Finite Group Databasebeta

← C2×#11724310 #13181079

Subgroups — C2×#11724310 #13181079

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×#89040C211yesmaximal, characteristic, complemented
2C2×C3×#3802C211yesmaximal, characteristic
3C2×#88988C211yesmaximal, characteristic, complemented
4C2×#88937C214yesmaximal, complemented
5#11724310C218yesmaximal, direct product factor
6C2×#64227—31nomaximal
7C22×C3×D32—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C3×#634—21no—
10C2×C32×D12—21no—
11C2×#6296C2212yescomplemented
12C2×#6294—21no—
13#89040—22no—
14#89040C2214yescomplemented
15C2×C3×#625C2212yes—
16C3×#3802—22no—
17C3×#3802C2214yes—
18C2×C3⋊#622C2212yescomplemented
19#88988C2214yescomplemented
20#88937C22116yescomplemented
21C2×C3×D3×D4—61no—
22C2×#3802D311yescharacteristic
23C2×#3797—31no—
24C2×D3×Dic6—61no—
25C2×#3787—31no—
26C2×#3775—34no—
27#64227—38no—
28C22×C3×C3⋊D3D411yescharacteristic
29C22×C32×D3—22no—
30C2×C3×D32—24no—
31C2×C3×D32—44no—
32C2×C33×C4D411yescharacteristic
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3C2311yescharacteristic
35C3×C4×C3⋊D3D412yes—
36C3×C4×C3⋊D3C2312yes—
37C3×#634—24no—
38C32×D12—24no—
39#6296—22no—
40#6296C2314yescomplemented
41#6294—24no—
42C3×#625—22no—
43C3×#625C2314yes—
44C3⋊#622C2314yescomplemented
45C22×#214—91nomaximal
46C23×C3×D3—62no—
47C22×D32—21no—
48C2×C32×D4—61no—
49C2×C4×C3⋊D3C2×D311yescharacteristic
50C2×#634—21no—
51C2×C3×#67—62no—
52C3×D3×D4—64no—
53C3×D3×D4—64no—
54C2×C3×D12—21no—
55C2×C3×D12—21no—
56C2×C3×D12—41no—
57C2×C3×C4×D3—61no—
58C2×C3×C4×D3—61no—
59C2×C3×Dic6—61no—
60C2×C3×Dic6—61no—
61C2×C3×Dic6—121no—
62C2×#627—32no—
63C2×#627—62no—
64C2×#625C2×D312yes—
65C2×#624—61no—
66C2×D3×Dic3—62no—
67#3802—22no—
68#3802C2×D314yes—
69#3797—34no—
70#3797—62no—
71D3×Dic6—64no—
72D3×Dic6—64no—
73C2×#622—32no—
74#3787—34no—
75#3775—316no—
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3SD1614yescomplemented
78C2×C3×C3⋊D3C2×D412yes—
79C2×C32×D3—24no—
80C2×C32×D3—44no—
81C3×D32—48no—
82C33×C4C2×D412yes—
83C3×C3⋊Dic3—24no—
84C3×C3⋊Dic3C2×D411yescharacteristic
85C3×C3⋊Dic3C2411yescharacteristic
86C22×C3×D4—91no—
87C2×D3×Q8—181no—
88C2×D3×D4—61no—
89C22×Dic6—91no—
90C2×#214—98no—
91C2×#214—94no—
92C22×C3⋊C8—91no—
93C23×C32—62no—
94C22×C3⋊D3#6711yescharacteristic
95C22×C3×D3—22no—
96C22×C3×D3—22no—
97C22×C3×D3—42no—
98C22×C3×D3—61no—
99C22×C3×D3—61no—
100C22×C3×D3—68no—
101C22×C3×D3—128no—
102C2×D32—24no—
103C2×D32—44no—
104C32×D4—64no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—41no—
108C2×C32×C4—41no—
109C2×C32×C4#6711yescharacteristic
110C2×C3⋊Dic3—22no—
111C2×C3⋊Dic3C22×D311yescharacteristic
112C4×C3⋊D3#6712yes—
113C4×C3⋊D3C22×D312yes—
114#634—24no—
115C3×#67—64no—
116C3×#67—64no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—122no—
122C3×D12—24no—
123C3×D12—24no—
124C3×D12—44no—
125C3×C4×D3—62no—
126C3×C4×D3—62no—
127C3×C4×D3—62no—
128C3×C4×D3—62no—
129C3×Dic6—64no—
130C3×Dic6—64no—
131C3×Dic6—124no—
132#627—34no—
133#627—64no—
134#627—64no—
135#627—62no—
136#625—22no—
137#625C22×D314yes—
138#624—64no—
139D3×Dic3—68no—
140#622—34no—
141C22×SD16—271noSylow
142C2×C33C2×SD1612yescomplemented
143C2×C33C22×D411yescharacteristic
144C3×C3⋊D3C2×SD1614yescomplemented
145C32×D3—44no—
146C24×C3—181no—
147C23×D3—62no—
148C2×C3×Q8—181no—
149C2×C3×D4—61no—
150C2×C3×D4—61no—
151C2×C3×D4—94no—
152C2×C3×D4—92no—
153C2×C3×D4—121no—
154C2×C3×D4—182no—
155C22×C3×C4—91no—
156C2×#67—62no—
157C22×Dic3—181no—
158D3×Q8—184no—
159D3×Q8—184no—
160D3×D4—64no—
161D3×D4—64no—
162C2×D12—21no—
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—
173#214—916no—
174C2×C3⋊C8—94no—
175C2×C3⋊C8—92no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32—41no—
180C22×C32—64no—
181C22×C32—124no—
182C22×C32C2×#6711yescharacteristic
183C2×C3⋊D3#21414yescomplemented
184C2×C3⋊D3C2×#6712yes—
185C2×C3×D3—24no—
186C2×C3×D3—24no—
187C2×C3×D3—44no—
188C2×C3×D3—44no—
189C2×C3×D3—44no—
190C2×C3×D3—64no—
191C2×C3×D3—64no—
192C2×C3×D3—62no—
193C2×C3×D3—62no—
194C2×C3×D3—84no—
195C2×C3×D3—1216no—
196D32—48no—
197C32×C4—22no—
198C32×C4—22no—
199C32×C4—42no—
200C32×C4—42no—
201C32×C4C2×#6712yes—
202C3⋊Dic3—24no—
203C3⋊Dic3C2×#6711yescharacteristic
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—91no—
214C2×SD16—278no—
215C2×SD16—274no—
216C22×C8—271no—
217C33C22×SD1611yescharacteristic, complemented, Sylow
218C23×C3—62no—
219C23×C3—62no—
220C23×C3—91no—
221C23×C3—122no—
222C23×C3—184no—
223C23×C3—364no—
224C22×D3—22no—
225C22×D3—61no—
226C22×D3—61no—
227C22×D3—68no—
228C22×D3—128no—
229C3×Q8—184no—
230C3×D4—64no—
231C3×D4—64no—
232C3×D4—94no—
233C3×D4—124no—
234C3×D4—182no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—91no—
240C2×C3×C4—92no—
241C2×C3×C4—92no—
242C2×C3×C4—182no—
243C2×C3×C4GO+(4,𝔽2)11yescharacteristic
244#67—64no—
245#67—64no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—62no—
249C2×Dic3—62no—
250C2×Dic3—122no—
251C2×Dic3—184no—
252D12—24no—
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×C32—124no—
275C2×C32C2×#21412yescomplemented
276C2×C32C22×#6711yescharacteristic
277C3⋊D3C2×#21414yescomplemented
278C3×D3—44no—
279C3×D3—44no—
280C3×D3—64no—
281C3×D3—64no—
282C3×D3—84no—
283C24—181no—
284C2×Q8—181no—
285C2×Q8—272no—
286C2×Q8—274no—
287C2×Q8—542no—
288C2×D4—61no—
289C2×D4—94no—
290C2×D4—92no—
291C2×D4—182no—
292C22×C4—91no—
293C22×C4—541no—
294SD16—2716no—
295C2×C8—274no—
296C2×C8—272no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—64no—
302C22×C3—64no—
303C22×C3—92no—
304C22×C3—94no—
305C22×C3—124no—
306C22×C3—124no—
307C22×C3—124no—
308C22×C3—244no—
309C22×C3—368no—
310C22×C3C2×GO+(4,𝔽2)11yescharacteristic
311C2×D3—24no—
312C2×D3—44no—
313C2×D3—62no—
314C2×D3—62no—
315C2×D3—64no—
316C2×D3—64no—
317C2×D3—1216no—
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×#21411yescharacteristic, complemented
338C23—62no—
339C23—91no—
340C23—184no—
341C23—364no—
342Q8—184no—
343Q8—274no—
344Q8—542no—
345D4—64no—
346D4—94no—
347D4—182no—
348C2×C4—91no—
349C2×C4—92no—
350C2×C4—92no—
351C2×C4—182no—
352C2×C4—544no—
353C2×C4D32⋊D311yescharacteristic
354C8—274no—
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—124no—
365C2×C3—124no—
366C2×C3—244no—
367C2×C3#6422712yescomplemented
368C2×C3C22×GO+(4,𝔽2)11yescharacteristic
369D3—44no—
370D3—64no—
371D3—64no—
372C22—64no—
373C22—94no—
374C22—92no—
375C22—124no—
376C22—368no—
377C22C2×D32⋊D311yescharacteristic
378C4—91no—
379C4—91no—
380C4—184no—
381C4C2×D32⋊D312yes—
382C3—21no—
383C3—21no—
384C3—41no—
385C3—41no—
386C3C2×#6422711yescharacteristic, complemented
387C2—94no—
388C2—124no—
389C2#1172431012yesdirect product factor
390C2C22×D32⋊D311yescharacteristic
subgroupG/H
centerC22C2×D32⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#3802C2
hypercenterC2×C4D32⋊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
1. C2×#89040C2semidirect
3. C2×#88988C2semidirect
4. C2×#88937C2semidirect
5. #11724310C2direct (N × H)
11. C2×#6296C22semidirect
14. #89040C22semidirect
18. C2×C3⋊#622C22semidirect
19. #88988C22semidirect
20. #88937C22semidirect
40. #6296C23semidirect
44. C3⋊#622C23semidirect
77. C2×C3×C3⋊D3SD16semidirect
142. C2×C33C2×SD16semidirect
144. C3×C3⋊D3C2×SD16semidirect
183. C2×C3⋊D3#214semidirect
217. C33C22×SD16semidirect
275. C2×C32C2×#214semidirect
277. C3⋊D3C2×#214semidirect
337. C32C22×#214semidirect
367. C2×C3#64227semidirect
386. C3C2×#64227semidirect
389. C2#11724310direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C3×#38022
C2×#889882
C2×#889372
#117243102
C2×#642273
C22×#2149