Finite Group Databasebeta

← C2×#11724267 #13181001

Subgroups — C2×#11724267 #13181001

Order 1728.

968 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 178 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×#89028C211yesmaximal, characteristic
2C22×C3⋊#622C211yesmaximal, characteristic, complemented
3C2×C3×#3792C211yesmaximal, characteristic, complemented
4C2×#88926C211yesmaximal, characteristic, complemented
5C2×#88925C211yesmaximal, characteristic, complemented
6C2×#88924C211yesmaximal, characteristic, complemented
7#11724267C218yesmaximal, direct product factor
8C2×D3×#622C211yesmaximal, characteristic, complemented
9C2×#64287—31nomaximal
10C2×#6312C411yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C2×C32×#67C411yescharacteristic
13C2×#6293C411yescharacteristic
14C2×#6291C411yescharacteristic
15C2×#6289C2211yescharacteristic
16#89028C414yes—
17#89028C2214yes—
18C2×D3×C3⋊Dic3C2211yescharacteristic
19C2×C3⋊#622C2211yescharacteristic
20C2×C3⋊#622C2211yescharacteristic
21C2×C3⋊#622C2214yescomplemented
22C3×#3792C2214yescomplemented
23C2×C3×#622C2211yescharacteristic
24C2×C3×#622C2211yescharacteristic
25#88926C2214yescomplemented
26#88925C2214yescomplemented
27#88924C2214yescomplemented
28D3×#622C2214yescomplemented
29C2×#3830—31no—
30C2×#3804—62no—
31C2×#3792—31no—
32C2×#3792D311yescharacteristic, complemented
33C22×#622—31no—
34C22×#622—31no—
35#64287—38no—
36C2×#3788—31no—
37C2×#3787—31no—
38C23×C33C2×C411yescharacteristic
39C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
40C22×C32×D3C2×C411yescharacteristic
41#6312C2×C414yes—
42C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
43C2×C3×C3⋊Dic3C2×C412yes—
44C2×C3×C3⋊Dic3C2×C411yescharacteristic
45C2×C3×C3⋊Dic3C2311yescharacteristic
46C2×C3×C3⋊Dic3C2312yes—
47C32×#67C2×C414yes—
48C2×C32×Dic3C2×C411yescharacteristic
49#6293C2×C414yes—
50#6291C2×C414yes—
51#6289C2×C412yes—
52#6289C2312yes—
53D3×C3⋊Dic3C2×C412yes—
54D3×C3⋊Dic3C2312yes—
55C3⋊#622C2312yes—
56C3⋊#622C2312yes—
57C3×#622C2312yes—
58C3×#622C2312yes—
59C2×#907—91nomaximal
60C2×C32×D4—31no—
61C2×#638—31no—
62C2×#638—31no—
63C2×#638—62no—
64C2×#638—122no—
65C22×C3⋊Dic3—31no—
66C22×C3⋊Dic3C2×D311yescharacteristic
67#3830—34no—
68#3830—34no—
69C2×C4×C3⋊D3—31no—
70C2×#634—31no—
71C2×C3×#67—22no—
72C22×C3×Dic3—62no—
73C2×#627—62no—
74C2×#626—62no—
75C2×#624—62no—
76#3804—68no—
77#3804—68no—
78C2×D3×Dic3—62no—
79#3792—34no—
80#3792C2×D314yescomplemented
81C2×#622—31no—
82C2×#622—31no—
83C2×#622—34no—
84C2×#622—34no—
85C2×#622C2×D311yescharacteristic
86C2×#622C2×D311yescharacteristic
87#3788—34no—
88#3787—34no—
89C22×C33—22no—
90C22×C33C22×C412yes—
91C22×C33C22×C411yescharacteristic
92C2×C3⋊(C3⋊D3)—22no—
93C2×C3⋊(C3⋊D3)C22×C412yes—
94C2×C32×D3—22no—
95C2×C32×D3C22×C412yes—
96C3⋊(C3⋊Dic3)C22×C412yes—
97C3×C3⋊Dic3C22×C411yescharacteristic
98C3×C3⋊Dic3C22×C412yes—
99C3×C3⋊Dic3C2411yescharacteristic
100C32×Dic3C22×C412yes—
101C2×#237—182no—
102C2×#235—91no—
103C2×C3×M16—91no—
104C22×C3⋊C8—91no—
105#907—98no—
106C2×ΓL(1,𝔽25)—91no—
107C2×ΓL(1,𝔽25)—91no—
108C2×D3×C8—91no—
109C2×D3×C8—91no—
110C23×C32—22no—
111C23×C32—31no—
112C23×C32—42no—
113C23×C32C4×D311yescharacteristic
114C22×C3⋊D3—31no—
115C22×C3⋊D3—62no—
116C22×C3⋊D3—122no—
117C22×C3×D3—22no—
118C32×D4—34no—
119C2×C32×C4—31no—
120#638—34no—
121#638—34no—
122#638—68no—
123#638—128no—
124C2×C3⋊Dic3—31no—
125C2×C3⋊Dic3—32no—
126C2×C3⋊Dic3—32no—
127C2×C3⋊Dic3—62no—
128C2×C3⋊Dic3—122no—
129C2×C3⋊Dic3C4×D311yescharacteristic
130C2×C3⋊Dic3C4×D312yes—
131C2×C3⋊Dic3C22×D311yescharacteristic
132C2×C3⋊Dic3C22×D312yes—
133C4×C3⋊D3—32no—
134C4×C3⋊D3—32no—
135#634—34no—
136C3×#67—28no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—64no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—64no—
142#627—68no—
143#626—68no—
144#624—64no—
145#624—64no—
146D3×Dic3—64no—
147D3×Dic3—64no—
148#622—32no—
149#622—32no—
150#622C22×D312yes—
151#622C22×D312yes—
152C2×#131—271noSylow
153C2×C33—22no—
154C2×C33#13112yescomplemented
155C2×C33C23×C411yescharacteristic
156C3⋊(C3⋊D3)—22no—
157C32×D3—22no—
158C2×C3×D4—62no—
159C22×C3×C4—91no—
160C2×#67—91no—
161C2×#67—182no—
162C2×#67—182no—
163C2×#67—362no—
164C2×#67#15711yescharacteristic
165C22×Dic3—62no—
166C22×Dic3—182no—
167#237—188no—
168#237—188no—
169#235—94no—
170#235—94no—
171C2×D12—91no—
172C2×C4×D3—91no—
173C2×C4×D3—91no—
174C2×C4×D3—182no—
175C2×Dic6—91no—
176C2×Dic6—182no—
177C3×M16—94no—
178C2×C3×C8—91no—
179C2×C3×C8—91no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—94no—
183ΓL(1,𝔽25)—94no—
184ΓL(1,𝔽25)—94no—
185D3×C8—94no—
186D3×C8—94no—
187C22×C32—22no—
188C22×C32—24no—
189C22×C32—22no—
190C22×C32—32no—
191C22×C32—42no—
192C22×C32—44no—
193C22×C32—44no—
194C22×C32—62no—
195C22×C32—84no—
196C22×C32C2×C4×D312yes—
197C22×C32C2×C4×D311yescharacteristic
198C2×C3⋊D3—32no—
199C2×C3⋊D3—64no—
200C2×C3⋊D3—62no—
201C2×C3⋊D3—124no—
202C2×C3⋊D3—124no—
203C2×C3⋊D3—244no—
204C2×C3×D3—24no—
205C2×C3×D3—44no—
206C32×C4—32no—
207C3⋊Dic3—32no—
208C3⋊Dic3—64no—
209C3⋊Dic3—124no—
210C3⋊Dic3C2×C4×D312yes—
211C3⋊Dic3C2×C4×D311yescharacteristic
212C3⋊Dic3C23×D311yescharacteristic
213C3×Dic3—24no—
214C3×Dic3—62no—
215C3×Dic3—64no—
216C3×Dic3—62no—
217C2×#41—271no—
218#131—278no—
219C2×M16—91no—
220C2×M16—271no—
221C2×M16—271no—
222C22×C8—271no—
223C22×C8—271no—
224C22×C8—271no—
225C33C2×#13111yescharacteristic, complemented, Sylow
226C23×C3—22no—
227C23×C3—42no—
228C23×C3—62no—
229C23×C3C2×#15711yescharacteristic
230C22×D3—91no—
231C22×D3—182no—
232C22×D3—362no—
233C22×D3C2×#15711yescharacteristic
234C3×D4—68no—
235C2×C3×C4—62no—
236C2×C3×C4—91no—
237C2×C3×C4—92no—
238C2×C3×C4—91no—
239C2×C3×C4—92no—
240#67—94no—
241#67—188no—
242#67—188no—
243#67—368no—
244#67C2×#15714yes—
245C2×Dic3—62no—
246C2×Dic3—64no—
247C2×Dic3—62no—
248C2×Dic3—64no—
249C2×Dic3—91no—
250C2×Dic3—184no—
251C2×Dic3—184no—
252C2×Dic3—182no—
253C2×Dic3—362no—
254C2×Dic3C2×#15711yescharacteristic
255D12—94no—
256C4×D3—92no—
257C4×D3—92no—
258C4×D3—92no—
259C4×D3—92no—
260C4×D3—184no—
261C4×D3—184no—
262Dic6—94no—
263Dic6—188no—
264C3×C8—92no—
265C3×C8—92no—
266C3⋊C8—92no—
267C3⋊C8—92no—
268C2×C32—24no—
269C2×C32—22no—
270C2×C32—22no—
271C2×C32—44no—
272C2×C32—42no—
273C2×C32—44no—
274C2×C32—62no—
275C2×C32—84no—
276C2×C32#90712yescomplemented
277C2×C32C22×C4×D311yescharacteristic
278C3⋊D3—62no—
279C3⋊D3—124no—
280C3⋊D3—244no—
281C3×D3—44no—
282#41—274no—
283#41—274no—
284C2×Q8—271no—
285C2×D4—31no—
286C2×D4—271no—
287C2×D4—271no—
288C22×C4—91no—
289C22×C4—271no—
290C22×C4—271no—
291M16—94no—
292M16—274no—
293M16—274no—
294C2×C8—91no—
295C2×C8—91no—
296C2×C8—274no—
297C2×C8—274no—
298C2×C8—271no—
299C2×C8—271no—
300C2×C8—274no—
301C22×C3—22no—
302C22×C3—24no—
303C22×C3—22no—
304C22×C3—44no—
305C22×C3—44no—
306C22×C3—42no—
307C22×C3—64no—
308C22×C3—84no—
309C22×C3—124no—
310C22×C3C22×#15712yes—
311C22×C3C22×#15711yescharacteristic
312C2×D3—22no—
313C2×D3—92no—
314C2×D3—184no—
315C2×D3—182no—
316C2×D3—364no—
317C2×D3—364no—
318C2×D3—724no—
319C2×D3C22×#15712yes—
320C3×C4—64no—
321C3×C4—91no—
322C3×C4—92no—
323C3×C4—91no—
324Dic3—64no—
325Dic3—62no—
326Dic3—62no—
327Dic3—92no—
328Dic3—184no—
329Dic3—364no—
330Dic3C22×#15712yes—
331C32—22no—
332C32—42no—
333C32C2×#90711yescharacteristic, complemented
334C23—31no—
335C23—271no—
336C23D3×#15711yescharacteristic
337Q8—274no—
338D4—34no—
339D4—274no—
340D4—274no—
341C2×C4—31no—
342C2×C4—92no—
343C2×C4—92no—
344C2×C4—91no—
345C2×C4—91no—
346C2×C4—272no—
347C2×C4—272no—
348C2×C4—272no—
349C2×C4—272no—
350C2×C4—271no—
351C8—92no—
352C8—92no—
353C8—272no—
354C8—272no—
355C2×C3—22no—
356C2×C3—22no—
357C2×C3—24no—
358C2×C3—44no—
359C2×C3—42no—
360C2×C3—44no—
361C2×C3—84no—
362C2×C3—124no—
363C2×C3#6428712yescomplemented
364C2×C3C23×#15711yescharacteristic
365D3—22no—
366D3—182no—
367D3—364no—
368D3—724no—
369C22—22no—
370C22—32no—
371C22—62no—
372C22—272no—
373C22—542no—
374C22C2×D3×#15712yes—
375C22C2×D3×#15711yescharacteristic
376C4—32no—
377C4—91no—
378C4—91no—
379C4—92no—
380C4—272no—
381C3—22no—
382C3—42no—
383C3C2×#6428711yescharacteristic, complemented
384C2—22no—
385C2—62no—
386C2—542no—
387C2#1172426712yesdirect product factor
388C2C22×D3×#15711yescharacteristic
subgroupG/H
centerC22C2×D3×#157
commutatorC2×C33C23×C4
FrattiniC2C22×D3×#157
FittingC23×C33C2×C4
socleC22×C33C22×C4
prime radicalC2×#89028C2
cosocleC3×C3⋊Dic3C24
hypercenterC23D3×#157
hypocenterC33C2×#131
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊#622C2semidirect
3. C2×C3×#3792C2semidirect
4. C2×#88926C2semidirect
5. C2×#88925C2semidirect
6. C2×#88924C2semidirect
7. #11724267C2direct (N × H)
8. C2×D3×#622C2semidirect
21. C2×C3⋊#622C22semidirect
22. C3×#3792C22semidirect
25. #88926C22semidirect
26. #88925C22semidirect
27. #88924C22semidirect
28. D3×#622C22semidirect
32. C2×#3792D3semidirect
80. #3792C2×D3semidirect
154. C2×C33#131semidirect
225. C33C2×#131semidirect
276. C2×C32#907semidirect
333. C32C2×#907semidirect
363. C2×C3#64287semidirect
383. C3C2×#64287semidirect
387. C2#11724267direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890282
C22×C3⋊#6222
C2×C3×#37922
C2×#889262
C2×#889252
C2×#889242
#117242672
C2×D3×#6222
C2×#642873
C2×#9079