Finite Group Databasebeta

← #10592876⋊C3 #13143378

Subgroups — #10592876⋊C3 #13143378

Order 1728.

432 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 86 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#64151⋊C3C211yesmaximal, characteristic, complemented
3#64141⋊C3C211yesmaximal, characteristic
4#64110⋊C3C211yesmaximal, characteristic, complemented
5#63663⋊C3C211yesmaximal, characteristic, complemented
6#63660⋊C3C211yesmaximal, characteristic, complemented
7#63646⋊C3C211yesmaximal, characteristic, complemented
8#10592876C311yesmaximal, characteristic, complemented
9C3×#5177—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12#3828⋊C3C2211yescharacteristic
13SD16×H(3,𝔽3)—21no—
14SD16×H(3,𝔽3)—21no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16#3755⋊C3—21no—
17#3756⋊C3—21no—
18#3753⋊C3—21no—
19#3754⋊C3—21no—
20#3748⋊C3C2211yescharacteristic
21#3748⋊C3C2211yescharacteristic
22#3748⋊C3C2211yescharacteristic
23#3743⋊C3—21no—
24C8×(C3⋊D3)⋊C3—21no—
25C2×C32×SD16—21no—
26C2×C32×SD16D311yescharacteristic, complemented
27C2×C32×SD16C2×C311yescharacteristic, complemented
28#64151C2×C311yescharacteristic, complemented
29#64141C2×C311yescharacteristic
30#64110C2×C311yescharacteristic, complemented
31C3×#942—31no—
32C3×#932—31no—
33C3×#901—31no—
34#63663C2×C311yescharacteristic, complemented
35#63660C2×C311yescharacteristic, complemented
36#63646C2×C311yescharacteristic, complemented
37C3×#836—31no—
38C3×#833—31no—
39C3×#819—31no—
40C23×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)—21no—
42Q8×H(3,𝔽3)—21no—
43D4×H(3,𝔽3)—21no—
44D4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47#638⋊C3—21no—
48C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
49C4×(C3⋊D3)⋊C3—21no—
50#634⋊C3D411yescharacteristic
51C8×H(3,𝔽3)D411yescharacteristic
52C8×H(3,𝔽3)D411yescharacteristic
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56C3×#471—91no—
57#5177—31no—
58#5177—91no—
59C2×C32×Q8—21no—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×Q8C22×C311yescharacteristic
62C2×C32×D4—21no—
63C2×C32×D4C2×D311yescharacteristic
64C2×C32×D4C22×C311yescharacteristic
65#3828C22×C311yescharacteristic
66C3×#235—31no—
67C32×SD16—22no—
68C32×SD16—22no—
69C32×SD16—21no—
70C32×SD16—21no—
71C32×SD16—21no—
72C32×SD16—21no—
73C2×C32×C8—21no—
74C2×C32×C8C2×D311yescharacteristic
75C2×C32×C8C22×C311yescharacteristic
76#3756—21no—
77#3755—21no—
78#3754—21no—
79#3753—21no—
80#3748C22×C311yescharacteristic
81#3748C22×C311yescharacteristic
82#3748C22×C311yescharacteristic
83#3743—21no—
84C8×C3⋊D3—21no—
85C3×#216—61no—
86C3×#215—61no—
87C3×#214—61no—
88C3×#213—61no—
89C3×#208—31no—
90C3×#208—31no—
91C3×#208—31no—
92C3×ΓL(1,𝔽25)—61no—
93C3×D3×C8—61no—
94C22×H(3,𝔽3)—41no—
95C22×H(3,𝔽3)—81no—
96C22×H(3,𝔽3)#4111yescharacteristic
97C2×(C3⋊D3)⋊C3—21no—
98C4×H(3,𝔽3)—41no—
99C4×H(3,𝔽3)C2×D411yescharacteristic
100C4×H(3,𝔽3)C2×D411yescharacteristic
101(C3⋊Dic3)⋊C3—21no—
102C3×#137—91no—
103C3×AGL(1,ℤ/8)—91no—
104C2×C3×SD16—31no—
105C2×C3×SD16—31no—
106C2×C3×SD16—61no—
107C2×C3×SD16C3×D311yescharacteristic
108C3×#131—91no—
109#942—31no—
110#942—91no—
111#932—31no—
112#932—91no—
113#901—31no—
114#901—91no—
115C3×#108—91no—
116C3×#101—91no—
117C3×#100—91no—
118#836—31no—
119#836—91no—
120#833—31no—
121#833—91no—
122#819—31no—
123#819—91no—
124C23×C32—21no—
125C23×C32—21no—
126C23×C32—41no—
127C32×Q8—22no—
128C32×Q8—21no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—41no—
133C32×D4—22no—
134C32×D4—21no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—21no—
138C32×D4—41no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—41no—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C23×C311yescharacteristic
145#638—21no—
146C4⋊(C3⋊D3)C3×D411yescharacteristic
147C4×C3⋊D3—21no—
148#634C3×D411yescharacteristic
149C3×#67—61no—
150C3×D12—31no—
151C3×C4×D3—61no—
152C3×Dic6—31no—
153C32×C8—21no—
154C32×C8—21no—
155C32×C8#6711yescharacteristic
156C32×C8#6711yescharacteristic
157C32×C8C3×D411yescharacteristic
158C32×C8C3×D411yescharacteristic
159C3⋊(C3⋊C8)—21no—
160C3⋊(C3⋊C8)—21no—
161C3⋊(C3⋊C8)—21no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—61no—
165#471—91noSylow
166C2×H(3,𝔽3)—21no—
167C2×H(3,𝔽3)—81no—
168C2×H(3,𝔽3)#12111yescharacteristic
169(C3⋊D3)⋊C3—41no—
170C3×#41—91no—
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×D4C2×C3×D311yescharacteristic
179#235—31no—
180#235—91no—
181C3×Q16—181no—
182C3×SD16—21no—
183C3×SD16—21no—
184C3×SD16—61no—
185C3×SD16—61no—
186C3×SD16—61no—
187C3×SD16—61no—
188C3×SD16—62no—
189C3×SD16—62no—
190C3×SD16—181no—
191C3×SD16—181no—
192C3×D8—181no—
193C3×M16—91no—
194C3×M16—91no—
195C3×M16—91no—
196C3×M16—181no—
197C2×C3×C8—31no—
198C2×C3×C8—31no—
199C2×C3×C8—61no—
200C2×C3×C8—181no—
201C2×C3×C8C2×C3×D311yescharacteristic
202#216—61no—
203#216—181no—
204#215—61no—
205#215—181no—
206#214—61no—
207#214—181no—
208#213—61no—
209#213—181no—
210#208—31no—
211#208—31no—
212#208—31no—
213#208—91no—
214#208—91no—
215#208—91no—
216ΓL(1,𝔽25)—61no—
217ΓL(1,𝔽25)—181no—
218D3×C8—61no—
219D3×C8—181no—
220C22×C32—21no—
221C22×C32—42no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32—81no—
225C22×C32—81no—
226C22×C32—82no—
227C22×C32#23711yescharacteristic
228C22×C32C3×#4111yescharacteristic
229C2×C3⋊D3—21no—
230C2×C3×D3—61no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—42no—
234C32×C4—41no—
235C32×C4—41no—
236C32×C4D3×D411yescharacteristic
237C32×C4C2×#6711yescharacteristic
238C32×C4C2×C3×D411yescharacteristic
239C32×C4C2×C3×D411yescharacteristic
240C3⋊Dic3—21no—
241C3×Dic3—61no—
242#137—91no—
243AGL(1,ℤ/8)—91no—
244C2×SD16(C3⋊D3)⋊C311yescharacteristic, complemented
245#131—91no—
246#108—91no—
247#101—91no—
248#100—91no—
249H(3,𝔽3)#47111yescharacteristic, complemented, Sylow
250C23×C3—21no—
251C23×C3—61no—
252C23×C3—61no—
253C23×C3—121no—
254C3×Q8—21no—
255C3×Q8—21no—
256C3×Q8—61no—
257C3×Q8—61no—
258C3×Q8—61no—
259C3×Q8—61no—
260C3×Q8—62no—
261C3×Q8—91no—
262C3×Q8—121no—
263C3×D4—21no—
264C3×D4—21no—
265C3×D4—61no—
266C3×D4—61no—
267C3×D4—61no—
268C3×D4—61no—
269C3×D4—62no—
270C3×D4—91no—
271C3×D4—121no—
272C3×D4—181no—
273C2×C3×C4—21no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—121no—
280C2×C3×C4—181no—
281C2×C3×C4C22×C3×D311yescharacteristic
282#67—61no—
283#67—181no—
284D12—31no—
285D12—91no—
286C4×D3—61no—
287C4×D3—181no—
288Dic6—31no—
289Dic6—91no—
290C3×C8—31no—
291C3×C8—31no—
292C3×C8—31no—
293C3×C8—31no—
294C3×C8—61no—
295C3×C8—61no—
296C3×C8—181no—
297C3×C8—181no—
298C3×C8—181no—
299C3×C8C3×#6711yescharacteristic
300C3×C8C3×#6711yescharacteristic
301C3⋊C8—61no—
302C3⋊C8—61no—
303C3⋊C8—61no—
304C3⋊C8—181no—
305C3⋊C8—181no—
306C3⋊C8—181no—
307C2×C32—22no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—82no—
312C2×C32—81no—
313C2×C32—81no—
314C2×C32#93811yescharacteristic
315C2×C32C3×#12111yescharacteristic
316C3⋊D3—41no—
317C3×D3—121no—
318#41—91no—
319C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
320C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
321Q16—181no—
322SD16—21no—
323SD16—21no—
324SD16—181no—
325SD16—181no—
326D8—181no—
327M16—91no—
328M16—91no—
329M16—91no—
330M16—181no—
331C2×C8—181no—
332C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—41no—
336C22×C3—61no—
337C22×C3—81no—
338C22×C3—121no—
339C22×C3—121no—
340C22×C3—122no—
341C22×C3—181no—
342C22×C3—241no—
343C22×C3—241no—
344C22×C3—242no—
345C22×C3C3×#23711yescharacteristic
346C2×D3—61no—
347C2×D3—181no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—31no—
352C3×C4—41no—
353C3×C4—61no—
354C3×C4—61no—
355C3×C4—121no—
356C3×C4—121no—
357C3×C4—122no—
358C3×C4—181no—
359C3×C4C3×D3×D411yescharacteristic
360C3×C4C2×C3×#6711yescharacteristic
361Dic3—61no—
362Dic3—181no—
363C32—21no—
364C32#517711yescharacteristic, complemented
365C32C3×#47111yescharacteristic, complemented
366C23—21no—
367Q8—21no—
368Q8—21no—
369Q8—91no—
370D4—21no—
371D4—21no—
372D4—91no—
373D4—181no—
374C2×C4—21no—
375C2×C4—181no—
376C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
377C8—181no—
378C8—181no—
379C8—181no—
380C8#638⋊C311yescharacteristic
381C8#638⋊C311yescharacteristic
382C2×C3—21no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—61no—
386C2×C3—61no—
387C2×C3—62no—
388C2×C3—61no—
389C2×C3—81no—
390C2×C3—241no—
391C2×C3—241no—
392C2×C3—242no—
393C2×C3—361no—
394C2×C3C3×#93811yescharacteristic
395D3—121no—
396D3—361no—
397C22—41no—
398C22—81no—
399C22—181no—
400C22#3830⋊C311yescharacteristic
401C4—41no—
402C4—181no—
403C4D4×(C3⋊D3)⋊C311yescharacteristic
404C4C2×#638⋊C311yescharacteristic
405C3—31no—
406C3—31no—
407C3—61no—
408C3C3×#517711yescharacteristic
409C2—21no—
410C2—81no—
411C2—361no—
412C2#64147⋊C311yescharacteristic
subgroupG/H
centerC2#64147⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×SD16×H(3,𝔽3)C2
socleC2×C3C3×#938
prime radical#64141⋊C3C2
hypercenterC2×SD16(C3⋊D3)⋊C3
hypocenterC32C3×#471
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ 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. #64151⋊C3C2semidirect
4. #64110⋊C3C2semidirect
5. #63663⋊C3C2semidirect
6. #63660⋊C3C2semidirect
7. #63646⋊C3C2semidirect
8. #10592876C3semidirect
26. C2×C32×SD16D3semidirect
27. C2×C32×SD16C2×C3semidirect
28. #64151C2×C3semidirect
30. #64110C2×C3semidirect
34. #63663C2×C3semidirect
35. #63660C2×C3semidirect
36. #63646C2×C3semidirect
244. C2×SD16(C3⋊D3)⋊C3semidirect
249. H(3,𝔽3)#471semidirect
364. C32#5177semidirect
365. C32C3×#471semidirect

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
#64151⋊C32
#64141⋊C32
#64110⋊C32
#63663⋊C32
#63660⋊C32
#63646⋊C32
#105928763
C3×#51773