Finite Group Databasebeta

← C3×#10593101 #13164437

Subgroups — C3×#10593101 #13164437

Order 1728.

568 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C3×2−1+4⋊C9C211yesmaximal, characteristic, complemented
2C3×#63692C211yesmaximal, characteristic, complemented
3C3×#63691C211yesmaximal, characteristic, complemented
4C3×#63689C212yesmaximal, complemented
5C2×C3×#3689C212yesmaximal, complemented
6C3×#5837—31nomaximal
7#10593101C311yesmaximal, characteristic, direct product factor
8C3×D4×D9—41nomaximal
9C3×#41⋊C9C2211yescharacteristic, complemented
10C2×C3×Q8⋊C9C2212yescomplemented
11C3×#3689C2211yescharacteristic, complemented
12C3×#3689C2212yescomplemented
13C3×#3688C2211yescharacteristic, complemented
14C32×2−1+4D311yescharacteristic
15C3×#1009—31no—
16C3×#950—32no—
17C3×#950—31no—
18C3×#949—32no—
19C3×#949—31no—
20C3×#948—31no—
21C3×#942—31no—
22C3×#942—32no—
23C2×C3×#215—31no—
24C2×C3×#215—32no—
252−1+4⋊C9—21no—
262−1+4⋊C9C2×C311yescharacteristic, complemented
27#63692C2×C311yescharacteristic, complemented
28#63691C2×C311yescharacteristic, complemented
29#63689C2×C312yescomplemented
30C2×#3689C2×C312yescomplemented
31C22×C3×D9—42no—
32C3×D4×C9—41no—
33C3×#611—42no—
34C3×D36—41no—
35C3×C4×D9—41no—
36C3×Q8⋊C9C2311yescharacteristic
37C3×#577—91no—
38#5837—31no—
39C32×#41—31no—
40C32×#41—32no—
41C32×#41C2×D311yescharacteristic
42C2×C32×Q8—31no—
43C2×C32×Q8C2×D312yes—
44C3×#239—61no—
45C3×D3×D4—62no—
46C3×D3×D4—61no—
47C3×#235—32no—
48C3×#235—31no—
49C2×C3×D12—32no—
50C2×C3×D12—31no—
51C3×#216—32no—
52C3×#216—31no—
53C3×#215—32no—
54C3×#215—32no—
55C3×#215—32no—
56C3×#215—31no—
57C3×#215—32no—
58C3×#214—31no—
59C3×#213—31no—
60C3×#213—32no—
61C3×#208—32no—
62C3×#208—31no—
63C2×C3×C3⋊C8—32no—
64C2×C3×C3⋊C8—31no—
65D4×D9—41no—
66#41⋊C9—21no—
67#41⋊C9C22×C311yescharacteristic, complemented
68C2×Q8⋊C9—22no—
69C2×Q8⋊C9C22×C312yescomplemented
70#3689C22×C311yescharacteristic, complemented
71#3689C22×C312yescomplemented
72#3688C22×C311yescharacteristic, complemented
73C22×C3×C9—42no—
74C2×C3×D9—41no—
75C2×C3×D9—42no—
76C2×C3×D9—84no—
77C3×C4×C9—41no—
78C3×Dic9—41no—
79C3×2−1+4—21no—
80C3×2−1+4D911yescharacteristic, complemented
81C3×2−1+4C3×D311yescharacteristic
82C3×H(5,𝔽2)—91no—
83#1009—31no—
84C3×#137—91no—
85C3×#137—92no—
86C3×AGL(1,ℤ/8)—92no—
87C3×AGL(1,ℤ/8)—91no—
88C3×#135—92no—
89C3×#135—91no—
90C2×C3×SD16—91no—
91C2×C3×SD16—92no—
92C3×#131—91no—
93#950—32no—
94#950—31no—
95#949—32no—
96#949—31no—
97#948—31no—
98#942—31no—
99#942—32no—
100C2×#215—31no—
101C2×#215—32no—
102C22×C3×D3—61no—
103C22×C3×D3—62no—
104C32×Q8—32no—
105C32×Q8—31no—
106C32×Q8C22×D311yescharacteristic
107C32×D4—31no—
108C32×D4—32no—
109C32×D4S411yescharacteristic
110C2×C32×C4—32no—
111C2×C32×C4—31no—
112C2×C32×C4—61no—
113C3×#67—61no—
114C3×#67—62no—
115C3×D12—32no—
116C3×D12—31no—
117C3×D12—62no—
118C3×D12—61no—
119C3×C4×D3—62no—
120C3×C4×D3—61no—
121C3×Dic6—31no—
122C22×D9—42no—
123C3×C3⋊C8—31no—
124C3×C3⋊C8—31no—
125C3×C3⋊C8—32no—
126D4×C9—41no—
127D4×C9—81no—
128#611—42no—
129D36—41no—
130C4×D9—41no—
131Q8⋊C9—21no—
132Q8⋊C9C23×C311yescharacteristic
133#577—91noSylow
134C2×C3×C9—41no—
135C2×C3×C9—82no—
136C3×D9—42no—
137C3×D9—82no—
138C3×#41—21no—
139C3×#41—31no—
140C3×#41—31no—
141C3×#41—32no—
142C3×#41—32no—
143C3×#41—61no—
144C3×#41—62no—
145C3×#41—92no—
146C3×#41—91no—
147C3×#41—181no—
148C3×#41C2×D911yescharacteristic, complemented
149C3×#41C2×C3×D311yescharacteristic
150C2×C3×Q8—22no—
151C2×C3×Q8—31no—
152C2×C3×Q8—31no—
153C2×C3×Q8—61no—
154C2×C3×Q8C2×D912yescomplemented
155C2×C3×Q8C2×C3×D312yes—
156C2×C3×D4—92no—
157C2×C3×D4—91no—
158C2×C3×D4—182no—
159C2×C3×D4—181no—
160#239—61no—
161D3×D4—62no—
162D3×D4—61no—
163#235—32no—
164#235—31no—
165C2×D12—32no—
166C2×D12—31no—
167C3×Q16—92no—
168C3×Q16—91no—
169C3×SD16—91no—
170C3×SD16—92no—
171C3×SD16—92no—
172C3×SD16—92no—
173C3×SD16—91no—
174C3×SD16—92no—
175C3×D8—92no—
176C3×D8—91no—
177C3×M16—92no—
178C3×M16—91no—
179C2×C3×C8—92no—
180C2×C3×C8—91no—
181#216—32no—
182#216—31no—
183#215—32no—
184#215—32no—
185#215—32no—
186#215—31no—
187#215—32no—
188#214—31no—
189#213—31no—
190#213—32no—
191#208—32no—
192#208—31no—
193C2×C3⋊C8—32no—
194C2×C3⋊C8—31no—
195C22×C32—31no—
196C22×C32C2×S412yes—
197C2×C3×D3—61no—
198C2×C3×D3—62no—
199C2×C3×D3—242no—
200C2×C3×D3—241no—
201C32×C4—31no—
202C32×C4—32no—
203C32×C4C2×S411yescharacteristic
204C3×Dic3—61no—
205C22×C9—42no—
206C22×C9—82no—
207C2×D9—41no—
208C2×D9—42no—
209C2×D9—84no—
210C4×C9—41no—
211C4×C9—81no—
212Dic9—41no—
2132−1+4C3×D911yescharacteristic, complemented
214H(5,𝔽2)—91no—
215#137—91no—
216#137—92no—
217AGL(1,ℤ/8)—92no—
218AGL(1,ℤ/8)—91no—
219#135—92no—
220#135—91no—
221C2×SD16—91no—
222C2×SD16—92no—
223#131—91no—
224C3×C9—41noSylow
225C23×C3—181no—
226C23×C3—182no—
227C22×D3—61no—
228C22×D3—62no—
229C3×Q8—21no—
230C3×Q8—32no—
231C3×Q8—32no—
232C3×Q8—31no—
233C3×Q8—31no—
234C3×Q8—61no—
235C3×Q8—62no—
236C3×Q8—91no—
237C3×Q8C22×D911yescharacteristic
238C3×Q8C22×C3×D311yescharacteristic
239C3×D4—21no—
240C3×D4—31no—
241C3×D4—31no—
242C3×D4—32no—
243C3×D4—32no—
244C3×D4—62no—
245C3×D4—61no—
246C3×D4—92no—
247C3×D4—91no—
248C3×D4—182no—
249C3×D4—181no—
250C3×D4—182no—
251C3×D4—181no—
252C3×D4(C22⋊C9)⋊C211yescharacteristic
253C3×D4C3×S411yescharacteristic
254C2×C3×C4—32no—
255C2×C3×C4—32no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—61no—
259C2×C3×C4—62no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—62no—
263C2×C3×C4—182no—
264C2×C3×C4—181no—
265#67—61no—
266#67—62no—
267D12—32no—
268D12—31no—
269D12—62no—
270D12—61no—
271C4×D3—62no—
272C4×D3—61no—
273Dic6—31no—
274C3×C8—91no—
275C3×C8—91no—
276C3×C8—92no—
277C3⋊C8—31no—
278C3⋊C8—31no—
279C3⋊C8—32no—
280C2×C32—22no—
281C2×C32—61no—
282C2×C32C22×S411yescharacteristic
283C3×D3—121no—
284C3×D3—122no—
285C2×C9—41no—
286C2×C9—82no—
287C2×C9—81no—
288C2×C9—162no—
289D9—42no—
290D9—82no—
291#41—31no—
292#41—32no—
293#41—92no—
294#41—91no—
295#41—181no—
296#41C2×C3×D911yescharacteristic, complemented
297C2×Q8—31no—
298C2×Q8C2×C3×D912yescomplemented
299C2×D4—92no—
300C2×D4—91no—
301C2×D4—182no—
302C2×D4—181no—
303Q16—92no—
304Q16—91no—
305SD16—91no—
306SD16—92no—
307SD16—92no—
308SD16—92no—
309SD16—91no—
310SD16—92no—
311D8—92no—
312D8—91no—
313M16—92no—
314M16—91no—
315C2×C8—92no—
316C2×C8—91no—
317C22×C3—22no—
318C22×C3—31no—
319C22×C3—31no—
320C22×C3—61no—
321C22×C3—182no—
322C22×C3—181no—
323C22×C3—722no—
324C22×C3—721no—
325C22×C3C2×(C22⋊C9)⋊C212yes—
326C22×C3C2×C3×S412yes—
327C2×D3—61no—
328C2×D3—62no—
329C2×D3—242no—
330C2×D3—241no—
331C3×C4—21no—
332C3×C4—31no—
333C3×C4—32no—
334C3×C4—31no—
335C3×C4—32no—
336C3×C4—61no—
337C3×C4—62no—
338C3×C4—181no—
339C3×C4C2×(C22⋊C9)⋊C211yescharacteristic
340C3×C4C2×C3×S411yescharacteristic
341Dic3—61no—
342C32#592611yescharacteristic
343C9—41no—
344C9—81no—
345C23—181no—
346C23—182no—
347Q8—32no—
348Q8—31no—
349Q8—91no—
350Q8C22×C3×D911yescharacteristic
351D4—31no—
352D4—32no—
353D4—92no—
354D4—91no—
355D4—182no—
356D4—181no—
357D4—182no—
358D4—181no—
359D4C3×(C22⋊C9)⋊C211yescharacteristic
360C2×C4—32no—
361C2×C4—31no—
362C2×C4—61no—
363C2×C4—182no—
364C2×C4—181no—
365C8—91no—
366C8—91no—
367C8—92no—
368C2×C3—22no—
369C2×C3—22no—
370C2×C3—21no—
371C2×C3—42no—
372C2×C3—61no—
373C2×C3—61no—
374C2×C3—121no—
375C2×C3—361no—
376C2×C3—362no—
377C2×C3C22×(C22⋊C9)⋊C211yescharacteristic
378C2×C3C22×C3×S411yescharacteristic
379D3—121no—
380D3—122no—
381C22—31no—
382C22—182no—
383C22—181no—
384C22—722no—
385C22—721no—
386C22C2×C3×(C22⋊C9)⋊C212yes—
387C4—31no—
388C4—32no—
389C4—181no—
390C4C2×C3×(C22⋊C9)⋊C211yescharacteristic
391C3—21no—
392C3#1059310111yescharacteristic, direct product factor
393C3C3×#592611yescharacteristic
394C2—22no—
395C2—61no—
396C2—361no—
397C2—362no—
398C2C22×C3×(C22⋊C9)⋊C211yescharacteristic
subgroupG/H
centerC2×C3C22×(C22⋊C9)⋊C2
commutator, cosocle, hypocenterQ8⋊C9C23×C3
FrattiniC2×C3C22×C3×S4
FittingC32×2−1+4D3
socleC2×C32C22×S4
hypercenterC3×D4(C22⋊C9)⋊C2
LCSG ▷ Q8⋊C9
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×2−1+4⋊C9C2semidirect
2. C3×#63692C2semidirect
3. C3×#63691C2semidirect
4. C3×#63689C2semidirect
5. C2×C3×#3689C2semidirect
7. #10593101C3direct (N × H)
9. C3×#41⋊C9C22semidirect
10. C2×C3×Q8⋊C9C22semidirect
11. C3×#3689C22semidirect
12. C3×#3689C22semidirect
13. C3×#3688C22semidirect
26. 2−1+4⋊C9C2×C3semidirect
27. #63692C2×C3semidirect
28. #63691C2×C3semidirect
29. #63689C2×C3semidirect
30. C2×#3689C2×C3semidirect
67. #41⋊C9C22×C3semidirect
69. C2×Q8⋊C9C22×C3semidirect
70. #3689C22×C3semidirect
71. #3689C22×C3semidirect
72. #3688C22×C3semidirect
80. C3×2−1+4D9semidirect
148. C3×#41C2×D9semidirect
154. C2×C3×Q8C2×D9semidirect
213. 2−1+4C3×D9semidirect
296. #41C2×C3×D9semidirect
298. C2×Q8C2×C3×D9semidirect
392. C3#10593101direct (N × H)

Maximal subgroups

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

subgroupindex
C3×2−1+4⋊C92
C3×#636922
C3×#636912
C3×#636892
C2×C3×#36892
C3×#58373
#105931013
C3×D4×D94