Finite Group Databasebeta

← #10592864⋊C3 #13143367

Subgroups — #10592864⋊C3 #13143367

Order 1728.

502 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 96 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#64156⋊C3C211yesmaximal, characteristic, complemented
3D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4C2×#3753⋊C3C211yesmaximal, characteristic
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic, complemented
7#63641⋊C3C211yesmaximal, characteristic, complemented
8#10592864C311yesmaximal, characteristic, complemented
9C3×#5165—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3—21no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14SD16×H(3,𝔽3)—22no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16#3757⋊C3—21no—
17#3753⋊C3—22no—
18#3752⋊C3—21no—
19#3752⋊C3—21no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23C2×C32×SD16—21no—
24C2×C32×SD16D311yescharacteristic, complemented
25C2×C32×SD16C2×C311yescharacteristic, complemented
26#64156C2×C311yescharacteristic, complemented
27D4×C3⋊Dic3C2×C311yescharacteristic, complemented
28C2×#3753C2×C311yescharacteristic
29C3×#947—31no—
30C3×D4×Dic3—31no—
31C2×C3×#213—31no—
32#63662C2×C311yescharacteristic, complemented
33#63648C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35C3×#835—31no—
36C3×#821—31no—
37C3×Dic3⋊C8—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43D4×H(3,𝔽3)D412yes—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—42no—
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C2×(C3⋊Dic3)⋊C3D411yescharacteristic
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52C8×H(3,𝔽3)—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54C3×#463—91no—
55#5165—31no—
56#5165—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
63C22×C3⋊Dic3—21no—
64C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
65C22×C3×Dic3—61no—
66C2×C3×D12—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#3753—22no—
75#3752—21no—
76#3752—21no—
77C4⋊(C3⋊Dic3)C22×C311yescharacteristic
78C4×C3⋊Dic3C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C3×#217—61no—
81C3×#213—62no—
82C3×(C2×D3)⋊C4—61no—
83C3×(C2×D3)⋊C4—61no—
84C3×C4⋊Dic3—31no—
85C3×C4×Dic3—31no—
86C2×C3×C3⋊C8—31no—
87C22×H(3,𝔽3)—22no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
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)—41no—
95C4×H(3,𝔽3)C2×D411yescharacteristic
96C4×H(3,𝔽3)#4111yescharacteristic
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—21no—
99(C3⋊Dic3)⋊C3—41no—
100C2×C3×SD16—31no—
101C2×C3×SD16—31no—
102C2×C3×SD16—61no—
103C2×C3×SD16C3×D311yescharacteristic
104C2×C3×D8—91no—
105C3×#124—91no—
106C3×C4×D4—91no—
107#947—31no—
108#947—91no—
109D4×Dic3—31no—
110D4×Dic3—91no—
111C2×#213—31no—
112C2×#213—91no—
113C3×C4⋊C8—91no—
114C3×#103—91no—
115C3×#102—91no—
116#835—31no—
117#835—91no—
118#821—31no—
119#821—91no—
120Dic3⋊C8—31no—
121Dic3⋊C8—91no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—41no—
125C22×C3⋊D3—21no—
126C22×C3×D3—61no—
127C32×Q8—22no—
128C32×Q8—21no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—41no—
133C32×D4—21no—
134C32×D4—21no—
135C32×D4—22no—
136C32×D4—41no—
137C32×D4#6712yes—
138C32×D4C3×D412yes—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—41no—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C23×C311yescharacteristic
145C2×C3⋊Dic3—21no—
146C2×C3⋊Dic3—42no—
147C2×C3⋊Dic3C3×D411yescharacteristic
148C2×C3⋊Dic3C3×D411yescharacteristic
149C4⋊(C3⋊D3)—21no—
150C4⋊(C3⋊D3)—21no—
151C2×C3×Dic3—31no—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—122no—
155C3×D12—61no—
156C3×D12—61no—
157C32×C8—22no—
158C32×C8—21no—
159C32×C8—21no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—61no—
162#463—91noSylow
163C2×H(3,𝔽3)—42no—
164C2×H(3,𝔽3)#12111yescharacteristic
165C2×H(3,𝔽3)#13511yescharacteristic
166C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
167(C3⋊D3)⋊C3—81no—
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—91no—
176C2×C3×D4C2×C3×D311yescharacteristic
177C22×C3×C4—181no—
178C22×Dic3—61no—
179C22×Dic3—181no—
180C2×D12—31no—
181C2×D12—91no—
182C3×SD16—22no—
183C3×SD16—62no—
184C3×SD16—62no—
185C3×SD16—64no—
186C3×D8—182no—
187C2×C3×C8—31no—
188C2×C3×C8—31no—
189C2×C3×C8—61no—
190C2×C3×C8—91no—
191C2×C3×C8C2×C3×D311yescharacteristic
192C3×C4⋊C4—91no—
193C3×C22⋊C4—181no—
194C3×C22⋊C4—181no—
195C3×C22⋊C4—181no—
196C3×C42—91no—
197#217—61no—
198#217—181no—
199#213—62no—
200#213—182no—
201(C2×D3)⋊C4—61no—
202(C2×D3)⋊C4—61no—
203(C2×D3)⋊C4—181no—
204(C2×D3)⋊C4—181no—
205C4⋊Dic3—31no—
206C4⋊Dic3—91no—
207C4×Dic3—31no—
208C4×Dic3—91no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—91no—
211C22×C32—22no—
212C22×C32—22no—
213C22×C32—21no—
214C22×C32—42no—
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—41no—
224C2×C3⋊D3—41no—
225C2×C3⋊D3—41no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
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—21no—
240C3⋊Dic3—41no—
241C3×Dic3—61no—
242C3×Dic3—61no—
243C3×Dic3—121no—
244C2×SD16(C3⋊D3)⋊C311yescharacteristic, complemented
245C2×D8—91no—
246#124—91no—
247C4×D4—91no—
248C4⋊C8—91no—
249#103—91no—
250#102—91no—
251H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
252C23×C3—21no—
253C23×C3—61no—
254C23×C3—61no—
255C23×C3—121no—
256C23×C3—181no—
257C22×D3—61no—
258C22×D3—181no—
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—32no—
269C3×D4—32no—
270C3×D4—61no—
271C3×D4—61no—
272C3×D4—62no—
273C3×D4—121no—
274C3×D4—181no—
275C3×D4—181no—
276C3×D4C3×#6712yes—
277C2×C3×C4—21no—
278C2×C3×C4—31no—
279C2×C3×C4—31no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—91no—
284C2×C3×C4—91no—
285C2×C3×C4—121no—
286C2×C3×C4—181no—
287C2×C3×C4—362no—
288C2×C3×C4C22×C3×D311yescharacteristic
289C2×Dic3—31no—
290C2×Dic3—31no—
291C2×Dic3—61no—
292C2×Dic3—91no—
293C2×Dic3—91no—
294C2×Dic3—122no—
295C2×Dic3—181no—
296C2×Dic3—362no—
297D12—61no—
298D12—61no—
299D12—181no—
300D12—181no—
301C3×C8—21no—
302C3×C8—61no—
303C3×C8—61no—
304C3×C8—62no—
305C3×C8—181no—
306C3⋊C8—61no—
307C3⋊C8—181no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—42no—
312C2×C32—42no—
313C2×C32—82no—
314C2×C32#91411yescharacteristic
315C2×C32#91611yescharacteristic
316C2×C32#93911yescharacteristic
317C2×C32C3×#12111yescharacteristic
318C2×C32C3×#13511yescharacteristic
319C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
320C3⋊D3—81no—
321C3×D3—241no—
322C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
323C2×D4—91no—
324C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
325C22×C4—181no—
326SD16—22no—
327D8—182no—
328C2×C8—91no—
329C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
330C4⋊C4—91no—
331C22⋊C4—181no—
332C22⋊C4—181no—
333C22⋊C4—181no—
334C42—91no—
335C22×C3—22no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—62no—
341C22×C3—62no—
342C22×C3—61no—
343C22×C3—122no—
344C22×C3—121no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—121no—
348C22×C3—241no—
349C22×C3—241no—
350C22×C3—361no—
351C22×C3—361no—
352C22×C3—361no—
353C22×C3C3×D3×D411yescharacteristic
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—121no—
357C2×D3—361no—
358C2×D3—361no—
359C2×D3—361no—
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—181no—
372C3×C4—361no—
373C3×C4C3×#23711yescharacteristic
374C3×C4C2×C3×#6711yescharacteristic
375Dic3—61no—
376Dic3—61no—
377Dic3—121no—
378Dic3—181no—
379Dic3—181no—
380Dic3—361no—
381C32—21no—
382C32#516511yescharacteristic, complemented
383C32C3×#46311yescharacteristic, complemented
384C23—21no—
385C23—181no—
386Q8—21no—
387Q8—21no—
388D4—21no—
389D4—181no—
390D4—181no—
391D4#638⋊C312yes—
392C2×C4—21no—
393C2×C4—91no—
394C2×C4—91no—
395C2×C4—181no—
396C2×C4—362no—
397C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
398C8—21no—
399C8—181no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—42no—
407C2×C3—61no—
408C2×C3—61no—
409C2×C3—61no—
410C2×C3—122no—
411C2×C3—122no—
412C2×C3—242no—
413C2×C3—721no—
414C2×C3C3×#91411yescharacteristic
415C2×C3C3×#91611yescharacteristic
416C2×C3C3×#93911yescharacteristic
417D3—241no—
418D3—721no—
419C22—22no—
420C22—41no—
421C22—41no—
422C22—361no—
423C22—361no—
424C22—361no—
425C22D4×(C3⋊D3)⋊C311yescharacteristic
426C4—41no—
427C4—181no—
428C4—181no—
429C4—361no—
430C4#3830⋊C311yescharacteristic
431C4C2×#638⋊C311yescharacteristic
432C3—31no—
433C3—31no—
434C3—61no—
435C3C3×#516511yescharacteristic
436C2—42no—
437C2—721no—
438C2#64123⋊C311yescharacteristic
439C2#64125⋊C311yescharacteristic
440C2#64148⋊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 radicalC2×#3753⋊C3C2
hypercenterC2×SD16(C3⋊D3)⋊C3
hypocenterC32C3×#463
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. #64156⋊C3C2semidirect
3. D4×(C3⋊Dic3)⋊C3C2semidirect
5. #63662⋊C3C2semidirect
6. #63648⋊C3C2semidirect
7. #63641⋊C3C2semidirect
8. #10592864C3semidirect
24. C2×C32×SD16D3semidirect
25. C2×C32×SD16C2×C3semidirect
26. #64156C2×C3semidirect
27. D4×C3⋊Dic3C2×C3semidirect
32. #63662C2×C3semidirect
33. #63648C2×C3semidirect
34. #63641C2×C3semidirect
244. C2×SD16(C3⋊D3)⋊C3semidirect
251. H(3,𝔽3)#463semidirect
382. C32#5165semidirect
383. C32C3×#463semidirect

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
#64156⋊C32
D4×(C3⋊Dic3)⋊C32
C2×#3753⋊C32
#63662⋊C32
#63648⋊C32
#63641⋊C32
#105928643
C3×#51653