Finite Group Databasebeta

← #10592756⋊C3 #13142924

Subgroups — #10592756⋊C3 #13142924

Order 1728.

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

normal subgroups only

This group has 510 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
1H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic, complemented
3C2×#3753⋊C3C211yesmaximal, characteristic
4C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#63659⋊C3C211yesmaximal, characteristic, complemented
7C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
8#10592756C311yesmaximal, characteristic, complemented
9C3×#5057—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3—21no—
14C22⋊C4×H(3,𝔽3)—21no—
15C22⋊C4×H(3,𝔽3)—21no—
16C42×H(3,𝔽3)C2211yescharacteristic
17#3755⋊C3—22no—
18#3753⋊C3—22no—
19#3752⋊C3—21no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23C32×#124—21no—
24C32×#124D311yescharacteristic, complemented
25C32×#124C2×C311yescharacteristic, complemented
26C2×#3755C2×C311yescharacteristic, complemented
27C2×#3753C2×C311yescharacteristic
28C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
29C2×C3×#215—31no—
30C2×C3×#213—31no—
31C3×C4×D12—31no—
32#63662C2×C311yescharacteristic, complemented
33#63659C2×C311yescharacteristic, complemented
34C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
35C3×#835—31no—
36C3×#832—31no—
37C3×C4⋊(C3⋊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)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)D411yescharacteristic
46C2×C4×H(3,𝔽3)D411yescharacteristic
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×(C3⋊Dic3)⋊C3—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)D412yes—
51C4×(C3⋊D3)⋊C3—42no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54C3×#463—91no—
55#5057—31no—
56#5057—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×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C2×C4×C3⋊D3—21no—
65C2×C3×D12—31no—
66C2×C3×C4×D3—61no—
67C32×C22⋊C4—22no—
68C32×C22⋊C4—22no—
69C32×C22⋊C4—21no—
70C32×C22⋊C4—21no—
71C32×C22⋊C4—21no—
72C32×C22⋊C4—21no—
73C32×C42—21no—
74C32×C42C2×D311yescharacteristic
75C32×C42C22×C311yescharacteristic
76#3755—22no—
77#3753—22no—
78#3752—21no—
79C4⋊(C3⋊Dic3)C22×C311yescharacteristic
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
82C3×#215—62no—
83C3×#213—62no—
84C3×(C2×D3)⋊C4—61no—
85C3×C4⋊Dic3—31no—
86C2×C3×C3⋊C8—31no—
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—22no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C4×H(3,𝔽3)—21no—
96C4×H(3,𝔽3)—21no—
97C4×H(3,𝔽3)—41no—
98C4×H(3,𝔽3)C2×D411yescharacteristic
99C4×H(3,𝔽3)#4111yescharacteristic
100(C3⋊Dic3)⋊C3—41no—
101C2×C3×SD16—91no—
102C2×C3×D8—91no—
103C3×#124—31no—
104C3×#124—31no—
105C3×#124—61no—
106C3×#124C3×D311yescharacteristic
107C3×C4×D4—91no—
108C2×#215—31no—
109C2×#215—91no—
110C2×#213—31no—
111C2×#213—91no—
112C4×D12—31no—
113C4×D12—91no—
114C3×C4⋊C8—91no—
115C3×#103—91no—
116C3×#102—91no—
117#835—31no—
118#835—91no—
119#832—31no—
120#832—91no—
121C4⋊(C3⋊C8)—31no—
122C4⋊(C3⋊C8)—91no—
123C23×C32—22no—
124C23×C32—21no—
125C23×C32—21no—
126C22×C3⋊D3—21no—
127C22×C3×D3—61no—
128C32×Q8—21no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—41no—
133C32×Q8—41no—
134C32×D4—21no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—21no—
138C32×D4—41no—
139C32×D4—41no—
140C2×C32×C4—22no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4#6711yescharacteristic
147C2×C32×C4#6711yescharacteristic
148C2×C32×C4C3×D411yescharacteristic
149C2×C32×C4C3×D411yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C23×C311yescharacteristic
152C2×C3⋊Dic3—21no—
153C4⋊(C3⋊D3)—21no—
154C4⋊(C3⋊D3)C3×D412yes—
155C4×C3⋊D3—42no—
156C2×C3×Dic3—61no—
157C3×D12—32no—
158C3×D12—61no—
159C3×C4×D3—122no—
160C3⋊(C3⋊C8)—21no—
161C3⋊(C3⋊C8)—21no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164#463—91noSylow
165C2×H(3,𝔽3)—81no—
166C2×H(3,𝔽3)#12111yescharacteristic
167C2×H(3,𝔽3)#13511yescharacteristic
168C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
169(C3⋊D3)⋊C3—42no—
170C2×C3×Q8—31no—
171C2×C3×Q8—31no—
172C2×C3×Q8—61no—
173C2×C3×Q8C2×C3×D311yescharacteristic
174C2×C3×D4—31no—
175C2×C3×D4—31no—
176C2×C3×D4—61no—
177C2×C3×D4—91no—
178C2×C3×D4C2×C3×D311yescharacteristic
179C22×C3×C4—181no—
180C2×D12—31no—
181C2×D12—91no—
182C2×C4×D3—61no—
183C2×C4×D3—181no—
184C3×SD16—182no—
185C3×D8—182no—
186C2×C3×C8—91no—
187C2×C3×C8—91no—
188C3×C4⋊C4—91no—
189C3×C22⋊C4—21no—
190C3×C22⋊C4—21no—
191C3×C22⋊C4—61no—
192C3×C22⋊C4—61no—
193C3×C22⋊C4—61no—
194C3×C22⋊C4—61no—
195C3×C22⋊C4—62no—
196C3×C22⋊C4—62no—
197C3×C22⋊C4—181no—
198C3×C42—31no—
199C3×C42—31no—
200C3×C42—61no—
201C3×C42C2×C3×D311yescharacteristic
202#215—62no—
203#215—182no—
204#213—62no—
205#213—182no—
206(C2×D3)⋊C4—61no—
207(C2×D3)⋊C4—181no—
208C4⋊Dic3—31no—
209C4⋊Dic3—91no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—91no—
213C2×C3⋊C8—91no—
214C22×C32—21no—
215C22×C32—42no—
216C22×C32—42no—
217C22×C32—42no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32C2×#6711yescharacteristic
225C22×C32C2×C3×D411yescharacteristic
226C2×C3⋊D3—22no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3—41no—
229C2×C3×D3—62no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—41no—
239C32×C4—41no—
240C32×C4—42no—
241C32×C4—41no—
242C32×C4—41no—
243C32×C4D3×D411yescharacteristic
244C32×C4#23711yescharacteristic
245C32×C4C2×C3×D411yescharacteristic
246C32×C4C3×#4111yescharacteristic
247C3⋊Dic3—41no—
248C3×Dic3—121no—
249C2×SD16—91no—
250C2×D8—91no—
251#124(C3⋊D3)⋊C311yescharacteristic, complemented
252C4×D4—91no—
253C4⋊C8—91no—
254#103—91no—
255#102—91no—
256H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
257C23×C3—21no—
258C23×C3—61no—
259C23×C3—61no—
260C23×C3—62no—
261C23×C3—181no—
262C22×D3—61no—
263C22×D3—181no—
264C3×Q8—21no—
265C3×Q8—21no—
266C3×Q8—61no—
267C3×Q8—61no—
268C3×Q8—61no—
269C3×Q8—61no—
270C3×Q8—121no—
271C3×Q8—121no—
272C3×D4—21no—
273C3×D4—21no—
274C3×D4—61no—
275C3×D4—61no—
276C3×D4—61no—
277C3×D4—61no—
278C3×D4—92no—
279C3×D4—121no—
280C3×D4—121no—
281C3×D4—181no—
282C2×C3×C4—21no—
283C2×C3×C4—31no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4—62no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—181no—
296C2×C3×C4—362no—
297C2×C3×C4C3×#6711yescharacteristic
298C2×C3×C4C3×#6711yescharacteristic
299C2×C3×C4C22×C3×D311yescharacteristic
300C2×Dic3—61no—
301C2×Dic3—181no—
302D12—32no—
303D12—61no—
304D12—92no—
305D12—181no—
306C4×D3—122no—
307C4×D3—362no—
308C3×C8—181no—
309C3×C8—181no—
310C3⋊C8—61no—
311C3⋊C8—61no—
312C3⋊C8—181no—
313C3⋊C8—181no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—82no—
318C2×C32—81no—
319C2×C32—81no—
320C2×C32#93811yescharacteristic
321C2×C32#94911yescharacteristic
322C2×C32#95011yescharacteristic
323C2×C32C3×#12111yescharacteristic
324C2×C32C3×#13511yescharacteristic
325C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
326C3⋊D3—42no—
327C3×D3—122no—
328C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
329C2×D4—91no—
330C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
331C22×C4—181no—
332SD16—182no—
333D8—182no—
334C2×C8—91no—
335C2×C8—91no—
336C4⋊C4—91no—
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—181no—
340C42C2×(C3⋊D3)⋊C311yescharacteristic
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—122no—
355C22×C3—122no—
356C22×C3—182no—
357C22×C3—361no—
358C22×C3—361no—
359C22×C3C2×C3×#6711yescharacteristic
360C2×D3—62no—
361C2×D3—121no—
362C2×D3—121no—
363C2×D3—182no—
364C2×D3—361no—
365C2×D3—361no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—31no—
371C3×C4—31no—
372C3×C4—41no—
373C3×C4—61no—
374C3×C4—61no—
375C3×C4—61no—
376C3×C4—61no—
377C3×C4—61no—
378C3×C4—61no—
379C3×C4—121no—
380C3×C4—121no—
381C3×C4—121no—
382C3×C4—121no—
383C3×C4—122no—
384C3×C4—361no—
385C3×C4C3×D3×D411yescharacteristic
386C3×C4C3×#23711yescharacteristic
387Dic3—121no—
388Dic3—361no—
389C32—21no—
390C32#505711yescharacteristic, complemented
391C32C3×#46311yescharacteristic, complemented
392C23—21no—
393C23—181no—
394Q8—21no—
395Q8—21no—
396D4—21no—
397D4—21no—
398D4—92no—
399D4—181no—
400C2×C4—21no—
401C2×C4—181no—
402C2×C4—362no—
403C2×C4#638⋊C311yescharacteristic
404C2×C4#638⋊C311yescharacteristic
405C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
406C8—181no—
407C8—181no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—61no—
415C2×C3—61no—
416C2×C3—61no—
417C2×C3—81no—
418C2×C3—241no—
419C2×C3—241no—
420C2×C3—242no—
421C2×C3—362no—
422C2×C3C3×#93811yescharacteristic
423C2×C3C3×#94911yescharacteristic
424C2×C3C3×#95011yescharacteristic
425D3—122no—
426D3—362no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—182no—
431C22—361no—
432C22—361no—
433C22C2×#638⋊C311yescharacteristic
434C4—21no—
435C4—21no—
436C4—41no—
437C4—361no—
438C4D4×(C3⋊D3)⋊C311yescharacteristic
439C4#3830⋊C311yescharacteristic
440C3—31no—
441C3—31no—
442C3—61no—
443C3C3×#505711yescharacteristic
444C2—81no—
445C2—362no—
446C2#64147⋊C311yescharacteristic
447C2#64158⋊C311yescharacteristic
448C2#64159⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#124C2
socleC22×C3C2×C3×#67
prime radicalC2×#3753⋊C3C2
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C3×#463
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#124C2semidirect
2. C2×#3755⋊C3C2semidirect
4. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
5. #63662⋊C3C2semidirect
6. #63659⋊C3C2semidirect
7. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
8. #10592756C3semidirect
24. C32×#124D3semidirect
25. C32×#124C2×C3semidirect
26. C2×#3755C2×C3semidirect
28. C4×C4⋊(C3⋊D3)C2×C3semidirect
32. #63662C2×C3semidirect
33. #63659C2×C3semidirect
34. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
251. #124(C3⋊D3)⋊C3semidirect
256. H(3,𝔽3)#463semidirect
390. C32#5057semidirect
391. C32C3×#463semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
C2×#3755⋊C32
C2×#3753⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#63662⋊C32
#63659⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105927563
C3×#50573