Finite Group Databasebeta

← C2×#11724308 #13181076

Subgroups — C2×#11724308 #13181076

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×C3×#3797C211yesmaximal, characteristic, complemented
3C2×#88988C211yesmaximal, characteristic, complemented
4C2×#88936C214yesmaximal, complemented
5#11724308C218yesmaximal, direct product factor
6C2×#64227—31nomaximal
7C22×#1113—21no—
8C2×C3×C4⋊(C3⋊D3)—21no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×Dic6—21no—
11C2×#6295C2212yes—
12C4⋊#1113—22no—
13C4⋊#1113C2214yes—
14C2×C3×#627C2212yescomplemented
15C2×C3×#624—21no—
16C3×#3797—22no—
17C3×#3797C2214yescomplemented
18C2×C3⋊#622C2212yescomplemented
19#88988C2214yescomplemented
20#88936C22116yescomplemented
21C2×C3×D3×Q8—61no—
22C2×#3802—31no—
23C2×D3×D12—61no—
24C2×#3797D311yescharacteristic, complemented
25C2×#3787—31no—
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⋊Dic3C2311yescharacteristic
34C3×C4⋊(C3⋊D3)—24no—
35C3×C4×C3⋊D3D412yes—
36C3×C4×C3⋊D3C2312yes—
37C2×C32×Dic3—22no—
38C32×Dic6—24no—
39#6295—22no—
40#6295C2314yes—
41C3×#627—22no—
42C3×#627C2314yescomplemented
43C3×#624—24no—
44C3⋊#622C2314yescomplemented
45C22×#215—91nomaximal
46C22×D32—62no—
47C22×D32—61no—
48C2×C32×Q8—61no—
49C2×C4⋊(C3⋊D3)—21no—
50C2×C4×C3⋊D3C2×D311yescharacteristic
51C3×D3×Q8—64no—
52C3×D3×Q8—64no—
53C2×C3×D12—61no—
54C2×C3×D12—61no—
55C2×C3×D12—121no—
56C2×C3×C4×D3—62no—
57C2×C3×C4×D3—61no—
58C2×C3×C4×D3—61no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6—41no—
62C2×C3×Dic6—62no—
63C2×#627C2×D312yescomplemented
64C2×#626—62no—
65C2×#625—32no—
66C2×#624—21no—
67#3802—34no—
68#3802—62no—
69D3×D12—64no—
70D3×D12—64no—
71#3797—22no—
72#3797C2×D314yescomplemented
73C2×#622—32no—
74#3787—34no—
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⋊Dic3C2×D411yescharacteristic
84C3×C3⋊Dic3C2411yescharacteristic
85C32×Dic3—24no—
86C22×C3×Q8—91no—
87C2×D3×Q8—61no—
88C2×D3×D4—181no—
89C22×D12—91no—
90C2×#215—98no—
91C2×#215—94no—
92C22×C3⋊C8—91no—
93C22×C3⋊D3—22no—
94C22×C3⋊D3#6711yescharacteristic
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—
104C32×Q8—64no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—41no—
108C2×C32×C4—41no—
109C2×C32×C4—62no—
110C2×C32×C4#6711yescharacteristic
111C2×C3⋊Dic3C22×D311yescharacteristic
112C4⋊(C3⋊D3)—24no—
113C4×C3⋊D3#6712yes—
114C4×C3⋊D3C22×D312yes—
115C2×C3×Dic3—22no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—42no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
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×C4×D3—68no—
128C3×Dic6—24no—
129C3×Dic6—24no—
130C3×Dic6—44no—
131C3×Dic6—64no—
132C3×Dic6—64no—
133#627—22no—
134#627C22×D314yescomplemented
135#626—64no—
136#626—64no—
137#625—34no—
138#625—62no—
139#624—24no—
140#622—34no—
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—61no—
149C2×C3×Q8—61no—
150C2×C3×Q8—92no—
151C2×C3×Q8—94no—
152C2×C3×Q8—121no—
153C2×C3×Q8—182no—
154C2×C3×D4—181no—
155C22×C3×C4—91no—
156C22×C3×C4—181no—
157C2×#67—182no—
158D3×Q8—64no—
159D3×Q8—64no—
160D3×D4—184no—
161D3×D4—184no—
162C2×D12—61no—
163C2×D12—61no—
164C2×D12—94no—
165C2×D12—92no—
166C2×D12—121no—
167C2×D12—182no—
168C2×C4×D3—61no—
169C2×C4×D3—61no—
170C2×C4×D3—62no—
171C2×Dic6—21no—
172C2×Dic6—62no—
173#215—916no—
174C2×C3⋊C8—94no—
175C2×C3⋊C8—92no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32—41no—
180C22×C32C2×#6711yescharacteristic
181C2×C3⋊D3—24no—
182C2×C3⋊D3—44no—
183C2×C3⋊D3#21514yescomplemented
184C2×C3⋊D3C2×#6712yes—
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×C4—64no—
202C32×C4C2×#6712yes—
203C3⋊Dic3C2×#6711yescharacteristic
204C3⋊Dic3C23×D311yescharacteristic
205C3×Dic3—24no—
206C3×Dic3—24no—
207C3×Dic3—44no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C3×Dic3—61no—
211C3×Dic3—61no—
212C22×Q8—91no—
213C22×D4—271no—
214C2×SD16—278no—
215C2×SD16—274no—
216C22×C8—271no—
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—64no—
230C3×Q8—64no—
231C3×Q8—94no—
232C3×Q8—124no—
233C3×Q8—182no—
234C3×D4—184no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—91no—
242C2×C3×C4—92no—
243C2×C3×C4—92no—
244C2×C3×C4—122no—
245C2×C3×C4—184no—
246C2×C3×C4GO+(4,𝔽2)11yescharacteristic
247#67—184no—
248#67—184no—
249C2×Dic3—22no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252D12—64no—
253D12—64no—
254D12—94no—
255D12—124no—
256D12—182no—
257C4×D3—62no—
258C4×D3—62no—
259C4×D3—62no—
260C4×D3—62no—
261C4×D3—68no—
262Dic6—24no—
263Dic6—64no—
264Dic6—64no—
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×#21512yescomplemented
275C2×C32C22×#6711yescharacteristic
276C3⋊D3—44no—
277C3⋊D3C2×#21514yescomplemented
278C3×D3—64no—
279C3×D3—64no—
280C3×D3—124no—
281C3×D3—124no—
282C3×D3—244no—
283C24—541no—
284C2×Q8—61no—
285C2×Q8—92no—
286C2×Q8—94no—
287C2×Q8—182no—
288C2×D4—181no—
289C2×D4—274no—
290C2×D4—272no—
291C2×D4—542no—
292C22×C4—91no—
293C22×C4—181no—
294SD16—2716no—
295C2×C8—274no—
296C2×C8—272no—
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—64no—
323C3×C4—64no—
324C3×C4—91no—
325C3×C4—91no—
326C3×C4—124no—
327C3×C4C2×GO+(4,𝔽2)12yes—
328Dic3—24no—
329Dic3—61no—
330Dic3—61no—
331Dic3—61no—
332Dic3—61no—
333C32—21no—
334C32—21no—
335C32—41no—
336C32—41no—
337C32C22×#21511yescharacteristic, complemented
338C23—91no—
339C23—182no—
340C23—544no—
341C23—1084no—
342Q8—64no—
343Q8—94no—
344Q8—182no—
345D4—184no—
346D4—274no—
347D4—542no—
348C2×C4—62no—
349C2×C4—91no—
350C2×C4—92no—
351C2×C4—92no—
352C2×C4—184no—
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—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×D32⋊D311yescharacteristic
378C4—64no—
379C4—91no—
380C4—91no—
381C4C2×D32⋊D312yes—
382C3—21no—
383C3—21no—
384C3—41no—
385C3—41no—
386C3C2×#6422711yescharacteristic, complemented
387C2—94no—
388C2—364no—
389C2#1172430812yesdirect 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×C4⋊#1113C2
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
2. C2×C3×#3797C2semidirect
3. C2×#88988C2semidirect
4. C2×#88936C2semidirect
5. #11724308C2direct (N × H)
14. C2×C3×#627C22semidirect
17. C3×#3797C22semidirect
18. C2×C3⋊#622C22semidirect
19. #88988C22semidirect
20. #88936C22semidirect
24. C2×#3797D3semidirect
42. C3×#627C23semidirect
44. C3⋊#622C23semidirect
63. C2×#627C2×D3semidirect
72. #3797C2×D3semidirect
79. C2×C3×C3⋊D3SD16semidirect
134. #627C22×D3semidirect
142. C2×C33C2×SD16semidirect
145. C3×C3⋊D3C2×SD16semidirect
183. C2×C3⋊D3#215semidirect
217. C33C22×SD16semidirect
274. C2×C32C2×#215semidirect
277. C3⋊D3C2×#215semidirect
337. C32C22×#215semidirect
365. C2×C3#64227semidirect
386. C3C2×#64227semidirect
389. C2#11724308direct (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×C3×#37972
C2×#889882
C2×#889362
#117243082
C2×#642273
C22×#2159