91no—
247#121—91no—
248#106—91no—
249#103—91no—
250#102—91no—
251H(3,𝔽3)#46511yescharacteristic, complemented, Sylow
252C23×C3—21no—
253C23×C3—61no—
254C23×C3—61no—
255C23×C3—91no—
256C23×C3—121no—
257C22×D3—31no—
258C22×D3—91no—
259C3×Q8—21no—
260C3×Q8—21no—
261C3×Q8—61no—
262C3×Q8—61no—
263C3×Q8—61no—
264C3×Q8—61no—
265C3×Q8—62no—
266C3×Q8—121no—
267C3×D4—21no—
268C3×D4—21no—
269C3×D4—61no—
270C3×D4—61no—
271C3×D4—61no—
272C3×D4—61no—
273C3×D4—62no—
274C3×D4—121no—
275C3×D4—362no—
276C2×C3×C4—21no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—91no—
283C2×C3×C4—121no—
284C2×C3×C4—181no—
285C2×C3×C4—181no—
286C2×C3×C4—181no—
287C2×C3×C4—181no—
288C2×C3×C4C22×C3×D311yescharacteristic
289#67—122no—
290#67—362no—
291C2×Dic3—31no—
292C2×Dic3—61no—
293C2×Dic3—61no—
294C2×Dic3—91no—
295C2×Dic3—181no—
296C2×Dic3—181no—
297C4×D3—61no—
298C4×D3—61no—
299C4×D3—181no—
300C4×D3—181no—
301C3×C8—32no—
302C3×C8—32no—
303C3×C8—62no—
304C3×C8—181no—
305C3×C8C3×#6712yes—
306C3⋊C8—61no—
307C3⋊C8—181no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—82no—
312C2×C32—81no—
313C2×C32—81no—
314C2×C32D3×SD1611yescharacteristic
315C2×C32#91611yescharacteristic
316C2×C32#93811yescharacteristic
317C2×C32C3×#12111yescharacteristic
318C2×C32C2×C3×SD1611yescharacteristic
319C2×C32C3×#13511yescharacteristic
320C3⋊D3—22no—
321C3×D3—62no—
322C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
323C2×D4—181no—
324C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
325C22×C4—91no—
326SD16—22no—
327C2×C8—91no—
328C2×C8—182no—
329C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
330C4⋊C4—91no—
331C4⋊C4—91no—
332C4⋊C4—181no—
333C22⋊C4—181no—
334C22⋊C4—181no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—61no—
341C22×C3—92no—
342C22×C3—121no—
343C22×C3—121no—
344C22×C3—121no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—121no—
348C22×C3—122no—
349C22×C3—181no—
350C22×C3—181no—
351C22×C3—241no—
352C22×C3—241no—
353C22×C3C3×D3×D411yescharacteristic
354C2×D3—32no—
355C2×D3—61no—
356C2×D3—61no—
357C2×D3—92no—
358C2×D3—181no—
359C2×D3—181no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—31no—
364C3×C4—41no—
365C3×C4—61no—
366C3×C4—61no—
367C3×C4—121no—
368C3×C4—121no—
369C3×C4—122no—
370C3×C4—181no—
371C3×C4—361no—
372C3×C4—361no—
373C3×C4C3×#23711yescharacteristic
374C3×C4C2×C3×#6711yescharacteristic
375Dic3—61no—
376Dic3—121no—
377Dic3—121no—
378Dic3—181no—
379Dic3—361no—
380Dic3—361no—
381C32—21no—
382C32#517111yescharacteristic, complemented
383C32C3×#46511yescharacteristic, complemented
384C23—21no—
385C23—91no—
386Q8—21no—
387Q8—21no—
388D4—21no—
389D4—21no—
390D4—362no—
391C2×C4—21no—
392C2×C4—91no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4—181no—
397C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
398C8—181no—
399C8#638⋊C312yescomplemented
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—61no—
407C2×C3—61no—
408C2×C3—61no—
409C2×C3—81no—
410C2×C3—182no—
411C2×C3—241no—
412C2×C3—241no—
413C2×C3—242no—
414C2×C3C3×D3×SD1611yescharacteristic
415C2×C3C3×#91611yescharacteristic
416C2×C3C3×#93811yescharacteristic
417D3—62no—
418D3—182no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—92no—
423C22—181no—
424C22—181no—
425C22D4×(C3⋊D3)⋊C311yescharacteristic
426C4—41no—
427C4—181no—
428C4—361no—
429C4—361no—
430C4#3830⋊C311yescharacteristic
431C4C2×#638⋊C311yescharacteristic
432C3—31no—
433C3—31no—
434C3—61no—
435C3C3×#517111yescharacteristic
436C2—81no—
437C2—182no—
438C2SD16×(C3⋊D3)⋊C311yescharacteristic
439C2#64125⋊C311yescharacteristic
440C2#64147⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×SD16×H(3,𝔽3)C2
socleC22×C3C3×D3×D4
prime radical#64147⋊C3C2
hypercenterC2×SD16(C3⋊D3)⋊C3
hypocenterC32C3×#465
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×SD16×H(3,𝔽3)C2semidirect
2. #64155⋊C3C2semidirect
4. C2×C8×(C3⋊D3)⋊C3C2semidirect
5. #63662⋊C3C2semidirect
6. #63659⋊C3C2semidirect
7. #63644⋊C3C2semidirect
8. #10592870C3semidirect
24. C2×C32×SD16D3semidirect
25. C2×C32×SD16C2×C3semidirect
26. #64155C2×C3semidirect
28. C2×C8×C3⋊D3C2×C3semidirect
32. #63662C2×C3semidirect
33. #63659C2×C3semidirect
34. #63644C2×C3semidirect
52. C8×H(3,𝔽3)D4semidirect
93. C2×(C3⋊D3)⋊C3SD16semidirect
158. C32×C8#67semidirect
159. C32×C8C3×D4semidirect
225. C2×C3⋊D3C3×SD16semidirect
244. C2×SD16(C3⋊D3)⋊C3semidirect
251. H(3,𝔽3)#465semidirect
382. C32#5171semidirect
383. C32C3×#465semidirect
399. C8#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×H(3,𝔽3)2
#64155⋊C32
#64147⋊C32
C2×C8×(C3⋊D3)⋊C32
#63662⋊C32
#63659⋊C32
#63644⋊C32
#105928703
C3×#51713