Finite Group Databasebeta

← C2×#11724199 #13180830

Subgroups — C2×#11724199 #13180830

Order 1728.

971 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 971 conjugacy classes of subgroups — too many to draw. Only its 118 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×Dic3×A4C211yesmaximal, characteristic
2C2×C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic
3C2×C3×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic
4#11724199C214yesmaximal, direct product factor
5C2×#64259—31nomaximal
6C2×#64204—33nomaximal
7C2×#63964—31nomaximal
8C22×C32×A4C2211yescharacteristic
9C2×#6293—41nomaximal
10C3×Dic3×A4C2212yes—
11C3⋊(C3⋊SL(2,ℤ/4))C2212yes—
12C3×C3⋊SL(2,ℤ/4)C2212yes—
13C23×C3×Dic3D311yescharacteristic
14C2×C3×C4×A4—31no—
15C22×#627—31no—
16C2×Dic3×A4D313yes—
17C2×C3⋊SL(2,ℤ/4)—31no—
18C2×C3⋊SL(2,ℤ/4)—33no—
19C2×C3⋊SL(2,ℤ/4)—61no—
20C2×C3⋊SL(2,ℤ/4)—63no—
21C2×C3⋊SL(2,ℤ/4)D311yescharacteristic
22#64259—34no—
23C2×C3×SL(2,ℤ/4)—33no—
24#64204—312no—
25C2×#3757—31no—
26C2×C3×#217—31no—
27C2×#3724—31no—
28C2×Dic3⋊Dic3—31no—
29C2×Dic3⋊Dic3—31no—
30#63964—34no—
31#63964—34no—
32C2×C32×A4Q812yescomplemented
33C2×C32×A4C2311yescharacteristic
34C2×C3⋊(C3⋊Dic3)—41no—
35C2×C3×C3⋊Dic3—41no—
36C2×C32×Dic3—41no—
37#6293—44no—
38C2×A4⋊Q8—93no—
39C2×#924—91no—
40C2×#878—91no—
41C24×C32C2×D311yescharacteristic
42C22×C3×A4—21no—
43C22×C3×A4—23no—
44C22×C3×A4C2×D313yes—
45C22×C3×A4C2×D311yescharacteristic
46C22×C3⋊Dic3—31no—
47C22×C3⋊Dic3—31no—
48C2×#634—121no—
49C22×C3×Dic3—31no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—31no—
52C22×C3×Dic3—31no—
53C22×C3×Dic3—31no—
54C22×C3×Dic3C2×D312yes—
55C3×C4×A4—32no—
56C2×#627—34no—
57C2×#627—61no—
58C2×#627—62no—
59C2×#627—61no—
60C2×#627—123no—
61Dic3×A4C2×D316yes—
62C3⋊SL(2,ℤ/4)—32no—
63C3⋊SL(2,ℤ/4)—36no—
64C3⋊SL(2,ℤ/4)—62no—
65C3⋊SL(2,ℤ/4)—66no—
66C3⋊SL(2,ℤ/4)C2×D312yes—
67C3×SL(2,ℤ/4)—36no—
68#3757—32no—
69#3757—32no—
70C3×#217—32no—
71C3×#217—32no—
72#3724—34no—
73Dic3⋊Dic3—34no—
74Dic3⋊Dic3—34no—
75C22×C33—41no—
76C32×A4C2×Q811yescharacteristic, complemented
77C3⋊(C3⋊Dic3)—42no—
78C3×C3⋊Dic3—42no—
79C32×Dic3—42no—
80C23×C3×C4—31no—
81C23×Dic3C3⋊D311yescharacteristic
82C22×Dic6—91no—
83C22×Dic6—91no—
84C2×C4×A4—33no—
85C2×SL(2,ℤ/4)—33no—
86C2×SL(2,ℤ/4)—93no—
87C2×SL(2,ℤ/4)—183no—
88A4⋊Q8—912no—
89C2×C3×C22⋊C4—91no—
90C2×#217—31no—
91C2×#217—91no—
92C2×#217—91no—
93C2×#217—181no—
94C2×C4⋊Dic3—91no—
95C2×C4⋊Dic3—91no—
96#924—94no—
97#924—94no—
98C2×Dic3⋊C4—91no—
99C2×Dic3⋊C4—91no—
100C2×Dic3⋊C4—91no—
101C2×Dic3⋊C4—91no—
102#878—94no—
103#878—94no—
104C23×C32—31no—
105C23×C32—32no—
106C23×C32—31no—
107C23×C32Dic612yescomplemented
108C23×C32C22×D311yescharacteristic
109C2×C3×A4—22no—
110C2×C3×A4—21no—
111C2×C3×A4—26no—
112C2×C3×A4—23no—
113C2×C3×A4Dic616yescomplemented
114C2×C3×A4Dic612yescomplemented
115C2×C3×A4C22×D313yes—
116C2×C3×A4C22×D311yescharacteristic
117C2×C32×C4—121no—
118C2×C3⋊Dic3—32no—
119C2×C3⋊Dic3—32no—
120C2×C3⋊Dic3—41no—
121C2×C3⋊Dic3—61no—
122C2×C3⋊Dic3—62no—
123C2×C3⋊Dic3—61no—
124C2×C3⋊Dic3—121no—
125C2×C3⋊Dic3—123no—
126C2×C3⋊Dic3—241no—
127C2×C3⋊Dic3—243no—
128#634—124no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—32no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—43no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—123no—
141C2×C3×Dic3S411yescharacteristic
142#627—64no—
143#627—64no—
144#627—1212no—
145C2×C22⋊Q8—271noSylow
146C2×C33—42no—
147C2×C33—41no—
148C24×C3—21no—
149C24×C3D3211yescharacteristic
150C24×C3C2×C3⋊D311yescharacteristic
151C22×A4—23no—
152C22×A4D3213yes—
153C22×C3×C4—32no—
154C22×C3×C4—91no—
155C22×C3×C4—91no—
156C22×C3×C4—92no—
157C22×C3×C4—91no—
158C22×C3×C4—91no—
159C22×Dic3—31no—
160C22×Dic3—32no—
161C22×Dic3—31no—
162C22×Dic3—31no—
163C22×Dic3—31no—
164C22×Dic3—91no—
165C22×Dic3—91no—
166C22×Dic3—91no—
167C22×Dic3—91no—
168C22×Dic3—181no—
169C22×Dic3—181no—
170C22×Dic3C2×C3⋊D312yes—
171C2×Dic6—94no—
172C2×Dic6—94no—
173C2×Dic6—181no—
174C2×Dic6—182no—
175C2×Dic6—181no—
176C2×Dic6—182no—
177C2×Dic6—181no—
178C2×Dic6—181no—
179C2×Dic6—363no—
180C4×A4—36no—
181SL(2,ℤ/4)—36no—
182SL(2,ℤ/4)—96no—
183SL(2,ℤ/4)—186no—
184C3×C22⋊C4—92no—
185C3×C22⋊C4—92no—
186#217—32no—
187#217—32no—
188#217—92no—
189#217—92no—
190#217—92no—
191#217—92no—
192#217—182no—
193#217—182no—
194C4⋊Dic3—94no—
195C4⋊Dic3—94no—
196Dic3⋊C4—94no—
197Dic3⋊C4—94no—
198Dic3⋊C4—94no—
199Dic3⋊C4—94no—
200C22×C32—32no—
201C22×C32—31no—
202C22×C32—32no—
203C22×C32—32no—
204C22×C32—31no—
205C22×C32—31no—
206C22×C32—41no—
207C22×C32—43no—
208C22×C32—61no—
209C22×C32—83no—
210C22×C32—81no—
211C22×C32C2×Dic611yescharacteristic, complemented
212C22×C32C2×S411yescharacteristic
213C3×A4—21no—
214C3×A4—23no—
215C3×A4C2×Dic613yescomplemented
216C3×A4C2×Dic611yescharacteristic, complemented
217C32×C4—122no—
218C3⋊Dic3—42no—
219C3⋊Dic3—62no—
220C3⋊Dic3—62no—
221C3⋊Dic3—122no—
222C3⋊Dic3—126no—
223C3⋊Dic3—242no—
224C3⋊Dic3—246no—
225C3×Dic3—32no—
226C3×Dic3—46no—
227C3×Dic3—62no—
228C3×Dic3—62no—
229C3×Dic3—126no—
230C3×Dic3C2×S412yes—
231C22×Q8—271no—
232C23×C4—31no—
233C22⋊Q8—274no—
234C22⋊Q8—274no—
235C2×C4⋊C4—271no—
236C2×C4⋊C4—271no—
237C2×C4⋊C4—271no—
238C2×C22⋊C4—91no—
239C2×C22⋊C4—271no—
240C33—41noSylow
241C23×C3—22no—
242C23×C3—21no—
243C23×C3—31no—
244C23×C3—31no—
245C23×C3—31no—
246C23×C3—31no—
247C23×C3—32no—
248C23×C3—32no—
249C23×C3—61no—
250C23×C3—61no—
251C23×C3—62no—
252C23×C3#62712yescomplemented
253C23×C3#63412yescomplemented
254C23×C3C2×D3211yescharacteristic
255C23×C3C22×C3⋊D311yescharacteristic
256C2×A4—26no—
257C2×A4—23no—
258C2×A4#62716yescomplemented
259C2×A4C2×D3213yes—
260C2×C3×C4—31no—
261C2×C3×C4—92no—
262C2×C3×C4—92no—
263C2×C3×C4—91no—
264C2×C3×C4—92no—
265C2×C3×C4—92no—
266C2×C3×C4—92no—
267C2×C3×C4—123no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—182no—
271C2×C3×C4—181no—
272C2×Dic3—31no—
273C2×Dic3—32no—
274C2×Dic3—32no—
275C2×Dic3—32no—
276C2×Dic3—32no—
277C2×Dic3—32no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—62no—
281C2×Dic3—61no—
282C2×Dic3—92no—
283C2×Dic3—92no—
284C2×Dic3—92no—
285C2×Dic3—92no—
286C2×Dic3—123no—
287C2×Dic3—182no—
288C2×Dic3—182no—
289C2×Dic3—181no—
290C2×Dic3—181no—
291C2×Dic3—182no—
292C2×Dic3—182no—
293C2×Dic3—181no—
294C2×Dic3—181no—
295C2×Dic3—361no—
296C2×Dic3—363no—
297C2×Dic3—361no—
298C2×Dic3—362no—
299C2×Dic3—723no—
300C2×Dic3C3⋊S411yescharacteristic
301Dic6—184no—
302Dic6—184no—
303Dic6—184no—
304Dic6—184no—
305Dic6—3612no—
306C2×C32—31no—
307C2×C32—31no—
308C2×C32—32no—
309C2×C32—41no—
310C2×C32—42no—
311C2×C32—46no—
312C2×C32—43no—
313C2×C32—83no—
314C2×C32—86no—
315C2×C32—81no—
316C2×C32—82no—
317C2×C32A4⋊Q812yescomplemented
318C2×C32C22×S411yescharacteristic
319C24D3×C3⋊D311yescharacteristic
320C2×Q8—274no—
321C2×Q8—541no—
322C2×Q8—542no—
323C2×Q8—541no—
324C22×C4—32no—
325C22×C4—91no—
326C22×C4—91no—
327C22×C4—92no—
328C22×C4—91no—
329C22×C4—91no—
330C22×C4—271no—
331C22×C4—271no—
332C4⋊C4—274no—
333C4⋊C4—274no—
334C4⋊C4—274no—
335C22⋊C4—92no—
336C22⋊C4—92no—
337C22⋊C4—272no—
338C22⋊C4—272no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—32no—
342C22×C3—32no—
343C22×C3—31no—
344C22×C3—31no—
345C22×C3—32no—
346C22×C3—32no—
347C22×C3—31no—
348C22×C3—31no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—32no—
352C22×C3—32no—
353C22×C3—43no—
354C22×C3—61no—
355C22×C3—62no—
356C22×C3—61no—
357C22×C3—62no—
358C22×C3—62no—
359C22×C3—61no—
360C22×C3—61no—
361C22×C3—61no—
362C22×C3—83no—
363C22×C3—121no—
364C22×C3C2×#62711yescharacteristic, complemented
365C22×C3C2×#63411yescharacteristic, complemented
366C22×C3D3×S411yescharacteristic
367C22×C3C2×C3⋊S411yescharacteristic
368A4—23no—
369A4C2×#62713yescomplemented
370C3×C4—32no—
371C3×C4—92no—
372C3×C4—126no—
373C3×C4—182no—
374C3×C4—182no—
375Dic3—32no—
376Dic3—62no—
377Dic3—62no—
378Dic3—126no—
379Dic3—182no—
380Dic3—182no—
381Dic3—182no—
382Dic3—182no—
383Dic3—362no—
384Dic3—366no—
385Dic3—362no—
386Dic3—726no—
387Dic3C2×C3⋊S412yes—
388C32—41no—
389C32—43no—
390C32—83no—
391C32—81no—
392C32C2×A4⋊Q811yescharacteristic, complemented
393C23—31no—
394C23—31no—
395C23—32no—
396C23#629312yescomplemented
397C23C2×D3×C3⋊D311yescharacteristic
398Q8—544no—
399Q8—544no—
400C2×C4—31no—
401C2×C4—92no—
402C2×C4—92no—
403C2×C4—91no—
404C2×C4—92no—
405C2×C4—92no—
406C2×C4—92no—
407C2×C4—181no—
408C2×C4—181no—
409C2×C4—182no—
410C2×C4—181no—
411C2×C4—272no—
412C2×C4—272no—
413C2×C4—541no—
414C2×C4—542no—
415C2×C4—541no—
416C2×C3—22no—
417C2×C3—21no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—32no—
421C2×C3—31no—
422C2×C3—31no—
423C2×C3—32no—
424C2×C3—46no—
425C2×C3—43no—
426C2×C3—61no—
427C2×C3—61no—
428C2×C3—62no—
429C2×C3—86no—
430C2×C3—83no—
431C2×C3#6420412yescomplemented
432C2×C3#6425912yescomplemented
433C2×C3C2×D3×S411yescharacteristic
434C2×C3C22×C3⋊S411yescharacteristic
435C22—32no—
436C22—31no—
437C22—32no—
438C22—32no—
439C22—31no—
440C22—31no—
441C22—61no—
442C22C2×#629311yescharacteristic, complemented
443C22D3×C3⋊S411yescharacteristic
444C4—32no—
445C4—92no—
446C4—182no—
447C4—182no—
448C4—542no—
449C4—542no—
450C3—21no—
451C3—43no—
452C3—83no—
453C3C2×#6420411yescharacteristic, complemented
454C3C2×#6425911yescharacteristic, complemented
455C2—31no—
456C2—31no—
457C2—32no—
458C2#1172419912yesdirect product factor
459C2C2×D3×C3⋊S411yescharacteristic
subgroupG/H
center, hypercenterC22D3×C3⋊S4
commutator, cosocleC2×C32×A4C23
FrattiniC2C2×D3×C3⋊S4
Fitting, socleC24×C32C2×D3
prime radicalC22×C32×A4C22
hypocenterC32×A4C2×Q8
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
4. #11724199C2direct (N × H)
32. C2×C32×A4Q8semidirect
76. C32×A4C2×Q8semidirect
107. C23×C32Dic6semidirect
113. C2×C3×A4Dic6semidirect
114. C2×C3×A4Dic6semidirect
211. C22×C32C2×Dic6semidirect
215. C3×A4C2×Dic6semidirect
216. C3×A4C2×Dic6semidirect
252. C23×C3#627semidirect
253. C23×C3#634semidirect
258. C2×A4#627semidirect
317. C2×C32A4⋊Q8semidirect
364. C22×C3C2×#627semidirect
365. C22×C3C2×#634semidirect
369. A4C2×#627semidirect
392. C32C2×A4⋊Q8semidirect
396. C23#6293semidirect
431. C2×C3#64204semidirect
432. C2×C3#64259semidirect
442. C22C2×#6293semidirect
453. C3C2×#64204semidirect
454. C3C2×#64259semidirect
458. C2#11724199direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Dic3×A42
C2×C3⋊(C3⋊SL(2,ℤ/4))2
C2×C3×C3⋊SL(2,ℤ/4)2
#117241992
C2×#642593
C2×#642043
C2×#639643
C2×#62934