Finite Group Databasebeta

← #10592862⋊C3 #13143364

Subgroups — #10592862⋊C3 #13143364

Order 1728.

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

normal subgroups only

This group has 516 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
2Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3C2×#3755⋊C3C211yesmaximal, characteristic, complemented
4#64149⋊C3C211yesmaximal, characteristic
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic, complemented
7#63641⋊C3C211yesmaximal, characteristic, complemented
8#10592862C311yesmaximal, characteristic, complemented
9C3×#5163—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—21no—
13C2×#638⋊C3—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15SD16×H(3,𝔽3)—22no—
16C2×C8×H(3,𝔽3)C2211yescharacteristic
17#3755⋊C3—22no—
18C4⋊((C3⋊Dic3)⋊C3)—21no—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C4×(C3⋊Dic3)⋊C3—21no—
21C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23C2×C32×SD16—21no—
24C2×C32×SD16D311yescharacteristic, complemented
25C2×C32×SD16C2×C311yescharacteristic, complemented
26Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
27C2×#3755C2×C311yescharacteristic, complemented
28#64149C2×C311yescharacteristic
29C3×Q8×Dic3—31no—
30C2×C3×#215—31no—
31C3×#940—31no—
32#63659C2×C311yescharacteristic, complemented
33#63648C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35C3×#832—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)D412yescomplemented
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—
47#638⋊C3—42no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3D411yescharacteristic
50C2×(C3⋊Dic3)⋊C3D411yescharacteristic
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53C8×H(3,𝔽3)—21no—
54(C3⋊(C3⋊C8))⋊C3—21no—
55C3×#460—91no—
56#5163—31no—
57#5163—91no—
58C2×C32×Q8—21no—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C32×Q8C22×C311yescharacteristic
61C2×C32×D4—21no—
62C2×C32×D4C2×D311yescharacteristic
63C2×C32×D4C22×C311yescharacteristic
64C2×#638—21no—
65C2×#638—21no—
66C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
67C2×C3×#67—61no—
68C2×C3×#67—61no—
69C2×C3×D12—31no—
70C32×SD16—24no—
71C32×SD16—22no—
72C32×SD16—22no—
73C2×C32×C8—21no—
74C2×C32×C8C2×D311yescharacteristic
75C2×C32×C8C22×C311yescharacteristic
76#3755—22no—
77C4⋊(C3⋊Dic3)—21no—
78C4⋊(C3⋊Dic3)C22×C311yescharacteristic
79C4×C3⋊Dic3—21no—
80C4×C3⋊Dic3C22×C311yescharacteristic
81C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
82C3×#215—62no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—61no—
85C3×C4×Dic3—31no—
86C3×C4×Dic3—61no—
87C2×C3×C3⋊C8—31no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C4×H(3,𝔽3)—22no—
96C4×H(3,𝔽3)C2×D411yescharacteristic
97C4×H(3,𝔽3)#4111yescharacteristic
98(C3⋊Dic3)⋊C3—21no—
99(C3⋊Dic3)⋊C3—41no—
100(C3⋊Dic3)⋊C3SD1612yescomplemented
101C2×C3×SD16—31no—
102C2×C3×SD16—31no—
103C2×C3×SD16—61no—
104C2×C3×SD16—91no—
105C2×C3×SD16C3×D311yescharacteristic
106C3×#127—91no—
107C3×C4×Q8—91no—
108Q8×Dic3—31no—
109Q8×Dic3—91no—
110C2×#215—31no—
111C2×#215—91no—
112#940—31no—
113#940—91no—
114C3×C4⋊C8—91no—
115C3×#102—91no—
116C3×#102—91no—
117#832—31no—
118#832—91no—
119#821—31no—
120#821—91no—
121Dic3⋊C8—31no—
122Dic3⋊C8—91no—
123C23×C32—21no—
124C23×C32—21no—
125C23×C32—41no—
126C22×C3⋊D3—21no—
127C22×C3×D3—61no—
128C32×Q8—21no—
129C32×Q8—21no—
130C32×Q8—22no—
131C32×Q8—41no—
132C32×Q8#6712yescomplemented
133C32×Q8C3×D412yescomplemented
134C32×D4—22no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—21no—
138C32×D4—21no—
139C32×D4—41no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—41no—
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C23×C311yescharacteristic
146#638—42no—
147#638—42no—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3C3×D411yescharacteristic
150C2×C3⋊Dic3C3×D411yescharacteristic
151C4⋊(C3⋊D3)—21no—
152C4⋊(C3⋊D3)—21no—
153C3×#67—122no—
154C3×#67—122no—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—31no—
157C2×C3×Dic3—61no—
158C3×D12—61no—
159C3×D12—61no—
160C32×C8—22no—
161C32×C8—21no—
162C32×C8—21no—
163C3⋊(C3⋊C8)—21no—
164C3×C3⋊C8—61no—
165#460—91noSylow
166C2×H(3,𝔽3)—81no—
167C2×H(3,𝔽3)#12111yescharacteristic
168C2×H(3,𝔽3)C2×SD1611yescharacteristic
169C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
170(C3⋊D3)⋊C3—81no—
171C2×C3×Q8—31no—
172C2×C3×Q8—31no—
173C2×C3×Q8—61no—
174C2×C3×Q8C2×C3×D311yescharacteristic
175C2×C3×D4—31no—
176C2×C3×D4—31no—
177C2×C3×D4—61no—
178C2×C3×D4—91no—
179C2×C3×D4—181no—
180C2×C3×D4—181no—
181C2×C3×D4C2×C3×D311yescharacteristic
182C2×#67—61no—
183C2×#67—61no—
184C2×#67—181no—
185C2×#67—181no—
186C2×D12—31no—
187C2×D12—91no—
188C3×SD16—22no—
189C3×SD16—62no—
190C3×SD16—62no—
191C3×SD16—64no—
192C3×SD16—182no—
193C2×C3×C8—31no—
194C2×C3×C8—31no—
195C2×C3×C8—61no—
196C2×C3×C8—91no—
197C2×C3×C8C2×C3×D311yescharacteristic
198C3×C4⋊C4—91no—
199C3×C4⋊C4—181no—
200C3×C42—91no—
201C3×C42—181no—
202#215—62no—
203#215—182no—
204C4⋊Dic3—31no—
205C4⋊Dic3—61no—
206C4⋊Dic3—91no—
207C4⋊Dic3—181no—
208C4×Dic3—31no—
209C4×Dic3—61no—
210C4×Dic3—91no—
211C4×Dic3—181no—
212C2×C3⋊C8—31no—
213C2×C3⋊C8—91no—
214C22×C32—21no—
215C22×C32—42no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—81no—
223C22×C32—81no—
224C22×C32D3×D411yescharacteristic
225C22×C32C2×C3×D411yescharacteristic
226C2×C3⋊D3—41no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3—41no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C32×C4—22no—
233C32×C4—22no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—42no—
237C32×C4#23711yescharacteristic
238C32×C4C2×#6711yescharacteristic
239C32×C4C2×C3×D411yescharacteristic
240C32×C4C3×#4111yescharacteristic
241C3⋊Dic3—21no—
242C3⋊Dic3—41no—
243C3⋊Dic3C3×SD1612yescomplemented
244C3×Dic3—32no—
245C3×Dic3—61no—
246C3×Dic3—121no—
247C2×SD16—91no—
248C2×SD16(C3⋊D3)⋊C311yescharacteristic, complemented
249#127—91no—
250C4×Q8—91no—
251C4⋊C8—91no—
252#102—91no—
253#102—91no—
254H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
255C23×C3—21no—
256C23×C3—61no—
257C23×C3—61no—
258C23×C3—121no—
259C23×C3—181no—
260C22×D3—61no—
261C22×D3—181no—
262C3×Q8—21no—
263C3×Q8—32no—
264C3×Q8—32no—
265C3×Q8—61no—
266C3×Q8—61no—
267C3×Q8—62no—
268C3×Q8—121no—
269C3×Q8C3×#6712yes—
270C3×D4—21no—
271C3×D4—21no—
272C3×D4—61no—
273C3×D4—61no—
274C3×D4—61no—
275C3×D4—61no—
276C3×D4—62no—
277C3×D4—121no—
278C3×D4—181no—
279C3×D4—181no—
280C3×D4—362no—
281C3×D4—362no—
282C2×C3×C4—21no—
283C2×C3×C4—31no—
284C2×C3×C4—31no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—61no—
288C2×C3×C4—91no—
289C2×C3×C4—91no—
290C2×C3×C4—121no—
291C2×C3×C4—181no—
292C2×C3×C4C22×C3×D311yescharacteristic
293#67—122no—
294#67—122no—
295#67—362no—
296#67—362no—
297C2×Dic3—31no—
298C2×Dic3—31no—
299C2×Dic3—61no—
300C2×Dic3—91no—
301C2×Dic3—91no—
302C2×Dic3—181no—
303D12—61no—
304D12—61no—
305D12—181no—
306D12—181no—
307C3×C8—21no—
308C3×C8—61no—
309C3×C8—61no—
310C3×C8—62no—
311C3×C8—181no—
312C3⋊C8—61no—
313C3⋊C8—181no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—82no—
318C2×C32—81no—
319C2×C32—81no—
320C2×C32D3×SD1611yescharacteristic
321C2×C32#91411yescharacteristic
322C2×C32#93911yescharacteristic
323C2×C32C3×#12111yescharacteristic
324C2×C32C2×C3×SD1611yescharacteristic
325C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
326C3⋊D3—81no—
327C3×D3—241no—
328C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
329C2×D4—91no—
330C2×D4—181no—
331C2×D4—181no—
332C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
333SD16—22no—
334SD16—182no—
335C2×C8—91no—
336C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
337C4⋊C4—91no—
338C4⋊C4—181no—
339C42—91no—
340C42—181no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—61no—
347C22×C3—121no—
348C22×C3—121no—
349C22×C3—121no—
350C22×C3—121no—
351C22×C3—121no—
352C22×C3—121no—
353C22×C3—122no—
354C22×C3—241no—
355C22×C3—241no—
356C22×C3—361no—
357C22×C3—361no—
358C22×C3—361no—
359C22×C3C3×D3×D411yescharacteristic
360C2×D3—121no—
361C2×D3—121no—
362C2×D3—121no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—361no—
366C3×C4—22no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—31no—
371C3×C4—62no—
372C3×C4—62no—
373C3×C4—61no—
374C3×C4—61no—
375C3×C4—92no—
376C3×C4—122no—
377C3×C4—181no—
378C3×C4—361no—
379C3×C4C3×#23711yescharacteristic
380C3×C4C2×C3×#6711yescharacteristic
381Dic3—32no—
382Dic3—61no—
383Dic3—92no—
384Dic3—121no—
385Dic3—181no—
386Dic3—361no—
387C32—21no—
388C32#516311yescharacteristic, complemented
389C32C3×#46011yescharacteristic, complemented
390C23—21no—
391C23—181no—
392Q8—21no—
393Q8#638⋊C312yescomplemented
394D4—21no—
395D4—21no—
396D4—181no—
397D4—181no—
398D4—362no—
399D4—362no—
400C2×C4—21no—
401C2×C4—91no—
402C2×C4—91no—
403C2×C4—181no—
404C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
405C8—21no—
406C8—181no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—61no—
414C2×C3—61no—
415C2×C3—61no—
416C2×C3—81no—
417C2×C3—241no—
418C2×C3—241no—
419C2×C3—242no—
420C2×C3—721no—
421C2×C3C3×D3×SD1611yescharacteristic
422C2×C3C3×#91411yescharacteristic
423C2×C3C3×#93911yescharacteristic
424D3—241no—
425D3—721no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—361no—
430C22—361no—
431C22—361no—
432C22D4×(C3⋊D3)⋊C311yescharacteristic
433C4—22no—
434C4—92no—
435C4—181no—
436C4—361no—
437C4#3830⋊C311yescharacteristic
438C4C2×#638⋊C311yescharacteristic
439C3—31no—
440C3—31no—
441C3—61no—
442C3C3×#516311yescharacteristic
443C2—81no—
444C2—721no—
445C2SD16×(C3⋊D3)⋊C311yescharacteristic
446C2#64123⋊C311yescharacteristic
447C2#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 radical#64149⋊C3C2
hypercenterC2×SD16(C3⋊D3)⋊C3
hypocenterC32C3×#460
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. Q8×(C3⋊Dic3)⋊C3C2semidirect
3. C2×#3755⋊C3C2semidirect
5. #63659⋊C3C2semidirect
6. #63648⋊C3C2semidirect
7. #63641⋊C3C2semidirect
8. #10592862C3semidirect
24. C2×C32×SD16D3semidirect
25. C2×C32×SD16C2×C3semidirect
26. Q8×C3⋊Dic3C2×C3semidirect
27. C2×#3755C2×C3semidirect
32. #63659C2×C3semidirect
33. #63648C2×C3semidirect
34. #63641C2×C3semidirect
41. Q8×H(3,𝔽3)D4semidirect
100. (C3⋊Dic3)⋊C3SD16semidirect
132. C32×Q8#67semidirect
133. C32×Q8C3×D4semidirect
243. C3⋊Dic3C3×SD16semidirect
248. C2×SD16(C3⋊D3)⋊C3semidirect
254. H(3,𝔽3)#460semidirect
388. C32#5163semidirect
389. C32C3×#460semidirect
393. Q8#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
Q8×(C3⋊Dic3)⋊C32
C2×#3755⋊C32
#64149⋊C32
#63659⋊C32
#63648⋊C32
#63641⋊C32
#105928623
C3×#51633