Finite Group Databasebeta

← #10592747⋊C3 #13142878

Subgroups — #10592747⋊C3 #13142878

Order 1728.

508 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 106 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)×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic, complemented
3#64139⋊C3C211yesmaximal, characteristic
4C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#63636⋊C3C211yesmaximal, characteristic, complemented
7#63635⋊C3C211yesmaximal, characteristic, complemented
8#10592747C311yesmaximal, characteristic, complemented
9C3×#5048—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—21no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)—21no—
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16C22⋊C4×H(3,𝔽3)—21no—
17#3755⋊C3—22no—
18#3752⋊C3—21no—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3—22no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23C32×C22⋊Q8—21no—
24C32×C22⋊Q8D311yescharacteristic, complemented
25C32×C22⋊Q8C2×C311yescharacteristic, complemented
26C2×#3755C2×C311yescharacteristic, complemented
27#64139C2×C311yescharacteristic
28C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
29C2×C3×#215—31no—
30C3×#930—31no—
31C22×C3×C3⋊C8—31no—
32#63662C2×C311yescharacteristic, complemented
33#63636C2×C311yescharacteristic, complemented
34#63635C2×C311yescharacteristic, complemented
35C3×#835—31no—
36C3×#809—31no—
37C3×#808—31no—
38C23×H(3,𝔽3)D411yescharacteristic
39C22×(C3⋊D3)⋊C3—21no—
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)D411yescharacteristic
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48#638⋊C3—42no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54C3×#465—91no—
55#5048—31no—
56#5048—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C22×C311yescharacteristic
60C22×C32×C4—21no—
61C22×C32×C4C2×D311yescharacteristic
62C22×C32×C4C22×C311yescharacteristic
63C2×#638—21no—
64C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
65C2×C3×#67—61no—
66C2×C3×D12—31no—
67C32×C4⋊C4—22no—
68C32×C4⋊C4—21no—
69C32×C4⋊C4—21no—
70C32×C4⋊C4—21no—
71C32×C4⋊C4C2×D311yescharacteristic
72C32×C4⋊C4C22×C311yescharacteristic
73C32×C22⋊C4—22no—
74C32×C22⋊C4—21no—
75C32×C22⋊C4—21no—
76#3755—22no—
77#3752—21no—
78C4⋊(C3⋊Dic3)C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)—22no—
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
82C3×#215—62no—
83C3×(C2×D3)⋊C4—61no—
84C3×C4⋊Dic3—31no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—62no—
88C22×H(3,𝔽3)—21no—
89C22×H(3,𝔽3)—21no—
90C22×H(3,𝔽3)SD1612yescomplemented
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)—41no—
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—
102C22×C3×C8—91no—
103C3×C22⋊Q8—31no—
104C3×C22⋊Q8—31no—
105C3×C22⋊Q8—61no—
106C3×C22⋊Q8C3×D311yescharacteristic
107C3×#121—91no—
108C2×#215—31no—
109C2×#215—91no—
110#930—31no—
111#930—91no—
112C22×C3⋊C8—31no—
113C22×C3⋊C8—91no—
114C3×#106—91no—
115C3×#103—91no—
116C3×#102—91no—
117#835—31no—
118#835—91no—
119#809—31no—
120#809—91no—
121#808—31no—
122#808—91no—
123C23×C32—21no—
124C23×C32#6711yescharacteristic
125C23×C32C3×D411yescharacteristic
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—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—41no—
147C2×C32×C4—41no—
148C2×C32×C4#6711yescharacteristic
149C2×C32×C4C3×D411yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C23×C311yescharacteristic
152#638—42no—
153C2×C3⋊Dic3—21no—
154C4⋊(C3⋊D3)—21no—
155C4⋊(C3⋊D3)—21no—
156C3×#67—122no—
157C2×C3×Dic3—61no—
158C3×D12—61no—
159C3×D12—61no—
160C3⋊(C3⋊C8)—21no—
161C3⋊(C3⋊C8)C3×D412yescomplemented
162C3×C3⋊C8—32no—
163C3×C3⋊C8—61no—
164#465—91noSylow
165C2×H(3,𝔽3)—22no—
166C2×H(3,𝔽3)#12111yescharacteristic
167C2×H(3,𝔽3)C2×SD1611yescharacteristic
168C2×H(3,𝔽3)#13511yescharacteristic
169(C3⋊D3)⋊C3—81no—
170C2×C3×Q8—31no—
171C2×C3×Q8—31no—
172C2×C3×Q8—61no—
173C2×C3×Q8C2×C3×D311yescharacteristic
174C2×C3×D4—91no—
175C2×C3×D4—181no—
176C22×C3×C4—31no—
177C22×C3×C4—31no—
178C22×C3×C4—61no—
179C22×C3×C4C2×C3×D311yescharacteristic
180C2×#67—61no—
181C2×#67—181no—
182C2×D12—31no—
183C2×D12—91no—
184C3×SD16—182no—
185C2×C3×C8—91no—
186C2×C3×C8—91no—
187C2×C3×C8—182no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—31no—
190C3×C4⋊C4—31no—
191C3×C4⋊C4—61no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4—62no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—91no—
196C3×C4⋊C4C2×C3×D311yescharacteristic
197C3×C22⋊C4—21no—
198C3×C22⋊C4—61no—
199C3×C22⋊C4—61no—
200C3×C22⋊C4—62no—
201C3×C22⋊C4—181no—
202#215—62no—
203#215—182no—
204(C2×D3)⋊C4—61no—
205(C2×D3)⋊C4—181no—
206C4⋊Dic3—31no—
207C4⋊Dic3—91no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—62no—
211C2×C3⋊C8—91no—
212C2×C3⋊C8—91no—
213C2×C3⋊C8—182no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—21no—
218C22×C32—22no—
219C22×C32—21no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32#21512yescomplemented
223C22×C32C3×SD1612yescomplemented
224C22×C32C2×#6711yescharacteristic
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—21no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—41no—
237C32×C4—42no—
238C32×C4—42no—
239C32×C4—41no—
240C32×C4—41no—
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—
250C22×C8—91no—
251C22⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
252#121—91no—
253#106—91no—
254#103—91no—
255#102—91no—
256H(3,𝔽3)#46511yescharacteristic, complemented, Sylow
257C23×C3—31no—
258C23×C3—31no—
259C23×C3—61no—
260C23×C3—181no—
261C23×C3C3×#6711yescharacteristic
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—181no—
273C3×D4—181no—
274C3×D4—362no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—31no—
280C2×C3×C4—31no—
281C2×C3×C4—31no—
282C2×C3×C4—31no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—61no—
288C2×C3×C4—61no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—121no—
296C2×C3×C4—121no—
297C2×C3×C4—181no—
298C2×C3×C4C3×#6711yescharacteristic
299C2×C3×C4C22×C3×D311yescharacteristic
300#67—122no—
301#67—362no—
302C2×Dic3—61no—
303C2×Dic3—181no—
304D12—61no—
305D12—61no—
306D12—181no—
307D12—181no—
308C3×C8—92no—
309C3×C8—181no—
310C3⋊C8—32no—
311C3⋊C8—61no—
312C3⋊C8—92no—
313C3⋊C8—181no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—42no—
320C2×C32#93911yescharacteristic
321C2×C32C2×#21511yescharacteristic
322C2×C32#95011yescharacteristic
323C2×C32C3×#12111yescharacteristic
324C2×C32C2×C3×SD1611yescharacteristic
325C2×C32C3×#13511yescharacteristic
326C3⋊D3—81no—
327C3×D3—241no—
328C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
329C2×D4—91no—
330C2×D4—181no—
331C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
332SD16—182no—
333C2×C8—91no—
334C2×C8—91no—
335C2×C8—182no—
336C4⋊C4—21no—
337C4⋊C4—91no—
338C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—181no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—31no—
344C22×C3—31no—
345C22×C3—32no—
346C22×C3—32no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—61no—
350C22×C3—61no—
351C22×C3—61no—
352C22×C3—62no—
353C22×C3—121no—
354C22×C3—121no—
355C22×C3—361no—
356C22×C3—361no—
357C22×C3—361no—
358C22×C3C3×#21512yes—
359C22×C3C2×C3×#6711yescharacteristic
360C2×D3—121no—
361C2×D3—121no—
362C2×D3—121no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—361no—
366C3×C4—21no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—31no—
371C3×C4—41no—
372C3×C4—41no—
373C3×C4—61no—
374C3×C4—61no—
375C3×C4—61no—
376C3×C4—61no—
377C3×C4—121no—
378C3×C4—121no—
379C3×C4—121no—
380C3×C4—121no—
381C3×C4—121no—
382C3×C4—122no—
383C3×C4—122no—
384C3×C4—361no—
385C3×C4C3×D3×D411yescharacteristic
386C3×C4C3×#23711yescharacteristic
387Dic3—121no—
388Dic3—361no—
389C32—21no—
390C32#504811yescharacteristic, complemented
391C32C3×#46511yescharacteristic, complemented
392C23—181no—
393C23#638⋊C311yescharacteristic
394Q8—21no—
395Q8—21no—
396D4—181no—
397D4—181no—
398D4—362no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—181no—
404C2×C4#638⋊C311yescharacteristic
405C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
406C8—92no—
407C8—181no—
408C2×C3—22no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—31no—
415C2×C3—62no—
416C2×C3—62no—
417C2×C3—61no—
418C2×C3—61no—
419C2×C3—61no—
420C2×C3—122no—
421C2×C3—721no—
422C2×C3C3×#93911yescharacteristic
423C2×C3C2×C3×#21511yescharacteristic
424C2×C3C3×#95011yescharacteristic
425D3—241no—
426D3—721no—
427C22—21no—
428C22—21no—
429C22—361no—
430C22—361no—
431C22—361no—
432C22#3755⋊C312yescomplemented
433C22C2×#638⋊C311yescharacteristic
434C4—21no—
435C4—41no—
436C4—41no—
437C4—361no—
438C4D4×(C3⋊D3)⋊C311yescharacteristic
439C4#3830⋊C311yescharacteristic
440C3—31no—
441C3—31no—
442C3—61no—
443C3C3×#504811yescharacteristic
444C2—22no—
445C2—721no—
446C2#64148⋊C311yescharacteristic
447C2C2×#3755⋊C311yescharacteristic
448C2#64159⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊Q8C2
socleC22×C3C2×C3×#67
prime radical#64139⋊C3C2
hypercenterC22⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#465
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C22⋊Q8C2semidirect
2. C2×#3755⋊C3C2semidirect
4. C22×(C3⋊(C3⋊C8))⋊C3C2semidirect
5. #63662⋊C3C2semidirect
6. #63636⋊C3C2semidirect
7. #63635⋊C3C2semidirect
8. #10592747C3semidirect
24. C32×C22⋊Q8D3semidirect
25. C32×C22⋊Q8C2×C3semidirect
26. C2×#3755C2×C3semidirect
28. C22×C3⋊(C3⋊C8)C2×C3semidirect
32. #63662C2×C3semidirect
33. #63636C2×C3semidirect
34. #63635C2×C3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
90. C22×H(3,𝔽3)SD16semidirect
161. C3⋊(C3⋊C8)C3×D4semidirect
222. C22×C32#215semidirect
223. C22×C32C3×SD16semidirect
251. C22⋊Q8(C3⋊D3)⋊C3semidirect
256. H(3,𝔽3)#465semidirect
390. C32#5048semidirect
391. C32C3×#465semidirect
432. C22#3755⋊C3semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C22⋊Q82
C2×#3755⋊C32
#64139⋊C32
C22×(C3⋊(C3⋊C8))⋊C32
#63662⋊C32
#63636⋊C32
#63635⋊C32
#105927473
C3×#50483