Finite Group Databasebeta

← #10592941⋊C3 #13143656

Subgroups — #10592941⋊C3 #13143656

Order 1728.

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

normal subgroups only

This group has 698 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
2#64160⋊C3C212yesmaximal, complemented
3#64145⋊C3C211yesmaximal, characteristic, complemented
4#63664⋊C3C212yesmaximal, complemented
5#63660⋊C3C211yesmaximal, characteristic, complemented
6#10592941C311yesmaximal, characteristic, complemented
7C3×#5242—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×#634⋊C3C2211yescharacteristic, complemented
14#3757⋊C3—22no—
15#3756⋊C3—22no—
16#3754⋊C3—22no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
18#3748⋊C3C2212yescomplemented
19C32×H(5,𝔽2)—21no—
20C32×H(5,𝔽2)D311yescharacteristic
21C32×H(5,𝔽2)C2×C311yescharacteristic
22#64160C2×C312yescomplemented
23#64145C2×C311yescharacteristic, complemented
24C3×#951—32no—
25C3×#936—31no—
26#63664C2×C312yescomplemented
27#63660C2×C311yescharacteristic, complemented
28C3×#837—32no—
29C3×#833—31no—
30C23×H(3,𝔽3)—22no—
31C23×H(3,𝔽3)D412yes—
32Q8×H(3,𝔽3)D412yes—
33D4×H(3,𝔽3)—24no—
34D4×H(3,𝔽3)—22no—
35D4×H(3,𝔽3)—22no—
36D4×H(3,𝔽3)D412yes—
37C2×C4×H(3,𝔽3)—22no—
38C2×C4×H(3,𝔽3)—22no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42#634⋊C3—22no—
43(C3⋊(C3⋊C8))⋊C3—22no—
44C3×#455—91no—
45#5242—31no—
46#5242—91no—
47C32×#41—24no—
48C32×#41—22no—
49C32×#41—22no—
50C32×#41—22no—
51C32×#41C2×D312yes—
52C32×#41C22×C312yes—
53C2×C32×D4—24no—
54C2×C32×D4—24no—
55C2×C32×D4—21no—
56C2×C32×D4—22no—
57C2×C32×D4—22no—
58C2×C32×D4—22no—
59C2×C32×D4—22no—
60C2×C32×D4C2×D311yescharacteristic
61C2×C32×D4C22×C311yescharacteristic
62C2×#634C22×C311yescharacteristic, complemented
63C2×C3×Dic6—31no—
64#3757—22no—
65#3756—22no—
66#3754—22no—
67C4×C3⋊Dic3C22×C311yescharacteristic, complemented
68#3748C22×C312yescomplemented
69C3×#217—62no—
70C3×#216—62no—
71C3×#214—62no—
72C3×C4×Dic3—31no—
73C3×#208—32no—
74C22×H(3,𝔽3)—22no—
75C22×H(3,𝔽3)—22no—
76C22×H(3,𝔽3)—42no—
77C22×H(3,𝔽3)—82no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C4×H(3,𝔽3)—22no—
80C4×H(3,𝔽3)C2×D412yes—
81(C3⋊Dic3)⋊C3—41no—
82(C3⋊Dic3)⋊C3—41no—
83C3×H(5,𝔽2)—31no—
84C3×H(5,𝔽2)—31no—
85C3×H(5,𝔽2)—61no—
86C3×H(5,𝔽2)C3×D311yescharacteristic
87C3×#137—92no—
88C3×#124—91no—
89#951—32no—
90#951—92no—
91#936—31no—
92#936—91no—
93C3×#104—92no—
94C3×#100—91no—
95#837—32no—
96#837—92no—
97#833—31no—
98#833—91no—
99C23×C32—24no—
100C23×C32—22no—
101C23×C32—22no—
102C23×C32—22no—
103C23×C32#6712yes—
104C23×C32C3×D412yes—
105C32×Q8—22no—
106C32×Q8#6712yes—
107C32×Q8C3×D412yes—
108C32×D4—28no—
109C32×D4—24no—
110C32×D4—24no—
111C32×D4—24no—
112C32×D4—24no—
113C32×D4—22no—
114C32×D4—22no—
115C32×D4—22no—
116C32×D4—22no—
117C32×D4—22no—
118C32×D4#6712yes—
119C32×D4C3×D412yes—
120C2×C32×C4—24no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—24no—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—21no—
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C23×C311yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—21no—
131#634—22no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C3×Dic6—62no—
135C3⋊(C3⋊C8)—22no—
136C3×C3⋊C8—62no—
137#455—91noSylow
138C2×H(3,𝔽3)—21no—
139C2×H(3,𝔽3)—42no—
140C2×H(3,𝔽3)—42no—
141C2×H(3,𝔽3)#12011yescharacteristic
142C3×#41—22no—
143C3×#41—32no—
144C3×#41—32no—
145C3×#41—62no—
146C3×#41—62no—
147C3×#41—64no—
148C3×#41—62no—
149C3×#41C2×C3×D312yes—
150C2×C3×Q8—91no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—31no—
154C2×C3×D4—31no—
155C2×C3×D4—62no—
156C2×C3×D4—62no—
157C2×C3×D4—62no—
158C2×C3×D4—62no—
159C2×C3×D4—64no—
160C2×C3×D4—64no—
161C2×C3×D4—61no—
162C2×C3×D4C2×C3×D311yescharacteristic
163C2×Dic6—31no—
164C2×Dic6—91no—
165C3×Q16—182no—
166C3×SD16—182no—
167C3×M16—92no—
168C3×C22⋊C4—182no—
169C3×C42—91no—
170#217—62no—
171#217—182no—
172#216—62no—
173#216—182no—
174#214—62no—
175#214—182no—
176C4×Dic3—31no—
177C4×Dic3—91no—
178#208—32no—
179#208—92no—
180C22×C32—24no—
181C22×C32—24no—
182C22×C32—22no—
183C22×C32—22no—
184C22×C32—22no—
185C22×C32—22no—
186C22×C32—21no—
187C22×C32—42no—
188C22×C32—42no—
189C22×C32—82no—
190C22×C32—84no—
191C22×C32—82no—
192C22×C32—82no—
193C22×C32C2×#6711yescharacteristic
194C22×C32C2×C3×D411yescharacteristic
195C32×C4—24no—
196C32×C4—22no—
197C32×C4—22no—
198C32×C4—22no—
199C32×C4C2×#6712yes—
200C32×C4C2×C3×D412yes—
201C3⋊Dic3—41no—
202C3⋊Dic3—41no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205H(5,𝔽2)(C3⋊D3)⋊C311yescharacteristic
206#137—92no—
207#124—91no—
208#104—92no—
209#100—91no—
210H(3,𝔽3)#45511yescharacteristic, complemented, Sylow
211C23×C3—22no—
212C23×C3—32no—
213C23×C3—32no—
214C23×C3—62no—
215C23×C3—62no—
216C23×C3—64no—
217C23×C3—62no—
218C23×C3C3×#6712yes—
219C3×Q8—32no—
220C3×Q8—32no—
221C3×Q8—62no—
222C3×Q8—182no—
223C3×Q8C3×#6712yes—
224C3×D4—24no—
225C3×D4—22no—
226C3×D4—22no—
227C3×D4—32no—
228C3×D4—32no—
229C3×D4—62no—
230C3×D4—62no—
231C3×D4—62no—
232C3×D4—62no—
233C3×D4—64no—
234C3×D4—64no—
235C3×D4—68no—
236C3×D4—64no—
237C3×D4—64no—
238C3×D4—62no—
239C3×D4C3×#6712yes—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—62no—
245C2×C3×C4—62no—
246C2×C3×C4—62no—
247C2×C3×C4—62no—
248C2×C3×C4—64no—
249C2×C3×C4—64no—
250C2×C3×C4—61no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4C22×C3×D311yescharacteristic
254C2×Dic3—61no—
255C2×Dic3—61no—
256C2×Dic3—181no—
257C2×Dic3—181no—
258Dic6—62no—
259Dic6—182no—
260C3×C8—182no—
261C3⋊C8—62no—
262C3⋊C8—182no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—41no—
267C2×C32—44no—
268C2×C32—44no—
269C2×C32—42no—
270C2×C32—42no—
271C2×C32—42no—
272C2×C32—42no—
273C2×C32#95311yescharacteristic
274C2×C32C3×#12011yescharacteristic
275#41—22no—
276#41C2×(C3⋊D3)⋊C312yes—
277C2×Q8—91no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
281Q16—182no—
282SD16—182no—
283M16—92no—
284C22⋊C4—182no—
285C42—91no—
286C22×C3—22no—
287C22×C3—22no—
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—42no—
291C22×C3—62no—
292C22×C3—62no—
293C22×C3—62no—
294C22×C3—62no—
295C22×C3—64no—
296C22×C3—64no—
297C22×C3—61no—
298C22×C3—82no—
299C22×C3—122no—
300C22×C3—122no—
301C22×C3—242no—
302C22×C3—242no—
303C22×C3—242no—
304C22×C3—244no—
305C22×C3C2×C3×#6711yescharacteristic
306C3×C4—22no—
307C3×C4—32no—
308C3×C4—32no—
309C3×C4—62no—
310C3×C4—62no—
311C3×C4—64no—
312C3×C4—62no—
313C3×C4—361no—
314C3×C4—361no—
315C3×C4C2×C3×#6712yes—
316Dic3—121no—
317Dic3—121no—
318Dic3—361no—
319Dic3—361no—
320C32—21no—
321C32#524211yescharacteristic, complemented
322C32C3×#45511yescharacteristic, complemented
323C23—22no—
324C23#638⋊C312yes—
325Q8—182no—
326Q8#638⋊C312yes—
327D4—24no—
328D4—22no—
329D4—22no—
330D4#638⋊C312yes—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—181no—
334C2×C4—181no—
335C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
336C8—182no—
337C2×C3—21no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—42no—
341C2×C3—42no—
342C2×C3—61no—
343C2×C3—61no—
344C2×C3—61no—
345C2×C3—121no—
346C2×C3—122no—
347C2×C3—122no—
348C2×C3—122no—
349C2×C3—122no—
350C2×C3—124no—
351C2×C3—124no—
352C2×C3C3×#95311yescharacteristic
353C22—22no—
354C22—22no—
355C22—42no—
356C22—82no—
357C22C2×#638⋊C311yescharacteristic
358C4—22no—
359C4—361no—
360C4—361no—
361C4C2×#638⋊C312yes—
362C3—31no—
363C3—31no—
364C3—61no—
365C3C3×#524211yescharacteristic
366C2—21no—
367C2—42no—
368C2—42no—
369C2#1172170211yescharacteristic
subgroupG/H
centerC2#11721702
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting, prime radicalH(3,𝔽3)×H(5,𝔽2)C2
socleC2×C3C3×#953
hypercenterH(5,𝔽2)(C3⋊D3)⋊C3
hypocenterC32C3×#455
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
2. #64160⋊C3C2semidirect
3. #64145⋊C3C2semidirect
4. #63664⋊C3C2semidirect
5. #63660⋊C3C2semidirect
6. #10592941C3semidirect
13. C2×#634⋊C3C22semidirect
17. C4×(C3⋊Dic3)⋊C3C22semidirect
18. #3748⋊C3C22semidirect
22. #64160C2×C3semidirect
23. #64145C2×C3semidirect
26. #63664C2×C3semidirect
27. #63660C2×C3semidirect
62. C2×#634C22×C3semidirect
67. C4×C3⋊Dic3C22×C3semidirect
68. #3748C22×C3semidirect
210. H(3,𝔽3)#455semidirect
321. C32#5242semidirect
322. C32C3×#455semidirect

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
#64160⋊C32
#64145⋊C32
#63664⋊C32
#63660⋊C32
#105929413
C3×#52423