Finite Group Databasebeta

← C2×#11720997 #13150225

Subgroups — C2×#11720997 #13150225

Order 1728.

1262 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊(C3⋊D9)C211yesmaximal, characteristic
2C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
3C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
4C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
5C2×#88445C211yesmaximal, characteristic, complemented
6C2×#88445C211yesmaximal, characteristic, complemented
7C2×#88445C211yesmaximal, characteristic, complemented
8#11720997C218yesmaximal, direct product factor
9C2×#63897—31nomaximal
10C2×#63456—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C23×C3⋊D9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C2×C4⋊(C3⋊D9)—24no—
15C2×C4⋊(C3⋊D9)C2214yes—
16C22×C9×Dic3C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×(C3⋊D9)⋊C4—24no—
21C2×(C3⋊D9)⋊C4C2214yescomplemented
22C9×Dic3⋊C4C2214yescomplemented
23C3×Dic9⋊C4C2214yescomplemented
24#88445C2214yescomplemented
25#88445C2214yescomplemented
26#88445C2214yescomplemented
27C22×C4⋊(C3⋊D3)—31no—
28C22×#624—31no—
29C2×C3×Dic3⋊C4—31no—
30C2×C3×Dic3⋊C4D311yescharacteristic
31C2×#3722—31no—
32C2×#3722—31no—
33C2×#3722—31no—
34#63897—38no—
35C22×D36—31no—
36C22×D36—61no—
37C22×C4×D9—31no—
38C2×C9×C4⋊C4—31no—
39C2×(C2×D9)⋊C4—31no—
40C2×(C2×D9)⋊C4—31no—
41C2×(C2×D9)⋊C4—31no—
42C2×Dic9⋊C4D311yescharacteristic, complemented
43#63456—38no—
44C23×C3×C9C2311yescharacteristic
45C22×C3⋊D9—22no—
46C22×C3⋊D9—22no—
47C22×C3⋊D9—22no—
48C22×C3⋊D9—22no—
49C22×C3⋊D9—22no—
50C22×C3⋊D9D414yes—
51C22×C3⋊D9C2312yes—
52C22×C3⋊D9C2312yes—
53C2×C3×C4×C9—22no—
54C2×C3×C4×C9C2312yes—
55C4⋊(C3⋊D9)—28no—
56C2×C9×Dic3—22no—
57C2×C9×Dic3—22no—
58C2×C9×Dic3C2312yes—
59C2×C9×Dic3C2312yes—
60C2×C3×Dic9—22no—
61C2×C3×Dic9—22no—
62C2×C3×Dic9C2312yes—
63C2×C3×Dic9C2312yes—
64(C3⋊D9)⋊C4—28no—
65C2×#894—91no—
66C2×#894—91no—
67C23×C3⋊D3—31no—
68C23×C3⋊D3—31no—
69C22×C32×C4C2×D311yescharacteristic
70C2×C4⋊(C3⋊D3)—34no—
71C2×C4⋊(C3⋊D3)—64no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3C2×D311yescharacteristic
75C22×C3×Dic3C2×D311yescharacteristic
76C2×#624—34no—
77C2×#624—64no—
78C23×D9—31no—
79C23×D9—31no—
80C23×D9—61no—
81C23×D9—61no—
82C3×Dic3⋊C4—34no—
83C3×Dic3⋊C4C2×D314yes—
84#3722—34no—
85#3722—34no—
86#3722—34no—
87C22×C4×C9—21no—
88C22×C4×C9—31no—
89C22×C4×C9—31no—
90C22×C4×C9C2×D311yescharacteristic
91C22×Dic9C2×D311yescharacteristic
92C22×Dic9C2×D311yescharacteristic
93C2×D36—34no—
94C2×D36—64no—
95C2×D36—64no—
96C2×D36—68no—
97C2×C4×D9—34no—
98C2×C4×D9—64no—
99C9×C4⋊C4—34no—
100(C2×D9)⋊C4—34no—
101(C2×D9)⋊C4—34no—
102(C2×D9)⋊C4—34no—
103Dic9⋊C4C2×D314yescomplemented
104C22×C3×C9C2×D412yes—
105C22×C3×C9#4112yes—
106C22×C3×C9#4112yes—
107C22×C3×C9C2411yescharacteristic
108C2×C3⋊D9—28no—
109C2×C3⋊D9—22no—
110C2×C3⋊D9—22no—
111C2×C3⋊D9—22no—
112C2×C3⋊D9—22no—
113C2×C3⋊D9—22no—
114C2×C3⋊D9—44no—
115C2×C3⋊D9C2×D414yes—
116C3×C4×C9—22no—
117C9×Dic3—22no—
118C9×Dic3—22no—
119C3×Dic9—22no—
120C3×Dic9—22no—
121C22×D12—91no—
122C22×D12—91no—
123C22×D12—181no—
124C22×C4×D3—91no—
125C22×C4×D3—91no—
126C2×C3×C4⋊C4—31no—
127C2×C3×C4⋊C4—91no—
128C2×(C2×D3)⋊C4—91no—
129C2×(C2×D3)⋊C4—91no—
130C2×(C2×D3)⋊C4—91no—
131C2×(C2×D3)⋊C4—91no—
132C2×(C2×D3)⋊C4—91no—
133C2×(C2×D3)⋊C4—91no—
134C2×Dic3⋊C4—31no—
135C2×Dic3⋊C4D911yescharacteristic, complemented
136#894—98no—
137#894—98no—
138C23×C32C22×D311yescharacteristic
139C22×C3⋊D3—34no—
140C22×C3⋊D3—32no—
141C22×C3⋊D3—32no—
142C22×C3⋊D3—62no—
143C22×C3⋊D3—62no—
144C22×C3⋊D3—62no—
145C22×C3⋊D3—62no—
146C22×C3⋊D3—62no—
147C2×C32×C4—22no—
148C2×C32×C4C22×D312yes—
149C4⋊(C3⋊D3)—68no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3C22×D312yes—
157C2×C3×Dic3C22×D312yes—
158#624—68no—
159C23×C9—21no—
160C23×C9C22×D311yescharacteristic
161C22×D9—34no—
162C22×D9—32no—
163C22×D9—32no—
164C22×D9—64no—
165C22×D9—62no—
166C22×D9—62no—
167C22×D9—62no—
168C22×D9—62no—
169C22×D9—64no—
170C22×D9—62no—
171C22×D9—62no—
172C22×D9—62no—
173C22×D9—122no—
174C22×D9—122no—
175C22×D9—122no—
176C22×D9—122no—
177C2×C4×C9—24no—
178C2×C4×C9—22no—
179C2×C4×C9—22no—
180C2×C4×C9—32no—
181C2×C4×C9—32no—
182C2×C4×C9—62no—
183C2×C4×C9—62no—
184C2×C4×C9C22×D312yes—
185C2×Dic9—22no—
186C2×Dic9—22no—
187C2×Dic9C22×D312yes—
188C2×Dic9C22×D312yes—
189D36—68no—
190D36—616no—
191C4×D9—68no—
192C2×#123—271noSylow
193C2×C3×C9#12314yescomplemented
194C2×C3×C9C22×D411yescharacteristic
195C2×C3×C9C2×#4111yescharacteristic
196C2×C3×C9C2×#4111yescharacteristic
197C3⋊D9—24no—
198C3⋊D9—42no—
199C23×D3—91no—
200C23×D3—91no—
201C23×D3—91no—
202C23×D3—91no—
203C23×D3—181no—
204C23×D3—181no—
205C22×C3×C4—21no—
206C22×C3×C4—31no—
207C22×C3×C4—31no—
208C22×C3×C4—91no—
209C22×C3×C4—91no—
210C22×C3×C4C2×D911yescharacteristic
211C22×C3×C4D3211yescharacteristic
212C22×Dic3—31no—
213C22×Dic3—31no—
214C22×Dic3C2×D911yescharacteristic
215C22×Dic3C2×D911yescharacteristic
216C2×D12—94no—
217C2×D12—94no—
218C2×D12—184no—
219C2×D12—184no—
220C2×D12—184no—
221C2×D12—188no—
222C2×C4×D3—94no—
223C2×C4×D3—94no—
224C2×C4×D3—184no—
225C2×C4×D3—184no—
226C3×C4⋊C4—34no—
227C3×C4⋊C4—94no—
228(C2×D3)⋊C4—94no—
229(C2×D3)⋊C4—94no—
230(C2×D3)⋊C4—94no—
231(C2×D3)⋊C4—94no—
232(C2×D3)⋊C4—94no—
233(C2×D3)⋊C4—94no—
234Dic3⋊C4—34no—
235Dic3⋊C4C2×D914yescomplemented
236C22×C32#23512yes—
237C22×C32D3×D412yes—
238C22×C32#23912yes—
239C22×C32C23×D311yescharacteristic
240C2×C3⋊D3—34no—
241C2×C3⋊D3—62no—
242C2×C3⋊D3—68no—
243C2×C3⋊D3—62no—
244C2×C3⋊D3—62no—
245C2×C3⋊D3—62no—
246C2×C3⋊D3—62no—
247C2×C3⋊D3—124no—
248C32×C4—22no—
249C3×Dic3—22no—
250C3×Dic3—22no—
251C3×Dic3—62no—
252C3×Dic3—62no—
253C22×C9—21no—
254C22×C9—22no—
255C22×C9—22no—
256C22×C9—22no—
257C22×C9#23512yes—
258C22×C9D3×D412yes—
259C22×C9#23912yes—
260C22×C9C23×D311yescharacteristic
261C2×D9—34no—
262C2×D9—64no—
263C2×D9—68no—
264C2×D9—62no—
265C2×D9—62no—
266C2×D9—64no—
267C2×D9—62no—
268C2×D9—62no—
269C2×D9—62no—
270C2×D9—122no—
271C2×D9—122no—
272C2×D9—128no—
273C2×D9—122no—
274C2×D9—122no—
275C2×D9—128no—
276C2×D9—124no—
277C4×C9—24no—
278C4×C9—22no—
279C4×C9—62no—
280C4×C9—62no—
281Dic9—22no—
282Dic9—22no—
283C22×D4—271no—
284C23×C4—271no—
285#123—278no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—91no—
288C2×C22⋊C4—271no—
289C2×C22⋊C4—271no—
290C2×C22⋊C4—271no—
291C3×C9C2×#12311yescharacteristic, complemented, Sylow
292C23×C3—21no—
293C23×C3C22×D911yescharacteristic
294C23×C3C2×D3211yescharacteristic
295C22×D3—92no—
296C22×D3—94no—
297C22×D3—92no—
298C22×D3—94no—
299C22×D3—92no—
300C22×D3—92no—
301C22×D3—182no—
302C22×D3—184no—
303C22×D3—182no—
304C22×D3—182no—
305C22×D3—182no—
306C22×D3—182no—
307C22×D3—182no—
308C22×D3—184no—
309C22×D3—182no—
310C22×D3—182no—
311C22×D3—182no—
312C22×D3—182no—
313C22×D3—182no—
314C22×D3—182no—
315C22×D3—362no—
316C22×D3—362no—
317C22×D3—362no—
318C22×D3—362no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—32no—
324C2×C3×C4—32no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—92no—
328C2×C3×C4—92no—
329C2×C3×C4—182no—
330C2×C3×C4—182no—
331C2×C3×C4C22×D912yes—
332C2×C3×C4C2×D3212yes—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—32no—
336C2×Dic3—32no—
337C2×Dic3—62no—
338C2×Dic3—62no—
339C2×Dic3C22×D912yes—
340C2×Dic3C22×D912yes—
341D12—188no—
342D12—188no—
343D12—1816no—
344C4×D3—188no—
345C4×D3—188no—
346C2×C32#89414yes—
347C2×C32C2×#23511yescharacteristic
348C2×C32C2×D3×D411yescharacteristic
349C2×C32C2×#23911yescharacteristic
350C3⋊D3—64no—
351C3⋊D3—122no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9—24no—
356C2×C9#89414yescomplemented
357C2×C9C2×#23511yescharacteristic
358C2×C9C2×D3×D411yescharacteristic
359C2×C9C2×#23911yescharacteristic
360D9—64no—
361D9—124no—
362D9—124no—
363D9—122no—
364C24—271no—
365C24—271no—
366C2×D4—274no—
367C2×D4—544no—
368C22×C4—31no—
369C22×C4—31no—
370C22×C4—91no—
371C22×C4—91no—
372C22×C4—274no—
373C22×C4—544no—
374C22×C4D3×D911yescharacteristic
375C4⋊C4—34no—
376C4⋊C4—94no—
377C22⋊C4—274no—
378C22⋊C4—274no—
379C22⋊C4—274no—
380C22×C3—22no—
381C22×C3—21no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3#369712yes—
385C22×C3D4×D912yes—
386C22×C3#370112yes—
387C22×C3C23×D911yescharacteristic
388C22×C3#379912yes—
389C22×C3#379912yes—
390C22×C3#381112yes—
391C22×C3C22×D3211yescharacteristic
392C2×D3—94no—
393C2×D3—94no—
394C2×D3—184no—
395C2×D3—182no—
396C2×D3—182no—
397C2×D3—188no—
398C2×D3—182no—
399C2×D3—188no—
400C2×D3—182no—
401C2×D3—184no—
402C2×D3—182no—
403C2×D3—182no—
404C2×D3—182no—
405C2×D3—182no—
406C2×D3—182no—
407C2×D3—182no—
408C2×D3—368no—
409C2×D3—362no—
410C2×D3—362no—
411C2×D3—362no—
412C2×D3—362no—
413C2×D3—368no—
414C2×D3—364no—
415C2×D3—364no—
416C3×C4—24no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4—62no—
420C3×C4—62no—
421C3×C4—182no—
422C3×C4—182no—
423Dic3—22no—
424Dic3—22no—
425Dic3—62no—
426Dic3—62no—
427C32C2×#89411yescharacteristic
428C9—21no—
429C9C2×#89411yescharacteristic, complemented
430C23—272no—
431C23—274no—
432C23—272no—
433C23—542no—
434C23—542no—
435C23—542no—
436C23—542no—
437C23—542no—
438C23C2×D3×D911yescharacteristic
439D4—548no—
440C2×C4—22no—
441C2×C4—32no—
442C2×C4—32no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—92no—
446C2×C4—92no—
447C2×C4—182no—
448C2×C4—182no—
449C2×C4—548no—
450C2×C4C2×D3×D912yes—
451C2×C3—24no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3#6345614yescomplemented
456C2×C3C2×#369711yescharacteristic
457C2×C3C2×D4×D911yescharacteristic
458C2×C3C2×#370111yescharacteristic
459C2×C3#6389714yes—
460C2×C3C2×#379911yescharacteristic
461C2×C3C2×#379911yescharacteristic
462C2×C3C2×#381111yescharacteristic
463D3—184no—
464D3—184no—
465D3—364no—
466D3—364no—
467D3—362no—
468D3—362no—
469C22—274no—
470C22—548no—
471C22—542no—
472C22—542no—
473C22—542no—
474C22—542no—
475C22—542no—
476C22—1084no—
477C22#8863512yes—
478C22#8864212yes—
479C22#8866812yes—
480C22C22×D3×D911yescharacteristic
481C4—22no—
482C4—62no—
483C4—62no—
484C4—182no—
485C4—182no—
486C3—21no—
487C3C2×#6345611yescharacteristic, complemented
488C3C2×#6389711yescharacteristic
489C2—544no—
490C2—1082no—
491C2#1172099714yesdirect product factor
492C2C2×#8863511yescharacteristic
493C2C2×#8864211yescharacteristic
494C2C2×#8866811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC22×C4⋊(C3⋊D9)C2
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#123
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×(C3⋊D9)⋊C4C2semidirect
3. C2×C9×Dic3⋊C4C2semidirect
4. C2×C3×Dic9⋊C4C2semidirect
5. C2×#88445C2semidirect
6. C2×#88445C2semidirect
7. C2×#88445C2semidirect
8. #11720997C2direct (N × H)
21. C2×(C3⋊D9)⋊C4C22semidirect
22. C9×Dic3⋊C4C22semidirect
23. C3×Dic9⋊C4C22semidirect
24. #88445C22semidirect
25. #88445C22semidirect
26. #88445C22semidirect
42. C2×Dic9⋊C4D3semidirect
103. Dic9⋊C4C2×D3semidirect
135. C2×Dic3⋊C4D9semidirect
193. C2×C3×C9#123semidirect
235. Dic3⋊C4C2×D9semidirect
291. C3×C9C2×#123semidirect
356. C2×C9#894semidirect
429. C9C2×#894semidirect
455. C2×C3#63456semidirect
487. C3C2×#63456semidirect
491. C2#11720997direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊(C3⋊D9)2
C22×(C3⋊D9)⋊C42
C2×C9×Dic3⋊C42
C2×C3×Dic9⋊C42
C2×#884452
#117209972
C2×#638973
C2×#634563