Finite Group Databasebeta

← C3×#10591622 #13158623

Subgroups — C3×#10591622 #13158623

Order 1728.

756 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C32×#215C212yesmaximal, complemented
3C3×#63914C211yesmaximal, characteristic
4C3×#63572C211yesmaximal, characteristic, complemented
5C3×#63563C212yesmaximal, complemented
6C3×#5163—32nomaximal
7#10591622C311yesmaximal, characteristic, direct product factor
8C2×C33×Q8C2211yescharacteristic
9C2×C32×D12C2212yes—
10C2×C3×#625—21no—
11C2×C3×#625—21no—
12C3×C4⋊(C3⋊Dic3)—21no—
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4×C3⋊Dic3—21no—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×#215—24no—
17C2×C32×C3⋊C8C2212yes—
18C2×C32×SD16—32no—
19Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
20C3×Q8×Dic3—32no—
21C3×Q8×Dic3—61no—
22C2×C3×#215—22no—
23C2×C3×#215D312yescomplemented
24C2×C3×#215C2×C312yescomplemented
25C3×#940—32no—
26#63914C2×C311yescharacteristic
27C3×#832—32no—
28C3×#821—32no—
29C3×Dic3⋊C8—32no—
30#63572C2×C311yescharacteristic, complemented
31#63563C2×C312yescomplemented
32C22×C32×D3—22no—
33C33×Q8—21no—
34C33×Q8D412yescomplemented
35C2×C33×C4—21no—
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3D411yescharacteristic
39C2×C3×C3⋊Dic3D411yescharacteristic
40C32×D12—22no—
41C32×D12—22no—
42C3×#625—42no—
43C3×#625—42no—
44C32×C3⋊C8—22no—
45C3×#460—91no—
46#5163—32no—
47C2×C32×Q8—21no—
48C2×C32×Q8—22no—
49C2×C32×Q8—41no—
50C2×C32×Q8C2×D312yes—
51C2×C32×Q8C22×C311yescharacteristic
52C2×C32×D4—32no—
53C2×C3×#67—62no—
54C2×C3×#67—62no—
55C2×C3×D12—22no—
56C2×C3×D12C2×D312yes—
57C2×C3×D12C22×C312yes—
58C2×#625—21no—
59C2×#625—21no—
60C32×SD16—64no—
61C2×C32×C8—32no—
62C4⋊(C3⋊Dic3)—21no—
63C4⋊(C3⋊Dic3)C22×C311yescharacteristic
64C4×C3⋊Dic3—21no—
65C4×C3⋊Dic3C22×C311yescharacteristic
66C3×#215—28no—
67C3×#215—24no—
68C3×#215—24no—
69C3×C4⋊Dic3—32no—
70C3×C4⋊Dic3—62no—
71C3×C4⋊Dic3—61no—
72C3×C4⋊Dic3—62no—
73C3×C4×Dic3—32no—
74C3×C4×Dic3—62no—
75C3×C4×Dic3—61no—
76C3×C4×Dic3—62no—
77C2×C3×C3⋊C8—22no—
78C2×C3×C3⋊C8C2×D312yes—
79C2×C3×C3⋊C8C22×C312yes—
80C22×C33C2×D411yescharacteristic
81C2×C32×D3—42no—
82C2×C32×D3—42no—
83C2×C32×D3—42no—
84C33×C4—22no—
85C33×C4C2×D411yescharacteristic
86C33×C4#4111yescharacteristic
87C3×C3⋊Dic3—21no—
88C3×C3⋊Dic3—41no—
89C3×C3⋊Dic3SD1612yescomplemented
90C2×C3×SD16—32no—
91C2×C3×SD16—32no—
92C2×C3×SD16—62no—
93C3×#127—91no—
94C3×C4×Q8—91no—
95Q8×Dic3—32no—
96Q8×Dic3—61no—
97C2×#215C3×D312yescomplemented
98#940—32no—
99C3×C4⋊C8—91no—
100C3×#102—92no—
101#832—32no—
102#821—32no—
103Dic3⋊C8—32no—
104C23×C32—62no—
105C22×C3×D3—22no—
106C22×C3×D3—22no—
107C22×C3×D3—42no—
108C32×Q8—21no—
109C32×Q8—22no—
110C32×Q8—22no—
111C32×Q8—22no—
112C32×Q8—24no—
113C32×Q8—42no—
114C32×Q8—42no—
115C32×Q8—42no—
116C32×Q8#6714yescomplemented
117C32×Q8C3×D412yescomplemented
118C32×D4—62no—
119C32×D4—62no—
120C2×C32×C4—21no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—21no—
124C2×C32×C4—22no—
125C2×C32×C4—41no—
126C2×C32×C4—42no—
127C2×C32×C4—42no—
128C2×C32×C4C22×D312yes—
129C2×C32×C4C23×C311yescharacteristic
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3C3×D411yescharacteristic
132C2×C3⋊Dic3C3×D411yescharacteristic
133C3×#67—124no—
134C3×#67—124no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141C3×D12—24no—
142C3×D12—22no—
143C3×D12—22no—
144C3×D12—22no—
145C3×D12—22no—
146C3×D12—42no—
147#625—42no—
148#625—42no—
149C32×C8—62no—
150C3×C3⋊C8—24no—
151C3×C3⋊C8—22no—
152C3×C3⋊C8—22no—
153#460—91noSylow
154C2×C33#12111yescharacteristic
155C2×C33C2×SD1611yescharacteristic
156C2×C33AGL(1,ℤ/8)11yescharacteristic
157C32×D3—82no—
158C2×C3×Q8—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—41no—
161C2×C3×Q8D3211yescharacteristic
162C2×C3×Q8C2×C3×D312yes—
163C2×C3×D4—32no—
164C2×C3×D4—32no—
165C2×C3×D4—62no—
166C2×C3×D4—181no—
167C2×C3×D4—181no—
168C2×#67—62no—
169C2×#67—62no—
170C2×D12C2×C3×D312yes—
171C3×SD16—68no—
172C3×SD16—64no—
173C3×SD16—64no—
174C2×C3×C8—32no—
175C2×C3×C8—32no—
176C2×C3×C8—62no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4—181no—
179C3×C42—91no—
180C3×C42—181no—
181#215—24no—
182C4⋊Dic3—32no—
183C4⋊Dic3—61no—
184C4⋊Dic3—62no—
185C4⋊Dic3—62no—
186C4×Dic3—32no—
187C4×Dic3—61no—
188C4×Dic3—62no—
189C4×Dic3—62no—
190C2×C3⋊C8C2×C3×D312yes—
191C22×C32—21no—
192C22×C32—22no—
193C22×C32—41no—
194C22×C32—122no—
195C22×C32—122no—
196C22×C32—122no—
197C22×C32D3×D412yes—
198C22×C32C2×C3×D411yescharacteristic
199C2×C3×D3—44no—
200C2×C3×D3—42no—
201C2×C3×D3—42no—
202C2×C3×D3—42no—
203C2×C3×D3—42no—
204C2×C3×D3—42no—
205C2×C3×D3—42no—
206C2×C3×D3—82no—
207C2×C3×D3—82no—
208C32×C4—22no—
209C32×C4—24no—
210C32×C4—24no—
211C32×C4—21no—
212C32×C4—21no—
213C32×C4—22no—
214C32×C4—22no—
215C32×C4—41no—
216C32×C4—41no—
217C32×C4—44no—
218C32×C4—44no—
219C32×C4#23712yes—
220C32×C4C2×#6712yes—
221C32×C4C2×C3×D411yescharacteristic
222C32×C4C3×#4111yescharacteristic
223C3⋊Dic3—21no—
224C3⋊Dic3—41no—
225C3⋊Dic3C3×SD1612yescomplemented
226C3×Dic3—34no—
227C3×Dic3—62no—
228C3×Dic3—62no—
229C3×Dic3—121no—
230C3×Dic3—122no—
231C3×Dic3—122no—
232C2×SD16—32no—
233#127—91no—
234C4×Q8—91no—
235C4⋊C8—91no—
236#102—92no—
237C33#46011yescharacteristic, complemented, Sylow
238C23×C3—62no—
239C23×C3—62no—
240C23×C3—122no—
241C22×D3—22no—
242C3×Q8—22no—
243C3×Q8—21no—
244C3×Q8—22no—
245C3×Q8—22no—
246C3×Q8—24no—
247C3×Q8—42no—
248C3×Q8—42no—
249C3×Q8—42no—
250C3×Q8#62512yescomplemented
251C3×Q8C3×#6714yescomplemented
252C3×D4—64no—
253C3×D4—62no—
254C3×D4—62no—
255C3×D4—62no—
256C3×D4—62no—
257C3×D4—122no—
258C3×D4—362no—
259C3×D4—362no—
260C2×C3×C4—22no—
261C2×C3×C4—21no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—21no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—91no—
269C2×C3×C4—91no—
270C2×C3×C4—181no—
271C2×C3×C4C2×D3211yescharacteristic
272C2×C3×C4C22×C3×D312yes—
273#67—124no—
274#67—124no—
275C2×Dic3—32no—
276C2×Dic3—32no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281D12—22no—
282D12—22no—
283C3×C8—64no—
284C3×C8—62no—
285C3×C8—62no—
286C3⋊C8—22no—
287C2×C32—22no—
288C2×C32—22no—
289C2×C32—22no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—41no—
294C2×C32—41no—
295C2×C32—41no—
296C2×C32—242no—
297C2×C32D3×SD1612yes—
298C2×C32#91412yes—
299C2×C32#93912yes—
300C2×C32C3×#12111yescharacteristic
301C2×C32C2×C3×SD1611yescharacteristic
302C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
303C3×D3—84no—
304C3×D3—82no—
305C3×D3—82no—
306C2×Q8C3×D3211yescharacteristic
307C2×D4—32no—
308C2×D4—181no—
309C2×D4—181no—
310SD16—64no—
311C2×C8—32no—
312C4⋊C4—91no—
313C4⋊C4—181no—
314C42—91no—
315C42—181no—
316C22×C3—21no—
317C22×C3—22no—
318C22×C3—41no—
319C22×C3—124no—
320C22×C3—122no—
321C22×C3—122no—
322C22×C3—122no—
323C22×C3—122no—
324C22×C3—122no—
325C22×C3—122no—
326C22×C3—242no—
327C22×C3—242no—
328C22×C3#381111yescharacteristic
329C22×C3C3×D3×D412yes—
330C2×D3—42no—
331C2×D3—42no—
332C2×D3—42no—
333C3×C4—24no—
334C3×C4—24no—
335C3×C4—22no—
336C3×C4—22no—
337C3×C4—21no—
338C3×C4—22no—
339C3×C4—21no—
340C3×C4—44no—
341C3×C4—44no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4—92no—
345C3×C4—181no—
346C3×C4—361no—
347C3×C4#380511yescharacteristic
348C3×C4C2×#62511yescharacteristic
349C3×C4C3×#23712yes—
350C3×C4C2×C3×#6712yes—
351Dic3—34no—
352Dic3—62no—
353Dic3—62no—
354Dic3—121no—
355Dic3—122no—
356Dic3—122no—
357C32—22no—
358C32—21no—
359C32—41no—
360C32#516312yescomplemented
361C32C3×#46011yescharacteristic, complemented
362C23—62no—
363Q8—21no—
364Q8C3×#62512yescomplemented
365D4—62no—
366D4—62no—
367D4—362no—
368D4—362no—
369C2×C4—21no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—181no—
373C2×C4C2×C3×D3211yescharacteristic
374C8—62no—
375C2×C3—21no—
376C2×C3—22no—
377C2×C3—21no—
378C2×C3—22no—
379C2×C3—21no—
380C2×C3—22no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—244no—
385C2×C3—242no—
386C2×C3—242no—
387C2×C3#6394011yescharacteristic
388C2×C3#6394111yescharacteristic
389C2×C3#6398011yescharacteristic
390C2×C3C3×D3×SD1612yes—
391C2×C3C3×#91412yes—
392C2×C3C3×#93912yes—
393D3—82no—
394C22—122no—
395C22—122no—
396C22—122no—
397C22C3×#381111yescharacteristic
398C4—22no—
399C4—92no—
400C4—181no—
401C4—361no—
402C4C3×#380511yescharacteristic
403C4C2×C3×#62511yescharacteristic
404C3—22no—
405C3—21no—
406C3—41no—
407C3#1059162211yescharacteristic, direct product factor
408C3C3×#516312yescomplemented
409C2—242no—
410C2C3×#6394011yescharacteristic
411C2C3×#6394111yescharacteristic
412C2C3×#6398011yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC3×#63914C2
hypercenterC2×C3×Q8D32
hypocenterC32C3×#460
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×C3⋊Dic3C2semidirect
2. C2×C32×#215C2semidirect
4. C3×#63572C2semidirect
5. C3×#63563C2semidirect
7. #10591622C3direct (N × H)
19. Q8×C3⋊Dic3C2×C3semidirect
23. C2×C3×#215D3semidirect
24. C2×C3×#215C2×C3semidirect
30. #63572C2×C3semidirect
31. #63563C2×C3semidirect
34. C33×Q8D4semidirect
89. C3×C3⋊Dic3SD16semidirect
97. C2×#215C3×D3semidirect
116. C32×Q8#67semidirect
117. C32×Q8C3×D4semidirect
225. C3⋊Dic3C3×SD16semidirect
237. C33#460semidirect
250. C3×Q8#625semidirect
251. C3×Q8C3×#67semidirect
360. C32#5163semidirect
361. C32C3×#460semidirect
364. Q8C3×#625semidirect
407. C3#10591622direct (N × H)
408. C3C3×#5163semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊Dic32
C2×C32×#2152
C3×#639142
C3×#635722
C3×#635632
C3×#51633
#105916223