Finite Group Databasebeta

← C2×#11724339 #13181139

Subgroups — C2×#11724339 #13181139

Order 1728.

1080 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 143 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×C4⋊#1113C211yesmaximal, characteristic
2C2×#89040C211yesmaximal, characteristic, complemented
3C2×C3×#3787C211yesmaximal, characteristic, complemented
4C2×#88942C214yesmaximal, complemented
5#11724339C218yesmaximal, direct product factor
6C2×#64227—31nomaximal
7C22×#1113—21no—
8C2×C3×C4⋊(C3⋊D3)—21no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#634—21no—
11C2×#6296C2212yescomplemented
12C2×#6295C2212yes—
13C2×#6294—21no—
14C4⋊#1113—22no—
15C4⋊#1113C2214yes—
16#89040—22no—
17#89040C2214yescomplemented
18C2×C3×#622C2212yescomplemented
19C3×#3787C2214yescomplemented
20#88942C22116yescomplemented
21C2×#3802—31no—
22C2×D3×D12—61no—
23C2×#3797—31no—
24C2×D3×Dic6—61no—
25C2×#3787D311yescharacteristic, complemented
26C2×#3775—34no—
27#64227—38no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3D411yescharacteristic
30C2×#1113—24no—
31C2×#1113—44no—
32C2×C33×C4D411yescharacteristic
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C3×C4⋊(C3⋊D3)—24no—
36C3×C4×C3⋊D3D412yes—
37C3×C4×C3⋊D3C2312yes—
38C3×#634—24no—
39#6296—22no—
40#6296C2314yescomplemented
41#6295—22no—
42#6295C2314yes—
43#6294—24no—
44C3×#622C2314yescomplemented
45C22×#204—91nomaximal
46C22×D32—62no—
47C22×D32—61no—
48C2×C4⋊(C3⋊D3)—21no—
49C2×C4×C3⋊D3C2×D311yescharacteristic
50C2×#634—21no—
51C2×C3×D12—61no—
52C2×C3×D12—61no—
53C2×C3×D12—121no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—61no—
56C2×C3×Dic6—61no—
57C2×C3×Dic6—61no—
58C2×C3×Dic6—121no—
59C2×#627—32no—
60C2×#627—62no—
61C2×#626—62no—
62C2×#625—32no—
63C2×#624—61no—
64C2×D3×Dic3—62no—
65#3802—34no—
66#3802—62no—
67D3×D12—64no—
68D3×D12—64no—
69#3797—34no—
70#3797—62no—
71D3×Dic6—64no—
72D3×Dic6—64no—
73C2×#622C2×D312yescomplemented
74#3787C2×D314yescomplemented
75#3775—316no—
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3—24no—
78C2×C3×C3⋊D3—44no—
79C2×C3×C3⋊D3SD1614yescomplemented
80C2×C3×C3⋊D3C2×D412yes—
81#1113—48no—
82C33×C4C2×D412yes—
83C3×C3⋊Dic3—24no—
84C3×C3⋊Dic3C2×D411yescharacteristic
85C3×C3⋊Dic3C2411yescharacteristic
86C2×D3×Q8—181no—
87C2×D3×D4—181no—
88C22×D12—91no—
89C22×Dic6—91no—
90C22×C3×C8—91no—
91C2×#204—98no—
92C2×#204—94no—
93C22×C3⋊D3—22no—
94C22×C3⋊D3D1211yescharacteristic
95C22×C3×D3—61no—
96C22×C3×D3—61no—
97C22×C3×D3—62no—
98C22×C3×D3—62no—
99C22×C3×D3—122no—
100C2×D32—68no—
101C2×D32—64no—
102C2×D32—128no—
103C2×D32—124no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—41no—
107C2×C32×C4—41no—
108C2×C32×C4D1211yescharacteristic
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3C22×D311yescharacteristic
111C4⋊(C3⋊D3)—24no—
112C4×C3⋊D3D1212yes—
113C4×C3⋊D3C22×D312yes—
114#634—24no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—122no—
120C3×D12—64no—
121C3×D12—64no—
122C3×D12—124no—
123C3×C4×D3—62no—
124C3×C4×D3—62no—
125C3×C4×D3—62no—
126C3×C4×D3—62no—
127C3×Dic6—64no—
128C3×Dic6—64no—
129C3×Dic6—124no—
130#627—34no—
131#627—64no—
132#627—64no—
133#627—62no—
134#626—64no—
135#626—64no—
136#625—34no—
137#625—62no—
138#624—64no—
139D3×Dic3—68no—
140#622C22×D314yescomplemented
141C22×SD16—271noSylow
142C2×C33C2×SD1612yescomplemented
143C2×C33C22×D411yescharacteristic
144C3×C3⋊D3—44no—
145C3×C3⋊D3C2×SD1614yescomplemented
146C23×D3—181no—
147C23×D3—182no—
148C2×C3×Q8—181no—
149C2×C3×D4—181no—
150C22×C3×C4—91no—
151C2×#67—182no—
152C22×Dic3—181no—
153D3×Q8—184no—
154D3×Q8—184no—
155D3×D4—184no—
156D3×D4—184no—
157C2×D12—61no—
158C2×D12—61no—
159C2×D12—94no—
160C2×D12—92no—
161C2×D12—121no—
162C2×D12—182no—
163C2×C4×D3—61no—
164C2×C4×D3—61no—
165C2×C4×D3—182no—
166C2×Dic6—61no—
167C2×Dic6—61no—
168C2×Dic6—94no—
169C2×Dic6—92no—
170C2×Dic6—121no—
171C2×Dic6—182no—
172C2×Dic6—182no—
173C2×C3×C8—94no—
174C2×C3×C8—92no—
175#204—916no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32—41no—
180C22×C32C2×D1211yescharacteristic
181C2×C3⋊D3—24no—
182C2×C3⋊D3—44no—
183C2×C3⋊D3#20414yescomplemented
184C2×C3⋊D3C2×D1212yes—
185C2×C3×D3—64no—
186C2×C3×D3—64no—
187C2×C3×D3—62no—
188C2×C3×D3—62no—
189C2×C3×D3—64no—
190C2×C3×D3—64no—
191C2×C3×D3—124no—
192C2×C3×D3—124no—
193C2×C3×D3—124no—
194C2×C3×D3—244no—
195D32—1216no—
196D32—128no—
197C32×C4—22no—
198C32×C4—22no—
199C32×C4—42no—
200C32×C4—42no—
201C32×C4C2×D1212yes—
202C3⋊Dic3—24no—
203C3⋊Dic3C2×D1211yescharacteristic
204C3⋊Dic3C23×D311yescharacteristic
205C3×Dic3—61no—
206C3×Dic3—61no—
207C3×Dic3—61no—
208C3×Dic3—61no—
209C3×Dic3—64no—
210C3×Dic3—64no—
211C3×Dic3—124no—
212C22×Q8—271no—
213C22×D4—271no—
214C2×SD16—278no—
215C2×SD16—274no—
216C22×C8—91no—
217C33C22×SD1611yescharacteristic, complemented, Sylow
218C23×C3—91no—
219C23×C3—182no—
220C22×D3—61no—
221C22×D3—61no—
222C22×D3—62no—
223C22×D3—62no—
224C22×D3—122no—
225C22×D3—184no—
226C22×D3—188no—
227C22×D3—368no—
228C22×D3—364no—
229C3×Q8—184no—
230C3×D4—184no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—91no—
236C2×C3×C4—92no—
237C2×C3×C4—92no—
238C2×C3×C4—182no—
239C2×C3×C4GO+(4,𝔽2)11yescharacteristic
240#67—184no—
241#67—184no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—62no—
245C2×Dic3—62no—
246C2×Dic3—122no—
247C2×Dic3—184no—
248D12—64no—
249D12—64no—
250D12—94no—
251D12—124no—
252D12—182no—
253C4×D3—62no—
254C4×D3—62no—
255C4×D3—62no—
256C4×D3—62no—
257C4×D3—188no—
258Dic6—64no—
259Dic6—64no—
260Dic6—94no—
261Dic6—124no—
262Dic6—184no—
263Dic6—182no—
264Dic6—184no—
265C3×C8—94no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—22no—
269C2×C32—22no—
270C2×C32—42no—
271C2×C32—42no—
272C2×C32—41no—
273C2×C32—41no—
274C2×C32C2×#20412yescomplemented
275C2×C32C22×D1211yescharacteristic
276C3⋊D3—44no—
277C3⋊D3C2×#20414yescomplemented
278C3×D3—64no—
279C3×D3—64no—
280C3×D3—124no—
281C3×D3—124no—
282C3×D3—244no—
283C24—541no—
284C2×Q8—181no—
285C2×Q8—272no—
286C2×Q8—274no—
287C2×Q8—542no—
288C2×D4—181no—
289C2×D4—274no—
290C2×D4—272no—
291C2×D4—542no—
292C22×C4—91no—
293C22×C4—541no—
294SD16—2716no—
295C2×C8—94no—
296C2×C8—92no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—92no—
302C22×C3—94no—
303C22×C3—184no—
304C22×C3—364no—
305C22×C3C2×GO+(4,𝔽2)11yescharacteristic
306C2×D3—62no—
307C2×D3—62no—
308C2×D3—64no—
309C2×D3—64no—
310C2×D3—64no—
311C2×D3—64no—
312C2×D3—124no—
313C2×D3—124no—
314C2×D3—124no—
315C2×D3—244no—
316C2×D3—3616no—
317C2×D3—368no—
318C3×C4—22no—
319C3×C4—22no—
320C3×C4—42no—
321C3×C4—42no—
322C3×C4—91no—
323C3×C4—91no—
324C3×C4—184no—
325C3×C4C2×GO+(4,𝔽2)12yes—
326Dic3—61no—
327Dic3—61no—
328Dic3—61no—
329Dic3—61no—
330Dic3—64no—
331Dic3—64no—
332Dic3—124no—
333C32—21no—
334C32—21no—
335C32—41no—
336C32—41no—
337C32C22×#20411yescharacteristic, complemented
338C23—91no—
339C23—182no—
340C23—544no—
341C23—1084no—
342Q8—184no—
343Q8—274no—
344Q8—542no—
345D4—184no—
346D4—274no—
347D4—542no—
348C2×C4—91no—
349C2×C4—92no—
350C2×C4—92no—
351C2×C4—182no—
352C2×C4—544no—
353C2×C4#157⋊D311yescharacteristic
354C8—94no—
355C2×C3—22no—
356C2×C3—22no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—42no—
360C2×C3—42no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3—94no—
364C2×C3—364no—
365C2×C3#6422712yescomplemented
366C2×C3C22×GO+(4,𝔽2)11yescharacteristic
367D3—64no—
368D3—64no—
369D3—124no—
370D3—124no—
371D3—244no—
372C22—94no—
373C22—92no—
374C22—184no—
375C22—364no—
376C22—1088no—
377C22C2×#157⋊D311yescharacteristic
378C4—91no—
379C4—91no—
380C4—184no—
381C4C2×#157⋊D312yes—
382C3—21no—
383C3—21no—
384C3—41no—
385C3—41no—
386C3C2×#6422711yescharacteristic, complemented
387C2—94no—
388C2—364no—
389C2#1172433912yesdirect product factor
390C2C22×#157⋊D311yescharacteristic
subgroupG/H
centerC22C2×#157⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C4⋊#1113C2
hypercenterC2×C4#157⋊D3
hypocenterC33C22×SD16
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#89040C2semidirect
3. C2×C3×#3787C2semidirect
4. C2×#88942C2semidirect
5. #11724339C2direct (N × H)
11. C2×#6296C22semidirect
17. #89040C22semidirect
18. C2×C3×#622C22semidirect
19. C3×#3787C22semidirect
20. #88942C22semidirect
25. C2×#3787D3semidirect
40. #6296C23semidirect
44. C3×#622C23semidirect
73. C2×#622C2×D3semidirect
74. #3787C2×D3semidirect
79. C2×C3×C3⋊D3SD16semidirect
140. #622C22×D3semidirect
142. C2×C33C2×SD16semidirect
145. C3×C3⋊D3C2×SD16semidirect
183. C2×C3⋊D3#204semidirect
217. C33C22×SD16semidirect
274. C2×C32C2×#204semidirect
277. C3⋊D3C2×#204semidirect
337. C32C22×#204semidirect
365. C2×C3#64227semidirect
386. C3C2×#64227semidirect
389. C2#11724339direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C2×#890402
C2×C3×#37872
C2×#889422
#117243392
C2×#642273
C22×#2049