Finite Group Databasebeta

← C2×#11723418 #13177261

Subgroups — C2×#11723418 #13177261

Order 1728.

2300 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C22×#6293C211yesmaximal, characteristic, complemented
3C2×#88857C211yesmaximal, characteristic, complemented
4C2×C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C2×#88807C211yesmaximal, characteristic, complemented
6C2×#88803C211yesmaximal, characteristic
7C2×#88803C211yesmaximal, characteristic
8#11723418C218yesmaximal, direct product factor
9C2×#64105—31nomaximal
10C2×#63851—34nomaximal
11C23×C3×C3⋊D3C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C2×C3×C4×C3⋊D3—24no—
17C2×C3×C4×C3⋊D3C2214yes—
18C22×C32×Dic3C2211yescharacteristic
19C22×C32×Dic3C2211yescharacteristic
20C2×#6293—24no—
21C2×#6293C2214yescomplemented
22#88857C2214yescomplemented
23C32×Dic3⋊C4C2214yescomplemented
24#88807C2214yescomplemented
25#88803C2214yes—
26#88803C2214yes—
27C22×C4×C3⋊D3D311yescharacteristic
28C22×#634—31no—
29C22×C3×C4×D3—34no—
30C22×#627—34no—
31C2×C32×C4⋊C4—31no—
32C2×#3752—31no—
33C2×#3752—31no—
34C2×#3750—31no—
35C2×#3750—31no—
36C2×#3750—34no—
37C2×#3750—64no—
38#64105—38no—
39C2×C3×Dic3⋊C4D314yescomplemented
40C2×Dic3⋊Dic3—34no—
41C2×(C2×D3)⋊Dic3—34no—
42C2×(C2×D3)⋊Dic3—34no—
43#63851—332no—
44C23×C33C2311yescharacteristic
45C22×C3×C3⋊D3—22no—
46C22×C3×C3⋊D3—22no—
47C22×C3×C3⋊D3Q814yescomplemented
48C22×C3×C3⋊D3C2312yes—
49C2×C33×C4—22no—
50C2×C33×C4C2312yes—
51C2×C3⋊(C3⋊Dic3)—22no—
52C2×C3⋊(C3⋊Dic3)—22no—
53C2×C3⋊(C3⋊Dic3)C2312yes—
54C2×C3⋊(C3⋊Dic3)C2312yes—
55C2×C3×C3⋊Dic3D414yes—
56C2×C3×C3⋊Dic3C2312yes—
57C3×C4×C3⋊D3—28no—
58C2×C32×Dic3—22no—
59C2×C32×Dic3—22no—
60C2×C32×Dic3C2312yes—
61C2×C32×Dic3C2312yes—
62#6293—28no—
63C2×#924—91no—
64C2×#896—94no—
65C23×C3⋊D3C2×D311yescharacteristic
66C23×C3×D3—34no—
67C22×C32×C4—24no—
68C22×C32×C4—31no—
69C22×C32×C4—31no—
70C22×C32×C4C2×D314yes—
71C22×C32×C4C2×D311yescharacteristic
72C22×C3⋊Dic3—31no—
73C22×C3⋊Dic3—31no—
74C22×C3⋊Dic3—34no—
75C22×C3⋊Dic3—34no—
76C22×C3⋊Dic3—64no—
77C22×C3⋊Dic3—64no—
78C22×C3⋊Dic3C2×D311yescharacteristic
79C2×C4×C3⋊D3—24no—
80C2×C4×C3⋊D3C2×D314yes—
81C2×#634—34no—
82C2×#634—64no—
83C22×C3×Dic3—34no—
84C22×C3×Dic3C2×D314yes—
85C22×C3×Dic3C2×D314yes—
86C2×C3×C4×D3—316no—
87C2×C3×C4×D3—616no—
88C2×#627—316no—
89C2×#627—616no—
90C32×C4⋊C4—34no—
91#3752—34no—
92#3752—34no—
93#3750—34no—
94#3750—316no—
95#3750—34no—
96#3750—616no—
97C3×Dic3⋊C4C2×D3116yescomplemented
98Dic3⋊Dic3—316no—
99(C2×D3)⋊Dic3—316no—
100(C2×D3)⋊Dic3—316no—
101C22×C33C2×D412yes—
102C22×C33C2×Q812yes—
103C22×C33#4112yes—
104C22×C33C2411yescharacteristic
105C2×C3×C3⋊D3—28no—
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3—22no—
108C2×C3×C3⋊D3C2×Q814yescomplemented
109C33×C4—22no—
110C3⋊(C3⋊Dic3)—22no—
111C3⋊(C3⋊Dic3)—22no—
112C3×C3⋊Dic3C2×D414yes—
113C32×Dic3—22no—
114C32×Dic3—22no—
115C23×C3×C4—91no—
116C22×C4×D3—34no—
117C22×Dic6—94no—
118C22×Dic6—91no—
119C2×C3×C4⋊C4—34no—
120C2×#217—91no—
121C2×#217—91no—
122C2×(C2×D3)⋊C4—94no—
123C2×(C2×D3)⋊C4—94no—
124C2×C4⋊Dic3—91no—
125#924—98no—
126C2×Dic3⋊C4—94no—
127C2×Dic3⋊C4—94no—
128C2×Dic3⋊C4—91no—
129C2×Dic3⋊C4—184no—
130C2×Dic3⋊C4C3⋊D311yescharacteristic, complemented
131#896—932no—
132C23×C32—24no—
133C23×C32C22×D314yes—
134C23×C32C22×D311yescharacteristic
135C22×C3⋊D3—22no—
136C22×C3⋊D3—22no—
137C22×C3⋊D3Dic614yescomplemented
138C22×C3⋊D3C22×D312yes—
139C22×C3×D3—316no—
140C22×C3×D3—38no—
141C22×C3×D3—68no—
142C22×C3×D3—68no—
143C2×C32×C4—22no—
144C2×C32×C4—28no—
145C2×C32×C4—28no—
146C2×C32×C4—32no—
147C2×C32×C4—32no—
148C2×C32×C4—48no—
149C2×C32×C4—62no—
150C2×C32×C4—62no—
151C2×C32×C4C22×D318yes—
152C2×C32×C4C22×D312yes—
153C2×C3⋊Dic3—32no—
154C2×C3⋊Dic3—32no—
155C2×C3⋊Dic3—38no—
156C2×C3⋊Dic3—38no—
157C2×C3⋊Dic3—68no—
158C2×C3⋊Dic3—68no—
159C2×C3⋊Dic3—616no—
160C2×C3⋊Dic3—68no—
161C2×C3⋊Dic3—62no—
162C2×C3⋊Dic3—62no—
163C2×C3⋊Dic3—68no—
164C2×C3⋊Dic3—128no—
165C2×C3⋊Dic3#6714yes—
166C2×C3⋊Dic3C22×D312yes—
167C4×C3⋊D3—28no—
168#634—68no—
169C2×C3×Dic3—28no—
170C2×C3×Dic3—28no—
171C2×C3×Dic3—316no—
172C2×C3×Dic3—38no—
173C2×C3×Dic3C22×D318yes—
174C2×C3×Dic3C22×D318yes—
175C3×C4×D3—632no—
176#627—632no—
177C2×C22⋊Q8—271noSylow
178C2×C33C22⋊Q814yescomplemented
179C2×C33C22×D411yescharacteristic
180C2×C33C22×Q811yescharacteristic
181C2×C33C2×#4111yescharacteristic
182C3×C3⋊D3—24no—
183C24×C3—91no—
184C23×D3—34no—
185C22×C3×C4—24no—
186C22×C3×C4—34no—
187C22×C3×C4—34no—
188C22×C3×C4—91no—
189C22×C3×C4—94no—
190C22×C3×C4—184no—
191C22×C3×C4D3214yes—
192C22×C3×C4C2×C3⋊D311yescharacteristic
193C22×Dic3—34no—
194C22×Dic3—94no—
195C22×Dic3—94no—
196C22×Dic3—91no—
197C22×Dic3—91no—
198C22×Dic3—184no—
199C22×Dic3—184no—
200C22×Dic3C2×C3⋊D311yescharacteristic
201C22×Dic3C2×C3⋊D311yescharacteristic
202C2×C4×D3—316no—
203C2×C4×D3—616no—
204C2×Dic6—916no—
205C2×Dic6—94no—
206C2×Dic6—184no—
207C2×Dic6—1816no—
208C3×C4⋊C4—316no—
209#217—94no—
210#217—94no—
211(C2×D3)⋊C4—916no—
212(C2×D3)⋊C4—916no—
213C4⋊Dic3—94no—
214Dic3⋊C4—916no—
215Dic3⋊C4—916no—
216Dic3⋊C4—94no—
217Dic3⋊C4—1816no—
218Dic3⋊C4C2×C3⋊D314yescomplemented
219C22×C32—24no—
220C22×C32—28no—
221C22×C32—28no—
222C22×C32—28no—
223C22×C32C2×Dic612yes—
224C22×C32#23518yes—
225C22×C32#23512yes—
226C22×C32D3×D418yes—
227C22×C32D3×Q818yes—
228C22×C32C2×#6712yes—
229C22×C32C23×D314yes—
230C22×C32C23×D311yescharacteristic
231C2×C3⋊D3—28no—
232C2×C3⋊D3—22no—
233C2×C3⋊D3—22no—
234C2×C3⋊D3C2×Dic614yescomplemented
235C2×C3×D3—316no—
236C2×C3×D3—632no—
237C2×C3×D3—68no—
238C2×C3×D3—68no—
239C32×C4—22no—
240C32×C4—28no—
241C32×C4—48no—
242C32×C4—62no—
243C32×C4—62no—
244C3⋊Dic3—616no—
245C3⋊Dic3—68no—
246C3⋊Dic3—62no—
247C3⋊Dic3—62no—
248C3⋊Dic3—68no—
249C3⋊Dic3—128no—
250C3⋊Dic3C2×#6714yes—
251C3×Dic3—28no—
252C3×Dic3—28no—
253C3×Dic3—316no—
254C22×Q8—271no—
255C23×C4—91no—
256C22⋊Q8—278no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—271no—
259C2×C4⋊C4—271no—
260C2×C22⋊C4—271no—
261C2×C22⋊C4—271no—
262C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
263C23×C3—24no—
264C23×C3—92no—
265C23×C3—94no—
266C23×C3—182no—
267C23×C3—182no—
268C23×C3C2×D3214yes—
269C23×C3C22×C3⋊D311yescharacteristic
270C22×D3—38no—
271C22×D3—316no—
272C22×D3—68no—
273C22×D3—68no—
274C2×C3×C4—28no—
275C2×C3×C4—22no—
276C2×C3×C4—28no—
277C2×C3×C4—38no—
278C2×C3×C4—38no—
279C2×C3×C4—48no—
280C2×C3×C4—68no—
281C2×C3×C4—68no—
282C2×C3×C4—92no—
283C2×C3×C4—94no—
284C2×C3×C4—188no—
285C2×C3×C4C2×D3218yes—
286C2×C3×C4C22×C3⋊D312yes—
287C2×Dic3—22no—
288C2×Dic3—22no—
289C2×Dic3—38no—
290C2×Dic3—316no—
291C2×Dic3—98no—
292C2×Dic3—98no—
293C2×Dic3—92no—
294C2×Dic3—92no—
295C2×Dic3—188no—
296C2×Dic3—188no—
297C2×Dic3—1816no—
298C2×Dic3—188no—
299C2×Dic3—188no—
300C2×Dic3—182no—
301C2×Dic3—182no—
302C2×Dic3—368no—
303C2×Dic3C22×C3⋊D312yes—
304C2×Dic3C22×C3⋊D312yes—
305C4×D3—632no—
306Dic6—188no—
307Dic6—1832no—
308C2×C32—24no—
309C2×C32—24no—
310C2×C32—24no—
311C2×C32—216no—
312C2×C32#896116yescomplemented
313C2×C32#92414yescomplemented
314C2×C32C22×Dic611yescharacteristic
315C2×C32C2×#23514yes—
316C2×C32C2×#23511yescharacteristic
317C2×C32C2×D3×D414yes—
318C2×C32C2×D3×Q814yes—
319C2×C32C22×#6711yescharacteristic
320C3⋊D3—24no—
321C3×D3—616no—
322C24—91no—
323C2×Q8—274no—
324C2×Q8—544no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—91no—
328C22×C4—94no—
329C22×C4—184no—
330C22×C4—271no—
331C22×C4—271no—
332C22×C4D3×C3⋊D311yescharacteristic
333C4⋊C4—34no—
334C4⋊C4—274no—
335C4⋊C4—274no—
336C22⋊C4—274no—
337C22⋊C4—274no—
338C22×C3—24no—
339C22×C3—28no—
340C22×C3—28no—
341C22×C3—28no—
342C22×C3—94no—
343C22×C3—182no—
344C22×C3—182no—
345C22×C3—188no—
346C22×C3D3×Dic618yes—
347C22×C3#379818yes—
348C22×C3D3×#6718yes—
349C22×C3#382812yes—
350C22×C3D4×C3⋊D312yes—
351C22×C3Q8×C3⋊D312yes—
352C22×C3C22×D3214yes—
353C22×C3C23×C3⋊D311yescharacteristic
354C2×D3—316no—
355C2×D3—68no—
356C2×D3—68no—
357C2×D3—632no—
358C3×C4—28no—
359C3×C4—22no—
360C3×C4—48no—
361C3×C4—68no—
362C3×C4—68no—
363C3×C4—94no—
364Dic3—22no—
365Dic3—22no—
366Dic3—316no—
367Dic3—1816no—
368Dic3—188no—
369Dic3—188no—
370Dic3—182no—
371Dic3—182no—
372Dic3—368no—
373C32—24no—
374C32C2×#89614yescomplemented
375C32C2×#92411yescharacteristic, complemented
376C23—92no—
377C23—94no—
378C23—182no—
379C23—182no—
380C23C2×D3×C3⋊D311yescharacteristic
381Q8—548no—
382C2×C4—22no—
383C2×C4—32no—
384C2×C4—32no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—92no—
388C2×C4—94no—
389C2×C4—188no—
390C2×C4—272no—
391C2×C4—272no—
392C2×C4—542no—
393C2×C4—542no—
394C2×C4C2×D3×C3⋊D312yes—
395C2×C3—24no—
396C2×C3—24no—
397C2×C3—24no—
398C2×C3—216no—
399C2×C3—184no—
400C2×C3#63851116yescomplemented
401C2×C3#6410514yescomplemented
402C2×C3C2×D3×Dic614yes—
403C2×C3C2×#379814yes—
404C2×C3C2×D3×#6714yes—
405C2×C3C2×#382811yescharacteristic
406C2×C3C2×D4×C3⋊D311yescharacteristic
407C2×C3C2×Q8×C3⋊D311yescharacteristic
408D3—616no—
409C22—94no—
410C22—182no—
411C22—182no—
412C22—188no—
413C22C3⋊D3×Dic612yes—
414C22#8902012yes—
415C22C3⋊D3×#6712yes—
416C22C22×D3×C3⋊D311yescharacteristic
417C4—22no—
418C4—62no—
419C4—62no—
420C4—94no—
421C4—542no—
422C4—542no—
423C3—24no—
424C3C2×#6385114yescomplemented
425C3C2×#6410511yescharacteristic, complemented
426C2—184no—
427C2#1172341814yesdirect product factor
428C2C2×C3⋊D3×Dic611yescharacteristic
429C2C2×#8902011yescharacteristic
430C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutator, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×C22⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×#6293C2semidirect
3. C2×#88857C2semidirect
4. C2×C32×Dic3⋊C4C2semidirect
5. C2×#88807C2semidirect
8. #11723418C2direct (N × H)
21. C2×#6293C22semidirect
22. #88857C22semidirect
23. C32×Dic3⋊C4C22semidirect
24. #88807C22semidirect
39. C2×C3×Dic3⋊C4D3semidirect
47. C22×C3×C3⋊D3Q8semidirect
97. C3×Dic3⋊C4C2×D3semidirect
108. C2×C3×C3⋊D3C2×Q8semidirect
130. C2×Dic3⋊C4C3⋊D3semidirect
137. C22×C3⋊D3Dic6semidirect
178. C2×C33C22⋊Q8semidirect
218. Dic3⋊C4C2×C3⋊D3semidirect
234. C2×C3⋊D3C2×Dic6semidirect
262. C33C2×C22⋊Q8semidirect
312. C2×C32#896semidirect
313. C2×C32#924semidirect
374. C32C2×#896semidirect
375. C32C2×#924semidirect
400. C2×C3#63851semidirect
401. C2×C3#64105semidirect
424. C3C2×#63851semidirect
425. C3C2×#64105semidirect
427. C2#11723418direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C22×#62932
C2×#888572
C2×C32×Dic3⋊C42
C2×#888072
C2×#888032
#117234182
C2×#641053
C2×#638513