Finite Group Databasebeta

← #10592715⋊C3 #13142750

Subgroups — #10592715⋊C3 #13142750

Order 1728.

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

normal subgroups only

This group has 528 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
1C4×D4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic
3C2×#3754⋊C3C211yesmaximal, characteristic, complemented
4#64084⋊C3C211yesmaximal, characteristic, complemented
5#63636⋊C3C211yesmaximal, characteristic, complemented
6#63637⋊C3C211yesmaximal, characteristic, complemented
7C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
8#10592715C311yesmaximal, characteristic, complemented
9C3×#5016—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C22⋊C4×H(3,𝔽3)—21no—
16C42×H(3,𝔽3)C2211yescharacteristic
17#3753⋊C3—22no—
18#3754⋊C3—22no—
19#3752⋊C3—21no—
20#3752⋊C3—21no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23C32×C4×D4—21no—
24C32×C4×D4D311yescharacteristic, complemented
25C32×C4×D4C2×C311yescharacteristic, complemented
26C2×#3754C2×C311yescharacteristic, complemented
27C2×#3753C2×C311yescharacteristic
28#64084C2×C311yescharacteristic, complemented
29C2×C3×#214—31no—
30C2×C3×#213—31no—
31C3×#875—31no—
32#63637C2×C311yescharacteristic, complemented
33#63636C2×C311yescharacteristic, complemented
34C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
35C3×#810—31no—
36C3×#809—31no—
37C3×C4⋊(C3⋊C8)—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)D412yes—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—42no—
44C2×C4×H(3,𝔽3)D411yescharacteristic
45C2×C4×H(3,𝔽3)D411yescharacteristic
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×(C3⋊Dic3)⋊C3—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54C3×#463—91no—
55#5016—31no—
56#5016—91no—
57C2×C32×D4—21no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C22×C311yescharacteristic
60C22×C32×C4—22no—
61C22×C32×C4—21no—
62C22×C32×C4—21no—
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C2×#634C22×C311yescharacteristic
65C2×C3×D12—31no—
66C2×C3×Dic6—31no—
67C32×C4⋊C4—21no—
68C32×C4⋊C4C2×D311yescharacteristic
69C32×C4⋊C4C22×C311yescharacteristic
70C32×C22⋊C4—22no—
71C32×C22⋊C4—21no—
72C32×C22⋊C4—21no—
73C32×C42—21no—
74C32×C42C2×D311yescharacteristic
75C32×C42C22×C311yescharacteristic
76#3754—22no—
77#3753—22no—
78#3752—21no—
79#3752—21no—
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
82C3×#214—62no—
83C3×#213—62no—
84C3×(C2×D3)⋊C4—61no—
85C3×(C2×D3)⋊C4—61no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—31no—
88C22×H(3,𝔽3)—22no—
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)—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—91no—
104C3×C4×D4—31no—
105C3×C4×D4—31no—
106C3×C4×D4—61no—
107C3×C4×D4C3×D311yescharacteristic
108C2×#214—31no—
109C2×#214—91no—
110C2×#213—31no—
111C2×#213—91no—
112#875—31no—
113#875—91no—
114C3×C4⋊C8—91no—
115C3×#103—91no—
116C3×#102—91no—
117#810—31no—
118#810—91no—
119#809—31no—
120#809—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×D4—22no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—22no—
132C32×D4D1212yes—
133C32×D4C3×D412yes—
134C2×C32×C4—22no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—44no—
141C2×C32×C4—42no—
142C2×C32×C4—42no—
143C2×C32×C4#6711yescharacteristic
144C2×C32×C4#6711yescharacteristic
145C2×C32×C4C3×D411yescharacteristic
146C2×C32×C4C3×D411yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C23×C311yescharacteristic
149C2×C3⋊Dic3—21no—
150C4⋊(C3⋊D3)—21no—
151C4⋊(C3⋊D3)—21no—
152#634—21no—
153#634—21no—
154C2×C3×Dic3—61no—
155C3×D12—61no—
156C3×D12—61no—
157C3×Dic6—61no—
158C3×Dic6—61no—
159C3⋊(C3⋊C8)—21no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163#463—91noSylow
164C2×H(3,𝔽3)—42no—
165C2×H(3,𝔽3)#12111yescharacteristic
166C2×H(3,𝔽3)#13511yescharacteristic
167C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
168(C3⋊D3)⋊C3—81no—
169C2×C3×Q8—91no—
170C2×C3×D4—31no—
171C2×C3×D4—31no—
172C2×C3×D4—61no—
173C2×C3×D4—91no—
174C2×C3×D4C2×C3×D311yescharacteristic
175C22×C3×C4—21no—
176C22×C3×C4—61no—
177C22×C3×C4—61no—
178C22×C3×C4—62no—
179C2×D12—31no—
180C2×D12—91no—
181C2×Dic6—31no—
182C2×Dic6—91no—
183C3×SD16—182no—
184C3×D8—182no—
185C2×C3×C8—91no—
186C2×C3×C8—91no—
187C3×C4⋊C4—31no—
188C3×C4⋊C4—31no—
189C3×C4⋊C4—61no—
190C3×C4⋊C4C2×C3×D311yescharacteristic
191C3×C22⋊C4—21no—
192C3×C22⋊C4—61no—
193C3×C22⋊C4—61no—
194C3×C22⋊C4—62no—
195C3×C22⋊C4—181no—
196C3×C22⋊C4—181no—
197C3×C42—31no—
198C3×C42—31no—
199C3×C42—61no—
200C3×C42C2×C3×D311yescharacteristic
201#214—62no—
202#214—182no—
203#213—62no—
204#213—182no—
205(C2×D3)⋊C4—61no—
206(C2×D3)⋊C4—61no—
207(C2×D3)⋊C4—181no—
208(C2×D3)⋊C4—181no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—91no—
212C2×C3⋊C8—91no—
213C22×C32—24no—
214C22×C32—22no—
215C22×C32—22no—
216C22×C32—21no—
217C22×C32—42no—
218C22×C32—42no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32C2×#6711yescharacteristic
224C22×C32C2×C3×D411yescharacteristic
225C2×C3⋊D3—41no—
226C2×C3⋊D3—41no—
227C2×C3⋊D3—41no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C32×C4—22no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—41no—
239C32×C4—42no—
240C32×C4—41no—
241C32×C4—41no—
242C32×C4C2×D1211yescharacteristic
243C32×C4#23511yescharacteristic
244C32×C4C2×C3×D411yescharacteristic
245C32×C4C3×#4111yescharacteristic
246C3⋊Dic3—41no—
247C3×Dic3—121no—
248C2×SD16—91no—
249C2×D8—91no—
250#124—91no—
251C4×D4(C3⋊D3)⋊C311yescharacteristic, complemented
252C4⋊C8—91no—
253#103—91no—
254#102—91no—
255H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
256C23×C3—21no—
257C23×C3—61no—
258C23×C3—61no—
259C23×C3—62no—
260C23×C3—181no—
261C22×D3—61no—
262C22×D3—181no—
263C3×Q8—181no—
264C3×Q8—181no—
265C3×D4—21no—
266C3×D4—32no—
267C3×D4—32no—
268C3×D4—61no—
269C3×D4—61no—
270C3×D4—62no—
271C3×D4—62no—
272C3×D4—181no—
273C3×D4—181no—
274C3×D4C3×D1212yes—
275C2×C3×C4—21no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—31no—
280C2×C3×C4—31no—
281C2×C3×C4—31no—
282C2×C3×C4—42no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—62no—
286C2×C3×C4—61no—
287C2×C3×C4—61no—
288C2×C3×C4—61no—
289C2×C3×C4—122no—
290C2×C3×C4—122no—
291C2×C3×C4—124no—
292C2×C3×C4—181no—
293C2×C3×C4C3×#6711yescharacteristic
294C2×C3×C4C3×#6711yescharacteristic
295C2×C3×C4C22×C3×D311yescharacteristic
296C2×Dic3—61no—
297C2×Dic3—181no—
298D12—61no—
299D12—61no—
300D12—181no—
301D12—181no—
302Dic6—61no—
303Dic6—61no—
304Dic6—181no—
305Dic6—181no—
306C3×C8—181no—
307C3×C8—181no—
308C3⋊C8—61no—
309C3⋊C8—61no—
310C3⋊C8—181no—
311C3⋊C8—181no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—44no—
316C2×C32—42no—
317C2×C32—42no—
318C2×C32#93011yescharacteristic
319C2×C32#93211yescharacteristic
320C2×C32#95011yescharacteristic
321C2×C32C3×#12111yescharacteristic
322C2×C32C3×#13511yescharacteristic
323C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
324C3⋊D3—81no—
325C3×D3—241no—
326C2×Q8—91no—
327C2×D4—91no—
328C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
329C22×C4—21no—
330SD16—182no—
331D8—182no—
332C2×C8—91no—
333C2×C8—91no—
334C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
335C22⋊C4—21no—
336C22⋊C4—181no—
337C22⋊C4—181no—
338C42C2×(C3⋊D3)⋊C311yescharacteristic
339C22×C3—22no—
340C22×C3—31no—
341C22×C3—31no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—62no—
345C22×C3—62no—
346C22×C3—64no—
347C22×C3—61no—
348C22×C3—121no—
349C22×C3—121no—
350C22×C3—121no—
351C22×C3—121no—
352C22×C3—122no—
353C22×C3—122no—
354C22×C3—361no—
355C22×C3—361no—
356C22×C3—361no—
357C22×C3C2×C3×#6711yescharacteristic
358C2×D3—121no—
359C2×D3—121no—
360C2×D3—121no—
361C2×D3—361no—
362C2×D3—361no—
363C2×D3—361no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—41no—
371C3×C4—61no—
372C3×C4—61no—
373C3×C4—61no—
374C3×C4—61no—
375C3×C4—62no—
376C3×C4—61no—
377C3×C4—61no—
378C3×C4—121no—
379C3×C4—121no—
380C3×C4—121no—
381C3×C4—122no—
382C3×C4—361no—
383C3×C4C2×C3×D1211yescharacteristic
384C3×C4C3×#23511yescharacteristic
385Dic3—121no—
386Dic3—361no—
387C32—21no—
388C32#501611yescharacteristic, complemented
389C32C3×#46311yescharacteristic, complemented
390C23—21no—
391C23—181no—
392Q8—181no—
393Q8—181no—
394D4—21no—
395D4—181no—
396D4—181no—
397D4C4⋊((C3⋊D3)⋊C3)12yes—
398C2×C4—21no—
399C2×C4—42no—
400C2×C4—181no—
401C2×C4#638⋊C311yescharacteristic
402C2×C4#638⋊C311yescharacteristic
403C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
404C8—181no—
405C8—181no—
406C2×C3—31no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—42no—
413C2×C3—61no—
414C2×C3—61no—
415C2×C3—61no—
416C2×C3—122no—
417C2×C3—122no—
418C2×C3—124no—
419C2×C3—721no—
420C2×C3C3×#93011yescharacteristic
421C2×C3C3×#93211yescharacteristic
422C2×C3C3×#95011yescharacteristic
423D3—241no—
424D3—721no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—361no—
429C22—361no—
430C22—361no—
431C22C2×#638⋊C311yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—361no—
436C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
437C4#3828⋊C311yescharacteristic
438C3—31no—
439C3—31no—
440C3—61no—
441C3C3×#501611yescharacteristic
442C2—42no—
443C2—721no—
444C2#64139⋊C311yescharacteristic
445C2#64141⋊C311yescharacteristic
446C2#64159⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×D4×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radicalC2×#3753⋊C3C2
hypercenterC4×D4(C3⋊D3)⋊C3
hypocenterC32C3×#463
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C4×D4×H(3,𝔽3)C2semidirect
3. C2×#3754⋊C3C2semidirect
4. #64084⋊C3C2semidirect
5. #63636⋊C3C2semidirect
6. #63637⋊C3C2semidirect
7. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
8. #10592715C3semidirect
24. C32×C4×D4D3semidirect
25. C32×C4×D4C2×C3semidirect
26. C2×#3754C2×C3semidirect
28. #64084C2×C3semidirect
32. #63637C2×C3semidirect
33. #63636C2×C3semidirect
34. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
251. C4×D4(C3⋊D3)⋊C3semidirect
255. H(3,𝔽3)#463semidirect
388. C32#5016semidirect
389. C32C3×#463semidirect

Maximal subgroups

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

subgroupindex
C4×D4×H(3,𝔽3)2
C2×#3753⋊C32
C2×#3754⋊C32
#64084⋊C32
#63636⋊C32
#63637⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105927153
C3×#50163