Finite Group Databasebeta

← C2×#11724215 #13180876

Subgroups — C2×#11724215 #13180876

Order 1728.

1370 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 1370 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C22×A4×C3⋊D3C211yesmaximal, characteristic
2C2×C3×C3⋊SL(2,ℤ/4)C212yesmaximal, complemented
3#11724215C214yesmaximal, direct product factor
4C2×D3×SL(2,ℤ/4)—32nomaximal
5C2×#63974—31nomaximal
6C22×C32×A4C2211yescharacteristic
7C2×A4×C3⋊D3C414yescomplemented
8C2×A4×C3⋊D3C2212yes—
9C2×#6294—41nomaximal
10C3×C3⋊SL(2,ℤ/4)C2214yescomplemented
11C24×C3⋊D3D311yescharacteristic
12C22×D3×A4—32no—
13C22×D3×A4—61no—
14C22×#624—31no—
15C22×#624—31no—
16C2×C3⋊SL(2,ℤ/4)D312yescomplemented
17C2×C3×SL(2,ℤ/4)—32no—
18C2×C3×SL(2,ℤ/4)—62no—
19D3×SL(2,ℤ/4)—38no—
20C2×C3×#217—32no—
21#63974—34no—
22#63974—34no—
23C2×#3722—32no—
24C22×C3×C3⋊D3—41no—
25C2×C32×A4C2×C412yescomplemented
26C2×C32×A4C2311yescharacteristic
27A4×C3⋊D3C2×C414yescomplemented
28C2×C3×C3⋊Dic3—42no—
29#6294—44no—
30C22×SL(2,ℤ/4)—91no—
31C2×D3×C22⋊C4—92no—
32C24×C32C2×D311yescharacteristic
33C23×C3⋊D3—31no—
34C23×C3⋊D3—34no—
35C23×C3⋊D3—31no—
36C23×C3⋊D3—32no—
37C23×C3⋊D3Dic314yescomplemented
38C23×C3⋊D3C2×D312yes—
39C22×C3×A4—22no—
40C22×C3×A4—21no—
41C22×C3×A4—41no—
42C22×C3×A4C2×D312yes—
43C2×D3×A4—38no—
44C2×D3×A4—34no—
45C2×D3×A4—64no—
46C2×D3×A4—62no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—32no—
49C2×#624—34no—
50C2×#624—34no—
51C2×#624—64no—
52C2×#624—61no—
53C2×#624—62no—
54C2×#624—61no—
55C2×D3×Dic3—122no—
56C3⋊SL(2,ℤ/4)C2×D314yescomplemented
57C3×SL(2,ℤ/4)—34no—
58C3×SL(2,ℤ/4)—64no—
59C3×#217—34no—
60C3×#217—34no—
61#3722—38no—
62C22×C33—41no—
63C2×C3×C3⋊D3—44no—
64C2×C3×C3⋊D3—42no—
65C32×A4C22×C411yescharacteristic, complemented
66C3×C3⋊Dic3—44no—
67C24×D3—32no—
68C24×D3—61no—
69C23×A4—91no—
70C22×C4×D3—92no—
71C22×C4×D3—92no—
72C2×SL(2,ℤ/4)—32no—
73C2×SL(2,ℤ/4)—62no—
74C2×SL(2,ℤ/4)—94no—
75C2×C3×C22⋊C4—92no—
76C2×#217—32no—
77C2×(C2×D3)⋊C4—92no—
78C2×(C2×D3)⋊C4—92no—
79D3×C22⋊C4—98no—
80D3×C22⋊C4—98no—
81C23×C32—31no—
82C23×C32—32no—
83C23×C32—31no—
84C23×C32C2×Dic312yescomplemented
85C23×C32C22×D311yescharacteristic
86C22×C3⋊D3—31no—
87C22×C3⋊D3—34no—
88C22×C3⋊D3—34no—
89C22×C3⋊D3—32no—
90C22×C3⋊D3—32no—
91C22×C3⋊D3—34no—
92C22×C3⋊D3—38no—
93C22×C3⋊D3—34no—
94C22×C3⋊D3—32no—
95C22×C3⋊D3—64no—
96C22×C3⋊D3—62no—
97C22×C3⋊D3—61no—
98C22×C3⋊D3C2×Dic314yescomplemented
99C22×C3⋊D3S411yescharacteristic
100C22×C3×D3—122no—
101C22×C3×D3—241no—
102C2×C3×A4—24no—
103C2×C3×A4—22no—
104C2×C3×A4—22no—
105C2×C3×A4—21no—
106C2×C3×A4—42no—
107C2×C3×A4—41no—
108C2×C3×A4C4×D314yescomplemented
109C2×C3×A4C22×D312yes—
110D3×A4—38no—
111D3×A4—64no—
112C2×C3⋊Dic3—42no—
113C2×C3×Dic3—34no—
114C2×C3×Dic3—34no—
115C2×C3×Dic3—64no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—122no—
119C2×C3×Dic3—242no—
120#624—68no—
121#624—64no—
122#624—64no—
123D3×Dic3—128no—
124C22×C22⋊C4—271noSylow
125C2×C33—42no—
126C2×C33—41no—
127C3×C3⋊D3—44no—
128C24×C3—21no—
129C24×C3D3212yes—
130C23×D3—38no—
131C23×D3—34no—
132C23×D3—64no—
133C23×D3—62no—
134C23×D3—92no—
135C23×D3—94no—
136C23×D3—92no—
137C23×D3—98no—
138C23×D3—181no—
139C23×D3—182no—
140C23×D3—181no—
141C23×D3—184no—
142C22×A4—22no—
143C22×A4—41no—
144C22×A4—94no—
145C22×A4—92no—
146C22×A4D3211yescharacteristic
147C22×C3×C4—92no—
148C22×C3×C4—92no—
149C22×Dic3—32no—
150C22×Dic3—32no—
151C22×Dic3—361no—
152C2×C4×D3—98no—
153C2×C4×D3—98no—
154C2×C4×D3—182no—
155C2×C4×D3—184no—
156C2×C4×D3—188no—
157C2×C4×D3—182no—
158SL(2,ℤ/4)—34no—
159SL(2,ℤ/4)—64no—
160C3×C22⋊C4—94no—
161C3×C22⋊C4—94no—
162#217—34no—
163#217—34no—
164(C2×D3)⋊C4—98no—
165(C2×D3)⋊C4—98no—
166C22×C32—32no—
167C22×C32—31no—
168C22×C32—32no—
169C22×C32—32no—
170C22×C32—31no—
171C22×C32—31no—
172C22×C32—42no—
173C22×C32—61no—
174C22×C32—81no—
175C22×C32—82no—
176C22×C32—161no—
177C22×C32C22×Dic311yescharacteristic, complemented
178C22×C32C2×S411yescharacteristic
179C2×C3⋊D3—34no—
180C2×C3⋊D3—32no—
181C2×C3⋊D3—34no—
182C2×C3⋊D3—34no—
183C2×C3⋊D3—38no—
184C2×C3⋊D3—34no—
185C2×C3⋊D3—62no—
186C2×C3⋊D3—64no—
187C2×C3⋊D3SL(2,ℤ/4)14yescomplemented
188C2×C3⋊D3C2×S412yes—
189C2×C3×D3—124no—
190C2×C3×D3—128no—
191C2×C3×D3—244no—
192C2×C3×D3—242no—
193C3×A4—22no—
194C3×A4—21no—
195C3×A4—41no—
196C3×A4C2×C4×D312yescomplemented
197C3⋊Dic3—44no—
198C3×Dic3—64no—
199C3×Dic3—64no—
200C3×Dic3—124no—
201C3×Dic3—244no—
202C25—91no—
203C23×C4—271no—
204C23×C4—271no—
205C2×C22⋊C4—92no—
206C2×C22⋊C4—274no—
207C2×C22⋊C4—274no—
208C2×C22⋊C4—272no—
209C33—41noSylow
210C23×C3—22no—
211C23×C3—21no—
212C23×C3—32no—
213C23×C3—32no—
214C23×C3—34no—
215C23×C3—61no—
216C23×C3—61no—
217C23×C3—62no—
218C23×C3—361no—
219C23×C3D3×Dic314yescomplemented
220C23×C3C2×D3212yes—
221C22×D3—32no—
222C22×D3—38no—
223C22×D3—61no—
224C22×D3—64no—
225C22×D3—94no—
226C22×D3—94no—
227C22×D3—98no—
228C22×D3—98no—
229C22×D3—92no—
230C22×D3—94no—
231C22×D3—98no—
232C22×D3—916no—
233C22×D3—98no—
234C22×D3—182no—
235C22×D3—184no—
236C22×D3—184no—
237C22×D3—182no—
238C22×D3—181no—
239C22×D3—182no—
240C22×D3—188no—
241C22×D3—184no—
242C22×D3—184no—
243C22×D3—188no—
244C22×D3—182no—
245C22×D3—184no—
246C22×D3—182no—
247C22×D3—184no—
248C22×D3—188no—
249C2×A4—24no—
250C2×A4—22no—
251C2×A4—42no—
252C2×A4—41no—
253C2×A4—94no—
254C2×A4#62412yescomplemented
255C2×A4C2×D3211yescharacteristic
256C2×C3×C4—94no—
257C2×C3×C4—94no—
258C2×C3×C4—184no—
259C2×C3×C4—182no—
260C2×C3×C4—182no—
261C2×Dic3—34no—
262C2×Dic3—34no—
263C2×Dic3—64no—
264C2×Dic3—62no—
265C2×Dic3—62no—
266C2×Dic3—122no—
267C2×Dic3—242no—
268C2×Dic3—364no—
269C4×D3—188no—
270C4×D3—1816no—
271C4×D3—188no—
272C2×C32—31no—
273C2×C32—31no—
274C2×C32—32no—
275C2×C32—42no—
276C2×C32—44no—
277C2×C32—81no—
278C2×C32—82no—
279C2×C32—82no—
280C2×C32—84no—
281C2×C32—161no—
282C2×C32—162no—
283C2×C32C2×SL(2,ℤ/4)12yescomplemented
284C2×C32C22×S411yescharacteristic
285C3⋊D3—34no—
286C3⋊D3C2×SL(2,ℤ/4)14yescomplemented
287C3×D3—128no—
288C3×D3—244no—
289C24—94no—
290C24—92no—
291C24—271no—
292C24—271no—
293C24—272no—
294C24—274no—
295C24#111311yescharacteristic
296C22×C4—92no—
297C22×C4—92no—
298C22×C4—274no—
299C22×C4—274no—
300C22×C4—541no—
301C22×C4—542no—
302C22×C4—544no—
303C22×C4—541no—
304C22⋊C4—94no—
305C22⋊C4—94no—
306C22⋊C4—278no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—34no—
310C22×C3—32no—
311C22×C3—32no—
312C22×C3—32no—
313C22×C3—34no—
314C22×C3—34no—
315C22×C3—41no—
316C22×C3—61no—
317C22×C3—62no—
318C22×C3—61no—
319C22×C3—62no—
320C22×C3—62no—
321C22×C3—61no—
322C22×C3—62no—
323C22×C3—62no—
324C22×C3—82no—
325C22×C3—161no—
326C22×C3—362no—
327C22×C3—364no—
328C22×C3C2×D3×Dic312yescomplemented
329C22×C3D3×S412yes—
330C2×D3—34no—
331C2×D3—38no—
332C2×D3—64no—
333C2×D3—62no—
334C2×D3—98no—
335C2×D3—98no—
336C2×D3—916no—
337C2×D3—94no—
338C2×D3—98no—
339C2×D3—98no—
340C2×D3—184no—
341C2×D3—184no—
342C2×D3—188no—
343C2×D3—182no—
344C2×D3—184no—
345C2×D3—184no—
346C2×D3—184no—
347C2×D3—188no—
348C2×D3—184no—
349C2×D3—188no—
350A4—22no—
351A4—41no—
352A4C2×#62411yescharacteristic, complemented
353C3×C4—184no—
354C3×C4—184no—
355Dic3—64no—
356Dic3—64no—
357Dic3—124no—
358Dic3—244no—
359C32—42no—
360C32—81no—
361C32—82no—
362C32—161no—
363C32C22×SL(2,ℤ/4)11yescharacteristic, complemented
364C23—31no—
365C23—31no—
366C23—32no—
367C23—91no—
368C23—94no—
369C23—272no—
370C23—272no—
371C23—274no—
372C23—274no—
373C23—272no—
374C23—278no—
375C23—274no—
376C23—274no—
377C23—271no—
378C23—541no—
379C23—542no—
380C23—544no—
381C23#629412yescomplemented
382C23C2×#111311yescharacteristic
383C2×C4—94no—
384C2×C4—94no—
385C2×C4—184no—
386C2×C4—182no—
387C2×C4—182no—
388C2×C4—544no—
389C2×C4—548no—
390C2×C4—544no—
391C2×C3—21no—
392C2×C3—22no—
393C2×C3—32no—
394C2×C3—32no—
395C2×C3—34no—
396C2×C3—41no—
397C2×C3—42no—
398C2×C3—61no—
399C2×C3—61no—
400C2×C3—62no—
401C2×C3—82no—
402C2×C3—84no—
403C2×C3—161no—
404C2×C3—162no—
405C2×C3—364no—
406C2×C3D3×SL(2,ℤ/4)14yescomplemented
407C2×C3C2×D3×S412yes—
408D3—38no—
409D3—64no—
410D3—98no—
411D3—184no—
412C22—32no—
413C22—31no—
414C22—31no—
415C22—31no—
416C22—32no—
417C22—32no—
418C22—61no—
419C22—92no—
420C22—94no—
421C22—274no—
422C22—274no—
423C22—278no—
424C22—274no—
425C22—272no—
426C22—274no—
427C22—542no—
428C22—544no—
429C22C2×#629411yescharacteristic, complemented
430C22#8910111yescharacteristic
431C4—184no—
432C4—184no—
433C3—21no—
434C3—41no—
435C3—82no—
436C3—161no—
437C3C2×D3×SL(2,ℤ/4)12yescomplemented
438C2—31no—
439C2—31no—
440C2—32no—
441C2—94no—
442C2—274no—
443C2#1172421512yesdirect product factor
444C2C2×#8910111yescharacteristic
subgroupG/H
center, hypercenterC22#89101
commutator, hypocenterC32×A4C22×C4
FrattiniC2C2×#89101
Fitting, socleC24×C32C2×D3
prime radicalC22×A4×C3⋊D3C2
cosocleC2×C32×A4C23
LCSG ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×C3×C3⋊SL(2,ℤ/4)C2semidirect
3. #11724215C2direct (N × H)
7. C2×A4×C3⋊D3C4semidirect
10. C3×C3⋊SL(2,ℤ/4)C22semidirect
16. C2×C3⋊SL(2,ℤ/4)D3semidirect
25. C2×C32×A4C2×C4semidirect
27. A4×C3⋊D3C2×C4semidirect
37. C23×C3⋊D3Dic3semidirect
56. C3⋊SL(2,ℤ/4)C2×D3semidirect
65. C32×A4C22×C4semidirect
84. C23×C32C2×Dic3semidirect
98. C22×C3⋊D3C2×Dic3semidirect
108. C2×C3×A4C4×D3semidirect
177. C22×C32C22×Dic3semidirect
187. C2×C3⋊D3SL(2,ℤ/4)semidirect
196. C3×A4C2×C4×D3semidirect
219. C23×C3D3×Dic3semidirect
254. C2×A4#624semidirect
283. C2×C32C2×SL(2,ℤ/4)semidirect
286. C3⋊D3C2×SL(2,ℤ/4)semidirect
328. C22×C3C2×D3×Dic3semidirect
352. A4C2×#624semidirect
363. C32C22×SL(2,ℤ/4)semidirect
381. C23#6294semidirect
406. C2×C3D3×SL(2,ℤ/4)semidirect
429. C22C2×#6294semidirect
437. C3C2×D3×SL(2,ℤ/4)semidirect
443. C2#11724215direct (N × H)

Maximal subgroups

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

subgroupindex
C22×A4×C3⋊D32
C2×C3×C3⋊SL(2,ℤ/4)2
#117242152
C2×D3×SL(2,ℤ/4)3
C2×#639743
C2×#62944