Finite Group Databasebeta

← #10592940⋊C3 #13143657

Subgroups — #10592940⋊C3 #13143657

Order 1728.

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

normal subgroups only

This group has 754 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
1H(3,𝔽3)×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2#64158⋊C3C212yesmaximal, complemented
3#64149⋊C3C211yesmaximal, characteristic, complemented
4#63664⋊C3C212yesmaximal, complemented
5#63660⋊C3C211yesmaximal, characteristic, complemented
6#10592940C311yesmaximal, characteristic, complemented
7C3×#5241—31nomaximal
8#41×H(3,𝔽3)—22no—
9#41×H(3,𝔽3)C2212yes—
10C2×D4×H(3,𝔽3)—22no—
11C2×D4×H(3,𝔽3)—22no—
12C2×D4×H(3,𝔽3)C2211yescharacteristic
13C2×#638⋊C3—22no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
15#3755⋊C3—22no—
16#3753⋊C3—22no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
18#3748⋊C3C2212yescomplemented
19C32×H(5,𝔽2)—21no—
20C32×H(5,𝔽2)D311yescharacteristic, complemented
21C32×H(5,𝔽2)C2×C311yescharacteristic, complemented
22#64158C2×C312yescomplemented
23#64149C2×C311yescharacteristic, complemented
24C3×#949—32no—
25C3×#940—31no—
26#63664C2×C312yescomplemented
27#63660C2×C311yescharacteristic, complemented
28C3×#837—32no—
29C3×#833—31no—
30C23×H(3,𝔽3)—22no—
31C23×H(3,𝔽3)D412yes—
32C22×(C3⋊D3)⋊C3—21no—
33Q8×H(3,𝔽3)D412yescomplemented
34D4×H(3,𝔽3)—24no—
35D4×H(3,𝔽3)—22no—
36D4×H(3,𝔽3)—22no—
37D4×H(3,𝔽3)D412yescomplemented
38C2×C4×H(3,𝔽3)—22no—
39C2×C4×H(3,𝔽3)—22no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41#638⋊C3—44no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C4⋊((C3⋊D3)⋊C3)—22no—
44(C3⋊(C3⋊C8))⋊C3—22no—
45C3×#453—91no—
46#5241—31no—
47#5241—91no—
48C32×#41—24no—
49C32×#41—22no—
50C32×#41—22no—
51C32×#41—22no—
52C32×#41C2×D312yes—
53C32×#41C22×C312yes—
54C2×C32×D4—24no—
55C2×C32×D4—24no—
56C2×C32×D4—21no—
57C2×C32×D4—22no—
58C2×C32×D4—22no—
59C2×C32×D4—22no—
60C2×C32×D4—22no—
61C2×C32×D4C2×D311yescharacteristic
62C2×C32×D4C22×C311yescharacteristic
63C2×#638—22no—
64C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
65C2×C3×#67—62no—
66C2×C3×D12—31no—
67#3755—22no—
68#3753—22no—
69C4×C3⋊Dic3C22×C311yescharacteristic, complemented
70#3748C22×C312yescomplemented
71C3×#215—62no—
72C3×#213—62no—
73C3×C4×Dic3—31no—
74C3×#208—32no—
75C22×H(3,𝔽3)—22no—
76C22×H(3,𝔽3)—22no—
77C22×H(3,𝔽3)—42no—
78C22×H(3,𝔽3)—82no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×(C3⋊D3)⋊C3—42no—
81C2×(C3⋊D3)⋊C3—41no—
82C4×H(3,𝔽3)—22no—
83C4×H(3,𝔽3)C2×D412yes—
84(C3⋊Dic3)⋊C3—22no—
85C3×H(5,𝔽2)—31no—
86C3×H(5,𝔽2)—31no—
87C3×H(5,𝔽2)—61no—
88C3×H(5,𝔽2)C3×D311yescharacteristic
89C3×AGL(1,ℤ/8)—92no—
90C3×#127—91no—
91#949—32no—
92#949—92no—
93#940—31no—
94#940—91no—
95C3×#104—92no—
96C3×#100—91no—
97#837—32no—
98#837—92no—
99#833—31no—
100#833—91no—
101C23×C32—24no—
102C23×C32—22no—
103C23×C32—22no—
104C23×C32—22no—
105C23×C32#6712yes—
106C23×C32C3×D412yes—
107C22×C3⋊D3—21no—
108C22×C3×D3—61no—
109C32×Q8—22no—
110C32×Q8#6712yescomplemented
111C32×Q8C3×D412yescomplemented
112C32×D4—28no—
113C32×D4—24no—
114C32×D4—24no—
115C32×D4—24no—
116C32×D4—24no—
117C32×D4—22no—
118C32×D4—22no—
119C32×D4—22no—
120C32×D4—22no—
121C32×D4—22no—
122C32×D4#6712yescomplemented
123C32×D4C3×D412yescomplemented
124C2×C32×C4—24no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—24no—
128C2×C32×C4—22no—
129C2×C32×C4—22no—
130C2×C32×C4—21no—
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C23×C311yescharacteristic
133#638—44no—
134C2×C3⋊Dic3—21no—
135C4⋊(C3⋊D3)—22no—
136C3×#67—124no—
137C2×C3×Dic3—61no—
138C3×D12—62no—
139C3⋊(C3⋊C8)—22no—
140C3×C3⋊C8—62no—
141#453—91noSylow
142C2×H(3,𝔽3)—21no—
143C2×H(3,𝔽3)—42no—
144C2×H(3,𝔽3)—42no—
145C2×H(3,𝔽3)#12011yescharacteristic
146(C3⋊D3)⋊C3—81no—
147C3×#41—22no—
148C3×#41—32no—
149C3×#41—32no—
150C3×#41—62no—
151C3×#41—62no—
152C3×#41—64no—
153C3×#41—62no—
154C3×#41C2×C3×D312yes—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—31no—
158C2×C3×D4—31no—
159C2×C3×D4—62no—
160C2×C3×D4—62no—
161C2×C3×D4—62no—
162C2×C3×D4—62no—
163C2×C3×D4—64no—
164C2×C3×D4—64no—
165C2×C3×D4—61no—
166C2×C3×D4—91no—
167C2×C3×D4—182no—
168C2×C3×D4C2×C3×D311yescharacteristic
169C2×#67—62no—
170C2×#67—182no—
171C2×D12—31no—
172C2×D12—91no—
173C3×SD16—182no—
174C3×D8—182no—
175C3×M16—92no—
176C3×C42—91no—
177#215—62no—
178#215—182no—
179#213—62no—
180#213—182no—
181C4×Dic3—31no—
182C4×Dic3—91no—
183#208—32no—
184#208—92no—
185C22×C32—24no—
186C22×C32—24no—
187C22×C32—22no—
188C22×C32—22no—
189C22×C32—22no—
190C22×C32—22no—
191C22×C32—21no—
192C22×C32—42no—
193C22×C32—42no—
194C22×C32—82no—
195C22×C32—84no—
196C22×C32—82no—
197C22×C32—82no—
198C22×C32C2×#6711yescharacteristic
199C22×C32C2×C3×D411yescharacteristic
200C2×C3⋊D3—42no—
201C2×C3⋊D3—41no—
202C2×C3×D3—122no—
203C2×C3×D3—121no—
204C32×C4—24no—
205C32×C4—22no—
206C32×C4—22no—
207C32×C4—22no—
208C32×C4C2×#6712yes—
209C32×C4C2×C3×D412yes—
210C3⋊Dic3—22no—
211C3×Dic3—62no—
212H(5,𝔽2)(C3⋊D3)⋊C311yescharacteristic, complemented
213AGL(1,ℤ/8)—92no—
214#127—91no—
215#104—92no—
216#100—91no—
217H(3,𝔽3)#45311yescharacteristic, complemented, Sylow
218C23×C3—22no—
219C23×C3—32no—
220C23×C3—32no—
221C23×C3—62no—
222C23×C3—62no—
223C23×C3—64no—
224C23×C3—62no—
225C23×C3—181no—
226C23×C3C3×#6712yes—
227C22×D3—61no—
228C22×D3—181no—
229C3×Q8—32no—
230C3×Q8—32no—
231C3×Q8—62no—
232C3×Q8C3×#6712yes—
233C3×D4—24no—
234C3×D4—22no—
235C3×D4—22no—
236C3×D4—32no—
237C3×D4—32no—
238C3×D4—62no—
239C3×D4—62no—
240C3×D4—62no—
241C3×D4—62no—
242C3×D4—64no—
243C3×D4—64no—
244C3×D4—68no—
245C3×D4—64no—
246C3×D4—64no—
247C3×D4—62no—
248C3×D4—182no—
249C3×D4—364no—
250C3×D4C3×#6712yes—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—62no—
256C2×C3×C4—62no—
257C2×C3×C4—62no—
258C2×C3×C4—62no—
259C2×C3×C4—64no—
260C2×C3×C4—64no—
261C2×C3×C4—61no—
262C2×C3×C4—181no—
263C2×C3×C4C22×C3×D311yescharacteristic
264#67—124no—
265#67—364no—
266C2×Dic3—61no—
267C2×Dic3—181no—
268D12—62no—
269D12—182no—
270C3×C8—182no—
271C3⋊C8—62no—
272C3⋊C8—182no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—41no—
277C2×C32—44no—
278C2×C32—44no—
279C2×C32—42no—
280C2×C32—42no—
281C2×C32—42no—
282C2×C32—42no—
283C2×C32#95311yescharacteristic
284C2×C32C3×#12011yescharacteristic
285C3⋊D3—81no—
286C3×D3—241no—
287#41—22no—
288#41C2×(C3⋊D3)⋊C312yes—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—91no—
292C2×D4—182no—
293C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
294SD16—182no—
295D8—182no—
296M16—92no—
297C42—91no—
298C22×C3—22no—
299C22×C3—22no—
300C22×C3—31no—
301C22×C3—31no—
302C22×C3—42no—
303C22×C3—62no—
304C22×C3—62no—
305C22×C3—62no—
306C22×C3—62no—
307C22×C3—64no—
308C22×C3—64no—
309C22×C3—61no—
310C22×C3—82no—
311C22×C3—122no—
312C22×C3—122no—
313C22×C3—242no—
314C22×C3—242no—
315C22×C3—242no—
316C22×C3—244no—
317C22×C3—361no—
318C22×C3—362no—
319C22×C3C2×C3×#6711yescharacteristic
320C2×D3—122no—
321C2×D3—121no—
322C2×D3—361no—
323C2×D3—362no—
324C3×C4—22no—
325C3×C4—32no—
326C3×C4—32no—
327C3×C4—62no—
328C3×C4—62no—
329C3×C4—64no—
330C3×C4—62no—
331C3×C4—182no—
332C3×C4C2×C3×#6712yes—
333Dic3—62no—
334Dic3—182no—
335C32—21no—
336C32#524111yescharacteristic, complemented
337C32C3×#45311yescharacteristic, complemented
338C23—22no—
339C23—181no—
340C23#638⋊C312yes—
341Q8#638⋊C312yescomplemented
342D4—24no—
343D4—22no—
344D4—22no—
345D4—182no—
346D4—364no—
347D4#638⋊C312yescomplemented
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—181no—
351C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
352C8—182no—
353C2×C3—21no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—42no—
357C2×C3—42no—
358C2×C3—61no—
359C2×C3—61no—
360C2×C3—61no—
361C2×C3—121no—
362C2×C3—122no—
363C2×C3—122no—
364C2×C3—122no—
365C2×C3—122no—
366C2×C3—124no—
367C2×C3—124no—
368C2×C3—721no—
369C2×C3C3×#95311yescharacteristic
370D3—241no—
371D3—721no—
372C22—22no—
373C22—22no—
374C22—42no—
375C22—82no—
376C22—362no—
377C22—361no—
378C22C2×#638⋊C311yescharacteristic
379C4—22no—
380C4—182no—
381C4C2×#638⋊C312yes—
382C3—31no—
383C3—31no—
384C3—61no—
385C3C3×#524111yescharacteristic
386C2—21no—
387C2—42no—
388C2—42no—
389C2—721no—
390C2#1172170211yescharacteristic
subgroupG/H
centerC2#11721702
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×H(5,𝔽2)C2
socleC2×C3C3×#953
hypercenterH(5,𝔽2)(C3⋊D3)⋊C3
hypocenterC32C3×#453
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×H(5,𝔽2)C2semidirect
2. #64158⋊C3C2semidirect
3. #64149⋊C3C2semidirect
4. #63664⋊C3C2semidirect
5. #63660⋊C3C2semidirect
6. #10592940C3semidirect
14. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
17. C4×(C3⋊Dic3)⋊C3C22semidirect
18. #3748⋊C3C22semidirect
20. C32×H(5,𝔽2)D3semidirect
21. C32×H(5,𝔽2)C2×C3semidirect
22. #64158C2×C3semidirect
23. #64149C2×C3semidirect
26. #63664C2×C3semidirect
27. #63660C2×C3semidirect
33. Q8×H(3,𝔽3)D4semidirect
37. D4×H(3,𝔽3)D4semidirect
64. C2×C4⋊(C3⋊D3)C22×C3semidirect
69. C4×C3⋊Dic3C22×C3semidirect
70. #3748C22×C3semidirect
110. C32×Q8#67semidirect
111. C32×Q8C3×D4semidirect
122. C32×D4#67semidirect
123. C32×D4C3×D4semidirect
212. H(5,𝔽2)(C3⋊D3)⋊C3semidirect
217. H(3,𝔽3)#453semidirect
336. C32#5241semidirect
337. C32C3×#453semidirect
341. Q8#638⋊C3semidirect
347. D4#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×H(5,𝔽2)2
#64158⋊C32
#64149⋊C32
#63664⋊C32
#63660⋊C32
#105929403
C3×#52413