Finite Group Databasebeta

← C3×#10591003 #13158020

Subgroups — C3×#10591003 #13158020

Order 1728.

748 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 144 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×#3797C211yesmaximal, characteristic
2C3×M16×C3⋊D3C211yesmaximal, characteristic
3C3×#63818C211yesmaximal, characteristic
4C32×#824C212yesmaximal, complemented
5C3×#63560C212yesmaximal, complemented
6C3×D3×#101—32nomaximal
7#10591003C311yesmaximal, characteristic, direct product factor
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×Dic6C2212yes—
10C2×C3×#627C412yes—
11C2×C3×#624C412yes—
12C3×#3797—21no—
13C3×#3797—21no—
14C3×#3797—21no—
15C3×#3797—21no—
16C33×M16C2211yescharacteristic
17C3×#3748C2211yescharacteristic
18C3×#3743—21no—
19C3×C8×C3⋊D3—21no—
20C32×#208C2212yes—
21C3×#3712—21no—
22C3×#3710—21no—
23C2×C3×D3×Q8—32no—
24C2×#3797C2×C311yescharacteristic
25M16×C3⋊D3C2×C311yescharacteristic
26C3×D3×M16—32no—
27C3×D3×M16—32no—
28C3×D3×M16—61no—
29#63818C2×C311yescharacteristic
30C32×#101—32no—
31C3×#836—32no—
32C3×#824—22no—
33C3×#824—32no—
34C3×#824D312yescomplemented
35C3×#824C2×C312yescomplemented
36#63560C2×C312yescomplemented
37C22×C3×C3⋊D3C2×C411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C3×C3⋊Dic3C2×C411yescharacteristic
40C3×C4×C3⋊D3D411yescharacteristic
41C3×C4×C3⋊D3D411yescharacteristic
42C3×C4×C3⋊D3D411yescharacteristic
43C3×C4×C3⋊D3D411yescharacteristic
44C2×C32×Dic3C2×C414yes—
45C32×Dic6—22no—
46C32×Dic6—22no—
47C3×#627—22no—
48C3×#627—22no—
49C3×#624—22no—
50C3×#624—22no—
51C33×C8—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C32×C3⋊C8—22no—
54C2×C3×#101—91no—
55D3×#101—32no—
56C2×C32×Q8—32no—
57C2×C4×C3⋊D3C22×C311yescharacteristic
58C3×D3×Q8—62no—
59C3×D3×Q8—62no—
60C3×D3×Q8—62no—
61C3×D3×Q8—62no—
62C2×C3×C4×D3—32no—
63C2×C3×C4×D3—34no—
64C2×C3×C4×D3—61no—
65C2×C3×Dic6—22no—
66C2×C3×Dic6—34no—
67C2×C3×Dic6C2×D312yes—
68C2×C3×Dic6C22×C312yes—
69C2×#627C3×C412yes—
70C2×#624C3×C412yes—
71#3797—21no—
72#3797—21no—
73#3797—21no—
74#3797—21no—
75C32×M16—21no—
76C32×M16—22no—
77C32×M16—32no—
78C32×M16—41no—
79C32×M16C2×D312yes—
80C32×M16C22×C311yescharacteristic
81#3748C22×C311yescharacteristic
82#3743—21no—
83C8×C3⋊D3—21no—
84C3×#208—22no—
85C3×#208—32no—
86C3×#208—61no—
87C3×#208C2×D312yes—
88C3×#208C22×C312yes—
89C3×ΓL(1,𝔽25)—62no—
90C3×ΓL(1,𝔽25)—62no—
91C3×ΓL(1,𝔽25)—62no—
92C3×D3×C8—62no—
93C3×D3×C8—62no—
94C3×D3×C8—62no—
95#3712—21no—
96#3710—21no—
97C22×C33C22×C411yescharacteristic
98C2×C3×C3⋊D3—22no—
99C2×C3×C3⋊D3C22⋊C411yescharacteristic
100C2×C3×C3⋊D3C22⋊C411yescharacteristic
101C33×C4C2×D411yescharacteristic
102C33×C4C2×D411yescharacteristic
103C3×C3⋊Dic3C22⋊C411yescharacteristic
104C3×C3⋊Dic3C22⋊C411yescharacteristic
105C32×Dic3—24no—
106C22×C3×Q8—91no—
107C2×D3×Q8—32no—
108C2×C3×M16—91no—
109C2×C3×M16—91no—
110D3×M16—32no—
111D3×M16—32no—
112D3×M16—61no—
113C3×#101—32no—
114C3×#101—32no—
115C3×#101—62no—
116C3×#101—92no—
117#836—32no—
118#824—32no—
119#824C3×D312yescomplemented
120C22×C3⋊D3C2×C3×C411yescharacteristic
121C22×C3×D3—32no—
122C22×C3×D3—61no—
123C32×Q8—62no—
124C32×Q8—62no—
125C2×C32×C4—21no—
126C2×C32×C4—22no—
127C2×C32×C4—34no—
128C2×C32×C4—41no—
129C2×C32×C4C22×D312yes—
130C2×C32×C4C23×C311yescharacteristic
131C2×C3⋊Dic3C2×C3×C411yescharacteristic
132C4×C3⋊D3C3×D411yescharacteristic
133C4×C3⋊D3C3×D411yescharacteristic
134C4×C3⋊D3C3×D411yescharacteristic
135C4×C3⋊D3C3×D411yescharacteristic
136C2×C3×Dic3—24no—
137C2×C3×Dic3—32no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3C4×D314yes—
140C2×C3×Dic3C2×C3×C414yes—
141C3×C4×D3—32no—
142C3×C4×D3—32no—
143C3×C4×D3—32no—
144C3×C4×D3—32no—
145C3×C4×D3—61no—
146C3×C4×D3—61no—
147C3×C4×D3—61no—
148C3×C4×D3—61no—
149C3×C4×D3—64no—
150C3×C4×D3—64no—
151C3×Dic6—22no—
152C3×Dic6—22no—
153C3×Dic6—22no—
154C3×Dic6—22no—
155C3×Dic6—42no—
156C3×Dic6—42no—
157C3×Dic6—64no—
158C3×Dic6—64no—
159#627—22no—
160#627—22no—
161#624—22no—
162#624—22no—
163C32×C8—21no—
164C32×C8—22no—
165C32×C8—22no—
166C32×C8—42no—
167C32×C8—42no—
168C32×C8—62no—
169C3⋊(C3⋊C8)—21no—
170C3×C3⋊C8—22no—
171C3×C3⋊C8—22no—
172C3×C3⋊C8—42no—
173C3×C3⋊C8—62no—
174C3×C3⋊C8—62no—
175C2×#101—91noSylow
176C2×C33—21no—
177C2×C33C2×C22⋊C411yescharacteristic
178C3×C3⋊D3—21no—
179C3×C3⋊D3#10112yescomplemented
180C2×C3×Q8—32no—
181C2×C3×Q8—32no—
182C2×C3×Q8—62no—
183C2×C3×Q8—92no—
184C2×C3×Q8—181no—
185C2×C3×Q8—181no—
186C2×C3×Q8—181no—
187C2×C3×Q8—181no—
188C22×C3×C4—92no—
189C22×C3×C4—91no—
190D3×Q8—62no—
191D3×Q8—62no—
192D3×Q8—62no—
193D3×Q8—62no—
194C2×C4×D3—32no—
195C2×C4×D3—34no—
196C2×C4×D3—61no—
197C2×Dic6—34no—
198C2×Dic6C2×C3×D312yes—
199C3×M16—21no—
200C3×M16—22no—
201C3×M16—32no—
202C3×M16—32no—
203C3×M16—41no—
204C3×M16—62no—
205C3×M16—91no—
206C3×M16—181no—
207C3×M16—181no—
208C3×M16D3211yescharacteristic
209C3×M16C2×C3×D312yes—
210C2×C3×C8—181no—
211C2×C3×C8—181no—
212#208—32no—
213#208—61no—
214#208C2×C3×D312yes—
215ΓL(1,𝔽25)—62no—
216ΓL(1,𝔽25)—62no—
217ΓL(1,𝔽25)—62no—
218D3×C8—62no—
219D3×C8—62no—
220D3×C8—62no—
221C22×C32—21no—
222C22×C32—22no—
223C22×C32—41no—
224C22×C32C2×C4×D312yes—
225C22×C32C22×C3×C411yescharacteristic
226C2×C3⋊D3—22no—
227C2×C3⋊D3C3×C22⋊C411yescharacteristic
228C2×C3⋊D3C3×C22⋊C411yescharacteristic
229C2×C3×D3—32no—
230C2×C3×D3—32no—
231C2×C3×D3—61no—
232C2×C3×D3—61no—
233C2×C3×D3—64no—
234C2×C3×D3—122no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—22no—
238C32×C4—22no—
239C32×C4—41no—
240C32×C4—41no—
241C32×C4—64no—
242C32×C4D3×D412yes—
243C32×C4D3×D412yes—
244C32×C4C2×C3×D411yescharacteristic
245C32×C4C2×C3×D411yescharacteristic
246C3⋊Dic3C3×C22⋊C411yescharacteristic
247C3⋊Dic3C3×C22⋊C411yescharacteristic
248C3×Dic3—24no—
249C3×Dic3—24no—
250C3×Dic3—32no—
251C3×Dic3—32no—
252C3×Dic3—44no—
253C3×Dic3—61no—
254C3×Dic3—61no—
255C22×Q8—91no—
256C2×M16—91no—
257C2×M16—91no—
258#101—32no—
259#101—92no—
260C33C2×#10111yescharacteristic, complemented, Sylow
261C23×C3—91no—
262C22×D3—32no—
263C22×D3—61no—
264C3×Q8—62no—
265C3×Q8—62no—
266C3×Q8—62no—
267C3×Q8—62no—
268C3×Q8—122no—
269C3×Q8—122no—
270C3×Q8—182no—
271C3×Q8—182no—
272C2×C3×C4—22no—
273C2×C3×C4—21no—
274C2×C3×C4—34no—
275C2×C3×C4—34no—
276C2×C3×C4—41no—
277C2×C3×C4—64no—
278C2×C3×C4—91no—
279C2×C3×C4—91no—
280C2×C3×C4—91no—
281C2×C3×C4—91no—
282C2×C3×C4—91no—
283C2×C3×C4—182no—
284C2×C3×C4—182no—
285C2×C3×C4C2×D3211yescharacteristic
286C2×C3×C4C22×C3×D312yes—
287C2×Dic3—32no—
288C2×Dic3—61no—
289C2×Dic3C3×C4×D314yes—
290C4×D3—32no—
291C4×D3—32no—
292C4×D3—32no—
293C4×D3—32no—
294C4×D3—61no—
295C4×D3—61no—
296C4×D3—61no—
297C4×D3—61no—
298C4×D3—64no—
299C4×D3—64no—
300Dic6—22no—
301Dic6—22no—
302Dic6—64no—
303Dic6—64no—
304C3×C8—22no—
305C3×C8—21no—
306C3×C8—22no—
307C3×C8—42no—
308C3×C8—42no—
309C3×C8—62no—
310C3×C8—62no—
311C3×C8—122no—
312C3×C8—181no—
313C3⋊C8—22no—
314C3⋊C8—62no—
315C3⋊C8—62no—
316C2×C32—22no—
317C2×C32—21no—
318C2×C32—22no—
319C2×C32—21no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—42no—
323C2×C32—81no—
324C2×C32D3×C22⋊C412yes—
325C2×C32C2×C3×C22⋊C411yescharacteristic
326C3⋊D3—21no—
327C3⋊D3C3×#10112yescomplemented
328C3×D3—34no—
329C3×D3—62no—
330C3×D3—62no—
331C3×D3—121no—
332C2×Q8—32no—
333C2×Q8—92no—
334C2×Q8—181no—
335C2×Q8—181no—
336C2×Q8—181no—
337C2×Q8—181no—
338C22×C4—92no—
339C22×C4—91no—
340M16—32no—
341M16—91no—
342M16—181no—
343M16—181no—
344M16C3×D3211yescharacteristic
345C2×C8—181no—
346C2×C8—181no—
347C22×C3—21no—
348C22×C3—22no—
349C22×C3—41no—
350C22×C3—91no—
351C22×C3—91no—
352C22×C3—182no—
353C22×C3C4×D3211yescharacteristic
354C22×C3C2×C3×C4×D312yes—
355C2×D3—32no—
356C2×D3—32no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—64no—
360C2×D3—122no—
361C3×C4—22no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—22no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—64no—
368C3×C4—64no—
369C3×C4—91no—
370C3×C4—91no—
371C3×C4—124no—
372C3×C4#380211yescharacteristic
373C3×C4#381111yescharacteristic
374C3×C4C3×D3×D412yes—
375C3×C4C3×D3×D412yes—
376Dic3—24no—
377Dic3—32no—
378Dic3—32no—
379Dic3—61no—
380Dic3—61no—
381C32—21no—
382C32—22no—
383C32—41no—
384C32D3×#10112yescomplemented
385C32C2×C3×#10111yescharacteristic, complemented
386C23—91no—
387Q8—62no—
388Q8—62no—
389Q8—182no—
390Q8—182no—
391C2×C4—34no—
392C2×C4—91no—
393C2×C4—91no—
394C2×C4—91no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—182no—
398C2×C4—182no—
399C2×C4C2×C3×D3211yescharacteristic
400C8—21no—
401C8—62no—
402C8—181no—
403C2×C3—21no—
404C2×C3—22no—
405C2×C3—21no—
406C2×C3—22no—
407C2×C3—41no—
408C2×C3—42no—
409C2×C3—41no—
410C2×C3—81no—
411C2×C3—92no—
412C2×C3—181no—
413C2×C3#6392111yescharacteristic
414C2×C3C3×D3×C22⋊C412yes—
415D3—34no—
416D3—62no—
417D3—62no—
418D3—121no—
419C22—91no—
420C22—91no—
421C22—182no—
422C22C3×C4×D3211yescharacteristic
423C4—64no—
424C4—91no—
425C4—91no—
426C4C3×#380211yescharacteristic
427C4C3×#381111yescharacteristic
428C3—21no—
429C3—22no—
430C3—41no—
431C3#1059100311yescharacteristic, direct product factor
432C3C3×D3×#10112yescomplemented
433C2—21no—
434C2—92no—
435C2—181no—
436C2C3×#6392111yescharacteristic
subgroupG/H
centerC2×C3#63921
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×M16C22
socleC2×C33C2×C22⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC3×M16D32
hypocenterC32C2×C3×#101
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
4. C32×#824C2semidirect
5. C3×#63560C2semidirect
7. #10591003C3direct (N × H)
34. C3×#824D3semidirect
35. C3×#824C2×C3semidirect
36. #63560C2×C3semidirect
119. #824C3×D3semidirect
179. C3×C3⋊D3#101semidirect
260. C33C2×#101semidirect
327. C3⋊D3C3×#101semidirect
384. C32D3×#101semidirect
385. C32C2×C3×#101semidirect
431. C3#10591003direct (N × H)
432. C3C3×D3×#101semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37972
C3×M16×C3⋊D32
C3×#638182
C32×#8242
C3×#635602
C3×D3×#1013
#105910033