Finite Group Databasebeta

← C22×#88939 #13181161

Subgroups — C22×#88939 #13181161

Order 1728.

2520 conjugacy classes of subgroups; only the 279 normal subgroups are shown.

normal subgroups only

This group has 2520 conjugacy classes of subgroups — too many to draw. Only its 279 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
1C23×#1113C211yesmaximal, characteristic
2C23×C3×#157C211yesmaximal, characteristic, complemented
3C22×#6294C211yesmaximal, characteristic, complemented
4C2×#88939C2112yesmaximal, direct product factor
5C22×D32⋊C4—31nomaximal
6C23×C3×C3⋊D3—21no—
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×#1113—23no—
9C22×#1113—24no—
10C22×#1113C418yescomplemented
11C22×#1113C2216yes—
12C22×C3×#157—24no—
13C22×C3×#157C2216yescomplemented
14C22×C3×C3⋊Dic3—21no—
15C2×#6294—23no—
16C2×#6294C2216yescomplemented
17#88939C22116yesdirect product factor
18C23×D32—31no—
19C23×D32—61no—
20C23×#157D311yescharacteristic, complemented
21C22×#624—31no—
22C22×D3×Dic3—61no—
23C2×D32⋊C4—312no—
24C23×C33D411yescharacteristic
25C22×C3×C3⋊D3—28no—
26C22×C3×C3⋊D3—26no—
27C22×C3×C3⋊D3C2×C414yes—
28C22×C3×C3⋊D3D413yes—
29C22×C3×C3⋊D3D414yes—
30C22×C3×C3⋊D3C2313yes—
31C2×#1113—224no—
32C2×#1113—26no—
33C2×#1113—212no—
34C2×#1113C2×C4124yescomplemented
35C2×#1113C2314yes—
36C2×C3×#157—212no—
37C2×C3×#157C2314yescomplemented
38C2×C3×C3⋊Dic3—26no—
39#6294—26no—
40#6294C2314yescomplemented
41C22×(C2×D3)⋊C4—91nomaximal
42C23×C3⋊D3—21no—
43C23×C3⋊D3C2×D311yescharacteristic
44C23×C3×D3—61no—
45C23×C3×D3—61no—
46C23×C3×D3—61no—
47C23×C3×D3—61no—
48C23×C3×D3—121no—
49C22×D32—38no—
50C22×D32—36no—
51C22×D32—68no—
52C22×D32—66no—
53C22×D32—66no—
54C22×D32—68no—
55C22×D32—63no—
56C22×D32—64no—
57C22×#157—24no—
58C22×#157C2×D316yescomplemented
59C22×C3⋊Dic3—21no—
60C22×C3×Dic3—61no—
61C22×C3×Dic3—61no—
62C22×C3×Dic3—121no—
63C2×#624—36no—
64C2×#624—63no—
65C2×D3×Dic3—66no—
66C2×D3×Dic3—66no—
67D32⋊C4—316no—
68C22×C33C22⋊C414yescomplemented
69C22×C33C2×D413yes—
70C2×C3×C3⋊D3—224no—
71C2×C3×C3⋊D3—24no—
72C2×C3×C3⋊D3C22⋊C4112yescomplemented
73C2×C3×C3⋊D3C22×C416yes—
74C2×C3×C3⋊D3C2×D413yes—
75C2×C3×C3⋊D3C2×D416yes—
76C2×C3×C3⋊D3C2411yescharacteristic
77#1113—224no—
78#1113—24no—
79#1113C22×C418yescomplemented
80C3×#157—24no—
81C3×C3⋊Dic3—24no—
82C24×D3—91no—
83C24×D3—181no—
84C23×C3×C4—91no—
85C23×Dic3—91no—
86C22×C4×D3—181no—
87C2×(C2×D3)⋊C4—912no—
88C23×C32—21no—
89C23×C32—21no—
90C23×C32—41no—
91C23×C32—41no—
92C23×C32D1211yescharacteristic
93C22×C3⋊D3—26no—
94C22×C3⋊D3—28no—
95C22×C3⋊D3C4×D314yes—
96C22×C3⋊D3D1213yes—
97C22×C3⋊D3#6714yes—
98C22×C3⋊D3C22×D313yes—
99C22×C3×D3—64no—
100C22×C3×D3—63no—
101C22×C3×D3—64no—
102C22×C3×D3—63no—
103C22×C3×D3—64no—
104C22×C3×D3—64no—
105C22×C3×D3—63no—
106C22×C3×D3—63no—
107C22×C3×D3—66no—
108C22×C3×D3—68no—
109C22×C3×D3—68no—
110C22×C3×D3—66no—
111C22×C3×D3—128no—
112C22×C3×D3—126no—
113C2×D32—324no—
114C2×D32—34no—
115C2×D32—624no—
116C2×D32—64no—
117C2×D32—648no—
118C2×D32—612no—
119C2×D32—624no—
120C2×D32—624no—
121C2×D32—66no—
122C2×D32—612no—
123C2×#157—212no—
124C2×#157C22×D314yescomplemented
125C2×C3⋊Dic3—26no—
126C2×C3×Dic3—66no—
127C2×C3×Dic3—66no—
128C2×C3×Dic3—126no—
129#624—34no—
130#624—66no—
131D3×Dic3—64no—
132D3×Dic3—612no—
133C22×C22⋊C4—271noSylow
134C2×C33C2×C22⋊C416yescomplemented
135C2×C33C22×D411yescharacteristic
136C3×C3⋊D3—28no—
137C3×C3⋊D3C2×C22⋊C416yescomplemented
138C3×C3⋊D3C23×C411yescharacteristic
139C3×C3⋊D3C22×D411yescharacteristic
140C24×C3—91no—
141C24×C3—181no—
142C23×D3—61no—
143C23×D3—61no—
144C23×D3—61no—
145C23×D3—61no—
146C23×D3—96no—
147C23×D3—98no—
148C23×D3—121no—
149C23×D3—186no—
150C23×D3—188no—
151C23×D3—183no—
152C23×D3—184no—
153C23×D3—186no—
154C23×D3—188no—
155C22×C3×C4—96no—
156C22×C3×C4—181no—
157C22×C3×C4—184no—
158C22×Dic3—61no—
159C22×Dic3—61no—
160C22×Dic3—96no—
161C22×Dic3—121no—
162C22×Dic3—183no—
163C2×C4×D3—186no—
164C2×C4×D3—186no—
165(C2×D3)⋊C4—916no—
166C22×C32—23no—
167C22×C32—23no—
168C22×C32—24no—
169C22×C32—24no—
170C22×C32—44no—
171C22×C32—44no—
172C22×C32—43no—
173C22×C32—43no—
174C22×C32(C2×D3)⋊C414yescomplemented
175C22×C32C2×D1213yes—
176C2×C3⋊D3—24no—
177C2×C3⋊D3—224no—
178C2×C3⋊D3(C2×D3)⋊C4112yescomplemented
179C2×C3⋊D3C2×C4×D316yes—
180C2×C3⋊D3C2×D1213yes—
181C2×C3⋊D3C2×#6716yes—
182C2×C3⋊D3C23×D311yescharacteristic
183C2×C3×D3—612no—
184C2×C3×D3—612no—
185C2×C3×D3—66no—
186C2×C3×D3—66no—
187C2×C3×D3—63no—
188C2×C3×D3—63no—
189C2×C3×D3—66no—
190C2×C3×D3—66no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C2×C3×D3—64no—
194C2×C3×D3—624no—
195C2×C3×D3—624no—
196C2×C3×D3—64no—
197C2×C3×D3—1224no—
198C2×C3×D3—124no—
199D32—38no—
200D32—68no—
201D32—648no—
202D32—68no—
203D32—624no—
204D32—64no—
205#157—24no—
206C3⋊Dic3—24no—
207C3×Dic3—64no—
208C3×Dic3—64no—
209C3×Dic3—124no—
210C25—271no—
211C23×C4—91no—
212C23×C4—271no—
213C2×C22⋊C4—2712no—
214C33C22×C22⋊C411yescharacteristic, complemented, Sylow
215C23×C3—21no—
216C23×C3—21no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—93no—
220C23×C3—93no—
221C23×C3—94no—
222C23×C3—94no—
223C23×C3—186no—
224C23×C3—188no—
225C23×C3GO+(4,𝔽2)11yescharacteristic
226C22×D3—64no—
227C22×D3—64no—
228C22×D3—63no—
229C22×D3—63no—
230C22×D3—64no—
231C22×D3—64no—
232C22×D3—63no—
233C22×D3—63no—
234C22×D3—66no—
235C22×D3—68no—
236C22×D3—66no—
237C22×D3—68no—
238C22×D3—94no—
239C22×D3—924no—
240C22×D3—126no—
241C22×D3—128no—
242C22×D3—184no—
243C22×D3—1824no—
244C22×D3—186no—
245C22×D3—1812no—
246C22×D3—1824no—
247C22×D3—1848no—
248C22×D3—1812no—
249C22×D3—1824no—
250C2×C3×C4—94no—
251C2×C3×C4—186no—
252C2×C3×C4—1812no—
253C2×Dic3—66no—
254C2×Dic3—66no—
255C2×Dic3—94no—
256C2×Dic3—126no—
257C2×Dic3—186no—
258C4×D3—184no—
259C4×D3—1812no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—26no—
263C2×C32—26no—
264C2×C32—46no—
265C2×C32—46no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32C2×(C2×D3)⋊C416yescomplemented
269C2×C32C22×D1211yescharacteristic
270C3⋊D3—28no—
271C3⋊D3C2×(C2×D3)⋊C416yescomplemented
272C3⋊D3C22×C4×D311yescharacteristic
273C3⋊D3C22×#6711yescharacteristic
274C3×D3—66no—
275C3×D3—66no—
276C3×D3—61no—
277C3×D3—61no—
278C3×D3—61no—
279C3×D3—61no—
280C3×D3—68no—
281C3×D3—68no—
282C3×D3—128no—
283C24—91no—
284C24—181no—
285C24—276no—
286C24—278no—
287C24—543no—
288C24—544no—
289C22×C4—96no—
290C22×C4—184no—
291C22×C4—181no—
292C22×C4—276no—
293C22×C4—543no—
294C22⋊C4—2716no—
295C22×C3—23no—
296C22×C3—23no—
297C22×C3—24no—
298C22×C3—24no—
299C22×C3—43no—
300C22×C3—43no—
301C22×C3—44no—
302C22×C3—44no—
303C22×C3—91no—
304C22×C3—93no—
305C22×C3—96no—
306C22×C3—96no—
307C22×C3—912no—
308C22×C3—1824no—
309C22×C3—184no—
310C22×C3D32⋊C414yescomplemented
311C22×C3C2×GO+(4,𝔽2)13yes—
312C2×D3—63no—
313C2×D3—61no—
314C2×D3—612no—
315C2×D3—66no—
316C2×D3—63no—
317C2×D3—66no—
318C2×D3—61no—
319C2×D3—612no—
320C2×D3—66no—
321C2×D3—66no—
322C2×D3—624no—
323C2×D3—64no—
324C2×D3—64no—
325C2×D3—624no—
326C2×D3—98no—
327C2×D3—1224no—
328C2×D3—124no—
329C2×D3—188no—
330C2×D3—184no—
331C2×D3—1824no—
332C2×D3—1848no—
333C2×D3—188no—
334C3×C4—184no—
335C3×C4—184no—
336Dic3—64no—
337Dic3—64no—
338Dic3—124no—
339C32—21no—
340C32—21no—
341C32—41no—
342C32—41no—
343C32C22×(C2×D3)⋊C411yescharacteristic, complemented
344C23—93no—
345C23—93no—
346C23—94no—
347C23—94no—
348C23—186no—
349C23—188no—
350C23—274no—
351C23—2724no—
352C23—5412no—
353C23—5424no—
354C23—546no—
355C23#157⋊D311yescharacteristic
356C2×C4—94no—
357C2×C4—1812no—
358C2×C4—186no—
359C2×C4—274no—
360C2×C4—546no—
361C2×C3—26no—
362C2×C3—26no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—46no—
366C2×C3—46no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3—91no—
370C2×C3—91no—
371C2×C3—96no—
372C2×C3—188no—
373C2×C3C2×D32⋊C416yescomplemented
374C2×C3C22×GO+(4,𝔽2)11yescharacteristic
375D3—66no—
376D3—61no—
377D3—61no—
378D3—66no—
379D3—61no—
380D3—61no—
381D3—68no—
382D3—68no—
383D3—128no—
384C22—96no—
385C22—96no—
386C22—912no—
387C22—91no—
388C22—93no—
389C22—1824no—
390C22—184no—
391C22—278no—
392C22—544no—
393C22—5424no—
394C22#8893914yesdirect product factor
395C22C2×#157⋊D313yes—
396C4—184no—
397C4—184no—
398C3—21no—
399C3—21no—
400C3—41no—
401C3—41no—
402C3C22×D32⋊C411yescharacteristic, complemented
403C2—91no—
404C2—91no—
405C2—96no—
406C2—188no—
407C2C2×#8893916yesdirect product factor
408C2C22×#157⋊D311yescharacteristic
subgroupG/H
center, hypercenterC23#157⋊D3
commutatorC3×C3⋊D3C23×C4
FrattiniC2C22×#157⋊D3
Fitting, socleC23×C33D4
prime radicalC23×#1113C2
cosocleC2×C3×C3⋊D3C24
hypocenterC33C22×C22⋊C4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C23×C3×#157C2semidirect
3. C22×#6294C2semidirect
4. C2×#88939C2direct (N × H)
10. C22×#1113C4semidirect
13. C22×C3×#157C22semidirect
16. C2×#6294C22semidirect
17. #88939C22direct (N × H)
20. C23×#157D3semidirect
34. C2×#1113C2×C4semidirect
37. C2×C3×#157C23semidirect
40. #6294C23semidirect
58. C22×#157C2×D3semidirect
68. C22×C33C22⋊C4semidirect
72. C2×C3×C3⋊D3C22⋊C4semidirect
79. #1113C22×C4semidirect
124. C2×#157C22×D3semidirect
134. C2×C33C2×C22⋊C4semidirect
137. C3×C3⋊D3C2×C22⋊C4semidirect
174. C22×C32(C2×D3)⋊C4semidirect
178. C2×C3⋊D3(C2×D3)⋊C4semidirect
214. C33C22×C22⋊C4semidirect
268. C2×C32C2×(C2×D3)⋊C4semidirect
271. C3⋊D3C2×(C2×D3)⋊C4semidirect
310. C22×C3D32⋊C4semidirect
343. C32C22×(C2×D3)⋊C4semidirect
373. C2×C3C2×D32⋊C4semidirect
394. C22#88939direct (N × H)
402. C3C22×D32⋊C4semidirect
407. C2C2×#88939direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#11132
C23×C3×#1572
C22×#62942
C2×#889392
C22×D32⋊C43
C22×(C2×D3)⋊C49