Finite Group Databasebeta

← C2×#11720841 #13149818

Subgroups — C2×#11720841 #13149818

Order 1728.

999 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88650C211yesmaximal, characteristic
2C2×#88527C211yesmaximal, characteristic, complemented
3C2×#3743⋊C3C211yesmaximal, characteristic, complemented
4C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
5C2×#88434C211yesmaximal, characteristic, complemented
6C2×#88430C211yesmaximal, characteristic, complemented
7C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
8#11720841C218yesmaximal, direct product factor
9C2×#63805—31nomaximal
10C2×#63804—31nomaximal
11C2×C4×#272C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×#6200C411yescharacteristic
15C2×#6200C411yescharacteristic
16C2×#6198C411yescharacteristic
17C2×#6196C411yescharacteristic
18#88650C414yes—
19#88650C2214yes—
20C2×C8×H(3,𝔽3)C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22#88527C2214yescomplemented
23C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
24C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
25#3743⋊C3C2214yescomplemented
26C8×(C3⋊D3)⋊C3C2214yescomplemented
27#88434C2214yescomplemented
28#88430C2214yescomplemented
29((C3⋊D3)⋊C3)⋊C8C2214yescomplemented
30C2×#3798—31no—
31C2×#3798—31no—
32C2×#3743D311yescharacteristic, complemented
33C2×C8×C3⋊D3D311yescharacteristic, complemented
34C2×C3×ΓL(1,𝔽25)—31no—
35C2×C3×ΓL(1,𝔽25)—31no—
36C2×C3×ΓL(1,𝔽25)—31no—
37C2×C3×ΓL(1,𝔽25)—61no—
38C2×C3×D3×C8—31no—
39C2×#3712—31no—
40C2×#3711—31no—
41C2×#3710—31no—
42C2×D3×C3⋊C8—31no—
43C2×D3×C3⋊C8—31no—
44C2×D3×C3⋊C8—31no—
45#63805—38no—
46#63804—38no—
47C22×#272C2×C411yescharacteristic
48C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
49C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×#1084C2×C411yescharacteristic
52C4×#272C2×C412yes—
53C4×#272C2312yes—
54C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
55C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
56C4×(C3⋊D3)⋊C3C2×C412yes—
57C4×(C3⋊D3)⋊C3C2×C412yes—
58C4×(C3⋊D3)⋊C3C2312yes—
59C4×(C3⋊D3)⋊C3C2312yes—
60#6200C2×C414yes—
61#6200C2×C414yes—
62#6198C2×C414yes—
63#6196C2×C414yes—
64C8×H(3,𝔽3)C2312yes—
65#6180C2312yes—
66(C3⋊(C3⋊C8))⋊C3C2312yes—
67(C3⋊(C3⋊C8))⋊C3C2312yes—
68C2×#907—91no—
69C2×#907—91no—
70C2×#901—91no—
71C2×C4×C3⋊D3C2×D311yescharacteristic
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×C3×C4×D3—31no—
74C2×C3×C4×D3—31no—
75C2×C3×C4×D3—31no—
76C2×C3×C4×D3—31no—
77C2×C3×C4×D3—61no—
78C2×#627—31no—
79C2×#627—31no—
80C2×#626—31no—
81C2×#626—31no—
82C2×#626—31no—
83C2×#626—31no—
84C2×#625—31no—
85C2×#625—31no—
86#3798—34no—
87#3798—34no—
88#3798—34no—
89#3798—34no—
90C2×C32×C8—21no—
91C2×C32×C8C2×D311yescharacteristic
92C2×C32×C8C2×D311yescharacteristic
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
94C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
95#3743C2×D314yescomplemented
96C8×C3⋊D3C2×D314yescomplemented
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C3×ΓL(1,𝔽25)—34no—
103C3×ΓL(1,𝔽25)—34no—
104C3×ΓL(1,𝔽25)—34no—
105C3×ΓL(1,𝔽25)—64no—
106C3×D3×C8—34no—
107#3712—34no—
108#3711—34no—
109#3710—34no—
110D3×C3⋊C8—34no—
111D3×C3⋊C8—34no—
112D3×C3⋊C8—34no—
113C22×H(3,𝔽3)C22×C411yescharacteristic
114C2×#272—22no—
115C2×#272C22×C412yes—
116C2×(C3⋊D3)⋊C3—22no—
117C2×(C3⋊D3)⋊C3—22no—
118C2×(C3⋊D3)⋊C3C22×C412yes—
119C2×(C3⋊D3)⋊C3C22×C412yes—
120C4×H(3,𝔽3)C22×C411yescharacteristic
121C4×H(3,𝔽3)C2411yescharacteristic
122#1084C22×C412yes—
123(C3⋊Dic3)⋊C3C22×C412yes—
124(C3⋊Dic3)⋊C3C22×C412yes—
125C2×#235—91no—
126C2×#235—91no—
127C2×#235—91no—
128C2×C3×M16—91no—
129C2×C3×M16—91no—
130C22×C3×C8—91no—
131C2×#208—91no—
132C22×C3⋊C8—91no—
133C22×C3⋊C8—91no—
134#907—98no—
135#907—98no—
136#901—98no—
137C2×ΓL(1,𝔽25)—31no—
138C2×ΓL(1,𝔽25)—31no—
139C2×ΓL(1,𝔽25)—31no—
140C2×ΓL(1,𝔽25)—61no—
141C2×ΓL(1,𝔽25)—91no—
142C2×ΓL(1,𝔽25)—91no—
143C2×ΓL(1,𝔽25)—91no—
144C2×D3×C8—31no—
145C2×D3×C8—91no—
146C2×D3×C8—91no—
147C2×D3×C8—91no—
148C2×D3×C8—91no—
149C2×D3×C8—91no—
150C22×C3⋊D3C4×D311yescharacteristic
151C22×C3⋊D3C4×D311yescharacteristic
152C22×C3×D3—31no—
153C22×C3×D3—31no—
154C22×C3×D3—31no—
155C22×C3×D3—31no—
156C22×C3×D3—61no—
157C2×C32×C4—21no—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C3⋊Dic3C4×D311yescharacteristic
161C2×C3⋊Dic3C4×D311yescharacteristic
162C4×C3⋊D3C4×D312yes—
163C4×C3⋊D3C4×D312yes—
164C4×C3⋊D3C22×D312yes—
165C4×C3⋊D3C22×D312yes—
166C2×C3×Dic3—31no—
167C2×C3×Dic3—31no—
168C2×C3×Dic3—31no—
169C2×C3×Dic3—31no—
170C2×C3×Dic3—61no—
171C3×C4×D3—32no—
172C3×C4×D3—32no—
173C3×C4×D3—32no—
174C3×C4×D3—32no—
175C3×C4×D3—32no—
176C3×C4×D3—32no—
177C3×C4×D3—32no—
178C3×C4×D3—32no—
179C3×C4×D3—62no—
180C3×C4×D3—62no—
181#627—34no—
182#627—34no—
183#626—34no—
184#626—34no—
185#626—34no—
186#626—34no—
187#625—34no—
188#625—34no—
189C32×C8—22no—
190C32×C8C22×D312yes—
191C32×C8C22×D312yes—
192C3⋊(C3⋊C8)C22×D312yes—
193C3⋊(C3⋊C8)C22×D312yes—
194C3×C3⋊C8—32no—
195C3×C3⋊C8—32no—
196C3×C3⋊C8—32no—
197C3×C3⋊C8—32no—
198C3×C3⋊C8—62no—
199C2×#131—271noSylow
200C2×H(3,𝔽3)#13112yescomplemented
201C2×H(3,𝔽3)C23×C411yescharacteristic
202#272—22no—
203(C3⋊D3)⋊C3—22no—
204(C3⋊D3)⋊C3—22no—
205C22×C3×C4—91no—
206C22×C3×C4—91no—
207C22×C3×C4—91no—
208C2×#67—91no—
209C2×#67—91no—
210C2×#67—91no—
211C2×#67—91no—
212C2×#67—91no—
213C2×#67—91no—
214#235—94no—
215#235—94no—
216#235—94no—
217#235—94no—
218#235—94no—
219#235—94no—
220C2×D12—91no—
221C2×D12—91no—
222C2×D12—91no—
223C2×C4×D3—31no—
224C2×C4×D3—31no—
225C2×C4×D3—31no—
226C2×C4×D3—31no—
227C2×C4×D3—61no—
228C2×C4×D3—91no—
229C2×C4×D3—91no—
230C2×Dic6—91no—
231C2×Dic6—91no—
232C2×Dic6—91no—
233C3×M16—94no—
234C3×M16—94no—
235C2×C3×C8—31no—
236C2×C3×C8—31no—
237C2×C3×C8—61no—
238C2×C3×C8—94no—
239C2×C3×C8—91no—
240C2×C3×C8—91no—
241C2×C3×C8—91no—
242C2×C3×C8D3211yescharacteristic
243#208—94no—
244C2×C3⋊C8—31no—
245C2×C3⋊C8—31no—
246C2×C3⋊C8—31no—
247C2×C3⋊C8—31no—
248C2×C3⋊C8—61no—
249C2×C3⋊C8—91no—
250C2×C3⋊C8—91no—
251C2×C3⋊C8—94no—
252C2×C3⋊C8—94no—
253ΓL(1,𝔽25)—34no—
254ΓL(1,𝔽25)—34no—
255ΓL(1,𝔽25)—34no—
256ΓL(1,𝔽25)—64no—
257ΓL(1,𝔽25)—94no—
258ΓL(1,𝔽25)—94no—
259ΓL(1,𝔽25)—94no—
260D3×C8—34no—
261D3×C8—94no—
262D3×C8—94no—
263D3×C8—94no—
264D3×C8—94no—
265D3×C8—94no—
266C22×C32—21no—
267C22×C32C2×C4×D311yescharacteristic
268C22×C32C2×C4×D311yescharacteristic
269C2×C3⋊D3—22no—
270C2×C3⋊D3—22no—
271C2×C3⋊D3C2×C4×D312yes—
272C2×C3⋊D3C2×C4×D312yes—
273C2×C3×D3—32no—
274C2×C3×D3—32no—
275C2×C3×D3—32no—
276C2×C3×D3—32no—
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—122no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4C2×C4×D311yescharacteristic
286C32×C4C2×C4×D311yescharacteristic
287C32×C4C23×D311yescharacteristic
288C32×C4C23×D311yescharacteristic
289C3⋊Dic3C2×C4×D312yes—
290C3⋊Dic3C2×C4×D312yes—
291C3×Dic3—32no—
292C3×Dic3—32no—
293C3×Dic3—32no—
294C3×Dic3—32no—
295C3×Dic3—62no—
296C2×#41—271no—
297#131—278no—
298C2×M16—91no—
299C2×M16—91no—
300C2×M16—271no—
301C22×C8—91no—
302C22×C8—271no—
303C22×C8—271no—
304H(3,𝔽3)C2×#13111yescharacteristic, complemented, Sylow
305C23×C3—91no—
306C23×C3—91no—
307C23×C3—91no—
308C22×D3—31no—
309C22×D3—31no—
310C22×D3—31no—
311C22×D3—31no—
312C22×D3—61no—
313C22×D3—91no—
314C22×D3—91no—
315C2×C3×C4—31no—
316C2×C3×C4—31no—
317C2×C3×C4—61no—
318C2×C3×C4—92no—
319C2×C3×C4—92no—
320C2×C3×C4—92no—
321C2×C3×C4—92no—
322C2×C3×C4—91no—
323C2×C3×C4—91no—
324C2×C3×C4—91no—
325C2×C3×C4—92no—
326C2×C3×C4—92no—
327C2×C3×C4C2×D3211yescharacteristic
328#67—94no—
329#67—94no—
330#67—94no—
331#67—94no—
332#67—94no—
333#67—94no—
334C2×Dic3—31no—
335C2×Dic3—31no—
336C2×Dic3—31no—
337C2×Dic3—31no—
338C2×Dic3—61no—
339C2×Dic3—91no—
340C2×Dic3—91no—
341D12—94no—
342D12—94no—
343D12—94no—
344C4×D3—32no—
345C4×D3—32no—
346C4×D3—32no—
347C4×D3—32no—
348C4×D3—32no—
349C4×D3—32no—
350C4×D3—32no—
351C4×D3—32no—
352C4×D3—62no—
353C4×D3—62no—
354C4×D3—92no—
355C4×D3—92no—
356C4×D3—92no—
357C4×D3—92no—
358Dic6—94no—
359Dic6—94no—
360Dic6—94no—
361C3×C8—32no—
362C3×C8—32no—
363C3×C8—62no—
364C3×C8—92no—
365C3×C8—92no—
366C3×C8—92no—
367C3×C8C2×D3212yes—
368C3⋊C8—32no—
369C3⋊C8—32no—
370C3⋊C8—32no—
371C3⋊C8—32no—
372C3⋊C8—62no—
373C3⋊C8—92no—
374C3⋊C8—92no—
375C2×C32—21no—
376C2×C32—22no—
377C2×C32#90112yescomplemented
378C2×C32#90712yescomplemented
379C2×C32C22×C4×D311yescharacteristic
380C2×C32C22×C4×D311yescharacteristic
381C3⋊D3—22no—
382C3⋊D3—22no—
383C3×D3—62no—
384C3×D3—62no—
385C3×D3—62no—
386C3×D3—62no—
387C3×D3—122no—
388#41—274no—
389#41—274no—
390C2×Q8—271no—
391C2×D4—271no—
392C2×D4—271no—
393C2×D4—271no—
394C22×C4—91no—
395C22×C4—91no—
396C22×C4—91no—
397M16—94no—
398M16—94no—
399M16—274no—
400C2×C8—91no—
401C2×C8—91no—
402C2×C8—91no—
403C2×C8—94no—
404C2×C8—274no—
405C2×C8—274no—
406C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
407C22×C3—31no—
408C22×C3—31no—
409C22×C3—61no—
410C22×C3—92no—
411C22×C3—92no—
412C22×C3—92no—
413C22×C3—182no—
414C22×C3—182no—
415C22×C3—182no—
416C22×C3C4×D3211yescharacteristic
417C2×D3—32no—
418C2×D3—32no—
419C2×D3—32no—
420C2×D3—32no—
421C2×D3—62no—
422C2×D3—62no—
423C2×D3—62no—
424C2×D3—62no—
425C2×D3—62no—
426C2×D3—92no—
427C2×D3—92no—
428C2×D3—122no—
429C2×D3—182no—
430C2×D3—182no—
431C3×C4—31no—
432C3×C4—31no—
433C3×C4—31no—
434C3×C4—31no—
435C3×C4—61no—
436C3×C4—61no—
437C3×C4—92no—
438C3×C4—92no—
439C3×C4—92no—
440C3×C4C4×D3211yescharacteristic
441C3×C4C22×D3211yescharacteristic
442Dic3—32no—
443Dic3—32no—
444Dic3—32no—
445Dic3—32no—
446Dic3—62no—
447Dic3—92no—
448Dic3—92no—
449C32—21no—
450C32C2×#90111yescharacteristic, complemented
451C32C2×#90711yescharacteristic, complemented
452C23—91no—
453C23—91no—
454C23—91no—
455Q8—274no—
456D4—274no—
457D4—274no—
458D4—274no—
459C2×C4—91no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—92no—
463C2×C4—92no—
464C2×C4—92no—
465C2×C4—92no—
466C2×C4—92no—
467C2×C4—92no—
468C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
469C8—92no—
470C8—92no—
471C8—92no—
472C8C2×((C3⋊D3)⋊C3)⋊C212yes—
473C2×C3—32no—
474C2×C3—32no—
475C2×C3—31no—
476C2×C3—31no—
477C2×C3—62no—
478C2×C3—61no—
479C2×C3—182no—
480C2×C3—182no—
481C2×C3—182no—
482C2×C3#6380412yes—
483C2×C3C2×C4×D3211yescharacteristic
484D3—62no—
485D3—62no—
486D3—62no—
487D3—62no—
488D3—122no—
489D3—182no—
490D3—182no—
491C22—92no—
492C22—92no—
493C22—92no—
494C22—182no—
495C22—182no—
496C22—182no—
497C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
498C4—92no—
499C4—92no—
500C4—92no—
501C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
502C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
503C3—31no—
504C3—31no—
505C3—61no—
506C3C2×#6380411yescharacteristic
507C2—182no—
508C2—182no—
509C2—182no—
510C2#1172084112yesdirect product factor
511C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC3×C4C22×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC2×#88650C2
cosocleC4×H(3,𝔽3)C24
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#131
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88527C2semidirect
3. C2×#3743⋊C3C2semidirect
4. C2×C8×(C3⋊D3)⋊C3C2semidirect
5. C2×#88434C2semidirect
6. C2×#88430C2semidirect
7. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
8. #11720841C2direct (N × H)
22. #88527C22semidirect
25. #3743⋊C3C22semidirect
26. C8×(C3⋊D3)⋊C3C22semidirect
27. #88434C22semidirect
28. #88430C22semidirect
29. ((C3⋊D3)⋊C3)⋊C8C22semidirect
32. C2×#3743D3semidirect
33. C2×C8×C3⋊D3D3semidirect
95. #3743C2×D3semidirect
96. C8×C3⋊D3C2×D3semidirect
200. C2×H(3,𝔽3)#131semidirect
304. H(3,𝔽3)C2×#131semidirect
377. C2×C32#901semidirect
378. C2×C32#907semidirect
450. C32C2×#901semidirect
451. C32C2×#907semidirect
510. C2#11720841direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886502
C2×#885272
C2×#3743⋊C32
C2×C8×(C3⋊D3)⋊C32
C2×#884342
C2×#884302
C2×((C3⋊D3)⋊C3)⋊C82
#117208412
C2×#638053
C2×#638043