Finite Group Databasebeta

← C3×#10591189 #13158369

Subgroups — C3×#10591189 #13158369

Order 1728.

710 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 132 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×C3×#3748C211yesmaximal, characteristic
2C32×#924C212yesmaximal
3C2×C3×#3718C211yesmaximal, characteristic, complemented
4C3×#63574C211yesmaximal, characteristic, complemented
5C3×#63565C212yesmaximal, complemented
6C3×#5052—32nomaximal
7#10591189C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C2×C32×Dic6C2212yes—
10C3×#3748—22no—
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C32×#217—22no—
14C32×C4⋊Dic3C2212yes—
15C32×Dic3⋊C4—22no—
16C3×#3718—22no—
17C32×C22⋊Q8—32no—
18C2×#3748C2×C311yescharacteristic
19C2×C3×#216—32no—
20C3×#924—22no—
21C3×#924D312yes—
22C3×#924C2×C312yes—
23C2×C3×#208—32no—
24C2×C3×#208—61no—
25C2×#3718C2×C311yescharacteristic, complemented
26C3×#835—32no—
27C3×#810—32no—
28C3×#808—32no—
29#63574C2×C311yescharacteristic, complemented
30#63565C2×C312yescomplemented
31C23×C33D411yescharacteristic
32C2×C33×C4—21no—
33C2×C33×C4—21no—
34C2×C33×C4D411yescharacteristic
35C2×C33×C4C2311yescharacteristic
36C2×C32×Dic3—22no—
37C2×C32×Dic3—22no—
38C32×Dic6—22no—
39C32×Dic6—22no—
40C3×C3⋊(C3⋊C8)—21no—
41C3×C3⋊(C3⋊C8)D412yes—
42C3×#470—91no—
43#5052—32no—
44C2×C32×Q8—32no—
45C22×C32×C4—21no—
46C22×C32×C4—22no—
47C22×C32×C4—41no—
48C22×C32×C4C2×D312yes—
49C22×C32×C4C22×C311yescharacteristic
50C2×C3×Dic6—22no—
51C2×C3×Dic6C2×D312yes—
52C2×C3×Dic6C22×C312yes—
53C32×C4⋊C4—32no—
54C32×C4⋊C4—62no—
55C32×C22⋊C4—62no—
56#3748—22no—
57C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
58C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
59C3×#217—24no—
60C3×#217—22no—
61C3×#217—22no—
62C3×#216—64no—
63C3×C4⋊Dic3—22no—
64C3×C4⋊Dic3C2×D312yes—
65C3×C4⋊Dic3C22×C312yes—
66C3×Dic3⋊C4—24no—
67C3×Dic3⋊C4—22no—
68C3×Dic3⋊C4—22no—
69C3×#208—64no—
70C3×#208—64no—
71C2×C3×C3⋊C8—32no—
72C2×C3×C3⋊C8—32no—
73C2×C3×C3⋊C8—61no—
74C2×C3×C3⋊C8—61no—
75#3718—22no—
76C22×C33—21no—
77C22×C33—22no—
78C22×C33C2×D411yescharacteristic
79C33×C4—21no—
80C33×C4C2×D411yescharacteristic
81C33×C4#4111yescharacteristic
82C32×Dic3—42no—
83C32×Dic3—42no—
84C2×C3×Q16—91no—
85C2×C3×M16—91no—
86C3×C22⋊Q8—32no—
87C3×C22⋊Q8—32no—
88C3×C22⋊Q8—62no—
89C2×#216—32no—
90#924C3×D312yes—
91C2×#208—32no—
92C2×#208—61no—
93C3×#106—91no—
94C3×#103—92no—
95#835—32no—
96#810—32no—
97#808—32no—
98C23×C32—21no—
99C23×C32—22no—
100C23×C32—41no—
101C23×C32#6712yes—
102C23×C32C3×D411yescharacteristic
103C32×Q8—62no—
104C32×Q8—62no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—22no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—21no—
112C2×C32×C4—22no—
113C2×C32×C4—21no—
114C2×C32×C4—22no—
115C2×C32×C4—41no—
116C2×C32×C4—41no—
117C2×C32×C4—42no—
118C2×C32×C4—42no—
119C2×C32×C4—42no—
120C2×C32×C4—42no—
121C2×C32×C4—62no—
122C2×C32×C4—62no—
123C2×C32×C4#6712yes—
124C2×C32×C4C3×D411yescharacteristic
125C2×C32×C4C22×D312yes—
126C2×C32×C4C23×C311yescharacteristic
127C2×C3×Dic3—24no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—22no—
133C3×Dic6—22no—
134C3×Dic6—22no—
135C3×Dic6—22no—
136C3×Dic6—22no—
137C3×Dic6—42no—
138C3×Dic6—42no—
139C3⋊(C3⋊C8)—21no—
140C3⋊(C3⋊C8)C3×D412yes—
141C3×C3⋊C8—34no—
142C3×C3⋊C8—62no—
143C3×C3⋊C8—62no—
144C3×C3⋊C8—62no—
145#470—91noSylow
146C2×C33—41no—
147C2×C33#12111yescharacteristic
148C2×C33#13712yes—
149C2×C3×Q8—32no—
150C2×C3×Q8—32no—
151C2×C3×Q8—62no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—41no—
155C22×C3×C4D3211yescharacteristic
156C22×C3×C4C2×C3×D312yes—
157C2×Dic6C2×C3×D312yes—
158C3×Q16—182no—
159C3×M16—182no—
160C2×C3×C8—91no—
161C2×C3×C8—91no—
162C3×C4⋊C4—32no—
163C3×C4⋊C4—32no—
164C3×C4⋊C4—62no—
165C3×C4⋊C4—64no—
166C3×C4⋊C4—62no—
167C3×C4⋊C4—62no—
168C3×C22⋊C4—64no—
169C3×C22⋊C4—62no—
170C3×C22⋊C4—62no—
171#217—22no—
172#216—64no—
173C4⋊Dic3C2×C3×D312yes—
174Dic3⋊C4—22no—
175#208—64no—
176#208—64no—
177C2×C3⋊C8—32no—
178C2×C3⋊C8—32no—
179C2×C3⋊C8—61no—
180C2×C3⋊C8—61no—
181C22×C32—22no—
182C22×C32—21no—
183C22×C32—22no—
184C22×C32—24no—
185C22×C32—22no—
186C22×C32—22no—
187C22×C32—22no—
188C22×C32—21no—
189C22×C32—22no—
190C22×C32—41no—
191C22×C32—42no—
192C22×C32—42no—
193C22×C32—42no—
194C22×C32—42no—
195C22×C32—44no—
196C22×C32C2×#6712yes—
197C22×C32C2×C3×D411yescharacteristic
198C32×C4—21no—
199C32×C4—22no—
200C32×C4—22no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—22no—
204C32×C4—22no—
205C32×C4—41no—
206C32×C4—41no—
207C32×C4—42no—
208C32×C4—42no—
209C32×C4—122no—
210C32×C4—122no—
211C32×C4D3×D412yes—
212C32×C4#23712yes—
213C32×C4C2×C3×D411yescharacteristic
214C32×C4C3×#4111yescharacteristic
215C3×Dic3—44no—
216C3×Dic3—44no—
217C3×Dic3—42no—
218C3×Dic3—42no—
219C3×Dic3—42no—
220C3×Dic3—42no—
221C2×Q16—91no—
222C2×M16—91no—
223C22⋊Q8—32no—
224#106—91no—
225#103—92no—
226C33#47011yescharacteristic, complemented, Sylow
227C23×C3—22no—
228C23×C3—21no—
229C23×C3—41no—
230C23×C3#62511yescharacteristic
231C23×C3C3×#6712yes—
232C3×Q8—62no—
233C3×Q8—62no—
234C3×Q8—62no—
235C3×Q8—62no—
236C3×Q8—122no—
237C3×Q8—122no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—21no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—21no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—64no—
255C2×C3×C4—64no—
256C2×C3×C4—62no—
257C2×C3×C4—62no—
258C2×C3×C4—62no—
259C2×C3×C4—62no—
260C2×C3×C4#62511yescharacteristic
261C2×C3×C4C3×#6712yes—
262C2×C3×C4C2×D3211yescharacteristic
263C2×C3×C4C22×C3×D312yes—
264C2×Dic3—22no—
265C2×Dic3—22no—
266Dic6—22no—
267Dic6—22no—
268C3×C8—92no—
269C3×C8—181no—
270C3⋊C8—34no—
271C3⋊C8—62no—
272C3⋊C8—62no—
273C3⋊C8—62no—
274C2×C32—22no—
275C2×C32—22no—
276C2×C32—22no—
277C2×C32—22no—
278C2×C32—21no—
279C2×C32—41no—
280C2×C32—42no—
281C2×C32—44no—
282C2×C32—42no—
283C2×C32—44no—
284C2×C32—42no—
285C2×C32—41no—
286C2×C32#93912yes—
287C2×C32#94212yes—
288C2×C32#95112yes—
289C2×C32C3×#12111yescharacteristic
290C2×C32C3×#13712yes—
291C2×Q8—32no—
292C22×C4C3×D3211yescharacteristic
293Q16—182no—
294M16—182no—
295C2×C8—91no—
296C2×C8—91no—
297C4⋊C4—32no—
298C4⋊C4—62no—
299C22⋊C4—62no—
300C22×C3—22no—
301C22×C3—24no—
302C22×C3—22no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—21no—
307C22×C3—22no—
308C22×C3—21no—
309C22×C3—42no—
310C22×C3—42no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—44no—
314C22×C3—41no—
315C22×C3C2×#62511yescharacteristic
316C22×C3C2×C3×#6712yes—
317C3×C4—22no—
318C3×C4—22no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—22no—
322C3×C4—21no—
323C3×C4—22no—
324C3×C4—42no—
325C3×C4—42no—
326C3×C4—41no—
327C3×C4—41no—
328C3×C4—124no—
329C3×C4—124no—
330C3×C4—122no—
331C3×C4—122no—
332C3×C4—122no—
333C3×C4—122no—
334C3×C4#380511yescharacteristic
335C3×C4#381111yescharacteristic
336C3×C4C3×D3×D412yes—
337C3×C4C3×#23712yes—
338Dic3—42no—
339Dic3—42no—
340C32—22no—
341C32—21no—
342C32—41no—
343C32#505212yescomplemented
344C32C3×#47011yescharacteristic, complemented
345C23C3×#62511yescharacteristic
346Q8—62no—
347Q8—62no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—62no—
351C2×C4—62no—
352C2×C4C3×#62511yescharacteristic
353C2×C4C2×C3×D3211yescharacteristic
354C8—92no—
355C8—181no—
356C2×C3—21no—
357C2×C3—22no—
358C2×C3—22no—
359C2×C3—22no—
360C2×C3—22no—
361C2×C3—44no—
362C2×C3—42no—
363C2×C3—44no—
364C2×C3—42no—
365C2×C3—41no—
366C2×C3—41no—
367C2×C3—42no—
368C2×C3#6382712yes—
369C2×C3#6398011yescharacteristic
370C2×C3C3×#93912yes—
371C2×C3C3×#94212yes—
372C2×C3C3×#95112yes—
373C22—21no—
374C22—22no—
375C22C2×C3×#62511yescharacteristic
376C4—21no—
377C4—122no—
378C4—122no—
379C4C3×#380511yescharacteristic
380C4C3×#381111yescharacteristic
381C3—22no—
382C3—21no—
383C3—41no—
384C3#1059118911yescharacteristic, direct product factor
385C3C3×#505212yescomplemented
386C2—41no—
387C2C3×#6382712yes—
388C2C3×#6398011yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC22×C3×C4D32
hypocenterC32C3×#470
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×#3718C2semidirect
4. C3×#63574C2semidirect
5. C3×#63565C2semidirect
7. #10591189C3direct (N × H)
25. C2×#3718C2×C3semidirect
29. #63574C2×C3semidirect
30. #63565C2×C3semidirect
226. C33#470semidirect
343. C32#5052semidirect
344. C32C3×#470semidirect
384. C3#10591189direct (N × H)
385. C3C3×#5052semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37482
C32×#9242
C2×C3×#37182
C3×#635742
C3×#635652
C3×#50523
#105911893