Finite Group Databasebeta

← C2×#11720835 #13149804

Subgroups — C2×#11720835 #13149804

Order 1728.

1111 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic
2C2×#88527C211yesmaximal, characteristic, complemented
3C2×#3743⋊C3C211yesmaximal, characteristic, complemented
4C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
5C2×#88435C211yesmaximal, characteristic, complemented
6C2×#88430C211yesmaximal, characteristic, complemented
7C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
8#11720835C218yesmaximal, direct product factor
9C2×#63791—31nomaximal
10C2×D3×ΓL(1,𝔽25)—31nomaximal
11C22×((C3⋊D3)⋊C3)⋊C2C411yescharacteristic
12C2×C4×#272C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×#6199C411yescharacteristic
16C2×((C3⋊D3)⋊C3)⋊C4C411yescharacteristic
17C2×((C3⋊D3)⋊C3)⋊C4C411yescharacteristic
18C4×((C3⋊D3)⋊C3)⋊C2C414yes—
19C4×((C3⋊D3)⋊C3)⋊C2C2214yes—
20C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
21C2×#6180C2211yescharacteristic, complemented
22#88527C2214yescomplemented
23C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
24C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
25#3743⋊C3C2214yescomplemented
26C8×(C3⋊D3)⋊C3C2214yescomplemented
27#88435C2214yescomplemented
28#88430C2214yescomplemented
29((C3⋊D3)⋊C3)⋊C8C2214yescomplemented
30C2×C4×D32—31no—
31C2×C4×D32—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×#3711—31no—
42C2×#3711—31no—
43C2×#3710—31no—
44C2×D3×C3⋊C8—31no—
45#63791—38no—
46D3×ΓL(1,𝔽25)—38no—
47C22×#272C2×C411yescharacteristic
48C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
49C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
50C2×((C3⋊D3)⋊C3)⋊C2—24no—
51C2×((C3⋊D3)⋊C3)⋊C2C2×C414yes—
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53C2×#1084C2×C411yescharacteristic
54C4×#272C2×C412yes—
55C4×#272C2312yes—
56C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
57C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
58C4×(C3⋊D3)⋊C3C2×C412yes—
59C4×(C3⋊D3)⋊C3C2×C412yes—
60C4×(C3⋊D3)⋊C3C2312yes—
61C4×(C3⋊D3)⋊C3C2312yes—
62#6199C2×C414yes—
63((C3⋊D3)⋊C3)⋊C4C2×C414yes—
64((C3⋊D3)⋊C3)⋊C4C2×C414yes—
65C8×H(3,𝔽3)C2312yescomplemented
66#6180C2312yescomplemented
67(C3⋊(C3⋊C8))⋊C3C2312yescomplemented
68(C3⋊(C3⋊C8))⋊C3C2312yescomplemented
69C2×D3×M16—91no—
70C2×D3×M16—91no—
71C22×ΓL(1,𝔽25)—91no—
72C22×D32—31no—
73C22×D32—31no—
74C2×C4×C3⋊D3C2×D311yescharacteristic
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C2×C3×C4×D3—31no—
77C2×C3×C4×D3—31no—
78C2×C3×C4×D3—31no—
79C2×C3×C4×D3—31no—
80C2×C3×C4×D3—61no—
81C2×#624—31no—
82C2×#624—31no—
83C2×D3×Dic3—31no—
84C2×D3×Dic3—31no—
85C2×D3×Dic3—31no—
86C2×D3×Dic3—31no—
87C4×D32—34no—
88C4×D32—34no—
89C4×D32—34no—
90C4×D32—34no—
91C2×C32×C8—21no—
92C2×C32×C8C2×D311yescharacteristic, complemented
93C2×C32×C8C2×D311yescharacteristic, complemented
94C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
95C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
96#3743C2×D314yescomplemented
97C8×C3⋊D3C2×D314yescomplemented
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C3×ΓL(1,𝔽25)—34no—
104C3×ΓL(1,𝔽25)—34no—
105C3×ΓL(1,𝔽25)—34no—
106C3×ΓL(1,𝔽25)—64no—
107C3×D3×C8—34no—
108#3712—34no—
109#3711—34no—
110#3711—34no—
111#3711—34no—
112#3710—34no—
113D3×C3⋊C8—34no—
114C22×H(3,𝔽3)C22×C411yescharacteristic
115C2×#272—22no—
116C2×#272C22×C412yes—
117C2×(C3⋊D3)⋊C3—22no—
118C2×(C3⋊D3)⋊C3M1614yescomplemented
119C2×(C3⋊D3)⋊C3C22×C412yes—
120C2×(C3⋊D3)⋊C3C22×C412yes—
121((C3⋊D3)⋊C3)⋊C2—28no—
122C4×H(3,𝔽3)C22×C411yescharacteristic
123C4×H(3,𝔽3)C2411yescharacteristic
124#1084C22×C412yes—
125(C3⋊Dic3)⋊C3C22×C412yes—
126(C3⋊Dic3)⋊C3C22×C412yes—
127C22×C4×D3—91no—
128C22×C4×D3—91no—
129C22×C4×D3—91no—
130C2×C3×M16—91no—
131C2×C3×M16—91no—
132C22×C3×C8—91no—
133C2×#208—91no—
134C2×#208—91no—
135C22×C3⋊C8—91no—
136D3×M16—98no—
137D3×M16—98no—
138C2×ΓL(1,𝔽25)—31no—
139C2×ΓL(1,𝔽25)—31no—
140C2×ΓL(1,𝔽25)—31no—
141C2×ΓL(1,𝔽25)—61no—
142C2×ΓL(1,𝔽25)—91no—
143C2×ΓL(1,𝔽25)—98no—
144C2×ΓL(1,𝔽25)—91no—
145C2×ΓL(1,𝔽25)—91no—
146C2×ΓL(1,𝔽25)—91no—
147C2×ΓL(1,𝔽25)—91no—
148C2×D3×C8—31no—
149C2×D3×C8—91no—
150C2×D3×C8—91no—
151C2×D3×C8—91no—
152C22×C3⋊D3C4×D311yescharacteristic
153C22×C3⋊D3C4×D311yescharacteristic
154C22×C3×D3—31no—
155C22×C3×D3—31no—
156C22×C3×D3—31no—
157C22×C3×D3—31no—
158C22×C3×D3—61no—
159C2×D32—34no—
160C2×D32—34no—
161C2×D32—64no—
162C2×D32—64no—
163C2×C32×C4—21no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C3⋊Dic3C4×D311yescharacteristic
167C2×C3⋊Dic3C4×D311yescharacteristic
168C4×C3⋊D3C4×D312yes—
169C4×C3⋊D3C4×D312yes—
170C4×C3⋊D3C22×D312yes—
171C4×C3⋊D3C22×D312yes—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—61no—
177C3×C4×D3—32no—
178C3×C4×D3—32no—
179C3×C4×D3—32no—
180C3×C4×D3—32no—
181C3×C4×D3—32no—
182C3×C4×D3—32no—
183C3×C4×D3—32no—
184C3×C4×D3—32no—
185C3×C4×D3—62no—
186C3×C4×D3—62no—
187#624—34no—
188#624—34no—
189D3×Dic3—34no—
190D3×Dic3—34no—
191D3×Dic3—34no—
192D3×Dic3—34no—
193C32×C8—22no—
194C32×C8C22×D312yescomplemented
195C32×C8C22×D312yescomplemented
196C3⋊(C3⋊C8)C22×D312yescomplemented
197C3⋊(C3⋊C8)C22×D312yescomplemented
198C3×C3⋊C8—32no—
199C3×C3⋊C8—32no—
200C3×C3⋊C8—32no—
201C3×C3⋊C8—32no—
202C3×C3⋊C8—62no—
203C22×M16—271noSylow
204C2×H(3,𝔽3)C2×M1612yescomplemented
205C2×H(3,𝔽3)C23×C411yescharacteristic
206#272—22no—
207(C3⋊D3)⋊C3—22no—
208(C3⋊D3)⋊C3C2×M1614yescomplemented
209C23×D3—91no—
210C23×D3—91no—
211C23×D3—91no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×Dic3—91no—
216C22×Dic3—91no—
217C22×Dic3—91no—
218C2×C4×D3—31no—
219C2×C4×D3—31no—
220C2×C4×D3—31no—
221C2×C4×D3—31no—
222C2×C4×D3—61no—
223C2×C4×D3—91no—
224C2×C4×D3—91no—
225C2×C4×D3—94no—
226C2×C4×D3—94no—
227C2×C4×D3—94no—
228C2×C4×D3—94no—
229C2×C4×D3—94no—
230C2×C4×D3—94no—
231C2×C4×D3—91no—
232C2×C4×D3—91no—
233C2×C4×D3—91no—
234C2×C4×D3—91no—
235C2×C4×D3—91no—
236C2×C4×D3—91no—
237C3×M16—94no—
238C3×M16—94no—
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—61no—
242C2×C3×C8—94no—
243C2×C3×C8—91no—
244C2×C3×C8—91no—
245C2×C3×C8—91no—
246C2×C3×C8D3211yescharacteristic
247#208—94no—
248#208—94no—
249C2×C3⋊C8—31no—
250C2×C3⋊C8—31no—
251C2×C3⋊C8—31no—
252C2×C3⋊C8—31no—
253C2×C3⋊C8—61no—
254C2×C3⋊C8—91no—
255C2×C3⋊C8—91no—
256C2×C3⋊C8—94no—
257ΓL(1,𝔽25)—34no—
258ΓL(1,𝔽25)—34no—
259ΓL(1,𝔽25)—34no—
260ΓL(1,𝔽25)—64no—
261ΓL(1,𝔽25)—94no—
262ΓL(1,𝔽25)—94no—
263ΓL(1,𝔽25)—94no—
264ΓL(1,𝔽25)—94no—
265ΓL(1,𝔽25)—94no—
266D3×C8—34no—
267D3×C8—94no—
268D3×C8—94no—
269D3×C8—94no—
270C22×C32—21no—
271C22×C32C2×C4×D311yescharacteristic
272C22×C32C2×C4×D311yescharacteristic
273C2×C3⋊D3—22no—
274C2×C3⋊D3ΓL(1,𝔽25)14yescomplemented
275C2×C3⋊D3C2×C4×D312yes—
276C2×C3⋊D3C2×C4×D312yes—
277C2×C3×D3—34no—
278C2×C3×D3—32no—
279C2×C3×D3—32no—
280C2×C3×D3—32no—
281C2×C3×D3—32no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—122no—
287D32—68no—
288D32—68no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4C2×C4×D311yescharacteristic
292C32×C4C2×C4×D311yescharacteristic
293C32×C4C23×D311yescharacteristic
294C32×C4C23×D311yescharacteristic
295C3⋊Dic3C2×C4×D312yes—
296C3⋊Dic3C2×C4×D312yes—
297C3×Dic3—32no—
298C3×Dic3—32no—
299C3×Dic3—32no—
300C3×Dic3—32no—
301C3×Dic3—62no—
302C23×C4—271no—
303C2×M16—91no—
304C2×M16—91no—
305C2×M16—271no—
306C2×M16—271no—
307C2×M16—278no—
308C22×C8—91no—
309C22×C8—271no—
310H(3,𝔽3)C22×M1611yescharacteristic, complemented, Sylow
311C23×C3—91no—
312C23×C3—91no—
313C23×C3—91no—
314C22×D3—31no—
315C22×D3—31no—
316C22×D3—31no—
317C22×D3—31no—
318C22×D3—61no—
319C22×D3—91no—
320C22×D3—91no—
321C22×D3—94no—
322C22×D3—94no—
323C22×D3—94no—
324C22×D3—184no—
325C22×D3—184no—
326C22×D3—184no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—61no—
330C2×C3×C4—92no—
331C2×C3×C4—92no—
332C2×C3×C4—92no—
333C2×C3×C4—92no—
334C2×C3×C4—91no—
335C2×C3×C4—91no—
336C2×C3×C4—91no—
337C2×C3×C4—92no—
338C2×C3×C4—92no—
339C2×C3×C4C2×D3211yescharacteristic
340C2×Dic3—31no—
341C2×Dic3—31no—
342C2×Dic3—31no—
343C2×Dic3—31no—
344C2×Dic3—61no—
345C2×Dic3—91no—
346C2×Dic3—91no—
347C2×Dic3—94no—
348C2×Dic3—94no—
349C2×Dic3—94no—
350C4×D3—32no—
351C4×D3—32no—
352C4×D3—32no—
353C4×D3—32no—
354C4×D3—32no—
355C4×D3—32no—
356C4×D3—32no—
357C4×D3—32no—
358C4×D3—62no—
359C4×D3—62no—
360C4×D3—92no—
361C4×D3—92no—
362C4×D3—92no—
363C4×D3—92no—
364C4×D3—94no—
365C4×D3—94no—
366C4×D3—94no—
367C4×D3—94no—
368C4×D3—94no—
369C4×D3—94no—
370C3×C8—32no—
371C3×C8—32no—
372C3×C8—62no—
373C3×C8—92no—
374C3×C8—92no—
375C3×C8—92no—
376C3×C8C2×D3212yes—
377C3⋊C8—32no—
378C3⋊C8—32no—
379C3⋊C8—32no—
380C3⋊C8—32no—
381C3⋊C8—62no—
382C3⋊C8—92no—
383C3⋊C8—92no—
384C2×C32—22no—
385C2×C32—21no—
386C2×C32C2×ΓL(1,𝔽25)12yescomplemented
387C2×C32D3×M1612yescomplemented
388C2×C32C22×C4×D311yescharacteristic
389C2×C32C22×C4×D311yescharacteristic
390C3⋊D3—22no—
391C3⋊D3C2×ΓL(1,𝔽25)14yescomplemented
392C3×D3—34no—
393C3×D3—62no—
394C3×D3—62no—
395C3×D3—62no—
396C3×D3—122no—
397C24—271no—
398C22×C4—91no—
399C22×C4—91no—
400C22×C4—91no—
401C22×C4—274no—
402C22×C4—274no—
403C22×C4—271no—
404C22×C4—271no—
405C22×C4—271no—
406M16—94no—
407M16—94no—
408M16—274no—
409M16—274no—
410C2×C8—91no—
411C2×C8—91no—
412C2×C8—91no—
413C2×C8—94no—
414C2×C8—274no—
415C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
416C22×C3—31no—
417C22×C3—31no—
418C22×C3—61no—
419C22×C3—92no—
420C22×C3—94no—
421C22×C3—92no—
422C22×C3—92no—
423C22×C3—182no—
424C22×C3—182no—
425C22×C3C4×D3211yescharacteristic
426C2×D3—34no—
427C2×D3—32no—
428C2×D3—32no—
429C2×D3—32no—
430C2×D3—32no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—62no—
435C2×D3—92no—
436C2×D3—94no—
437C2×D3—92no—
438C2×D3—122no—
439C2×D3—182no—
440C2×D3—188no—
441C2×D3—188no—
442C2×D3—188no—
443C3×C4—31no—
444C3×C4—31no—
445C3×C4—31no—
446C3×C4—31no—
447C3×C4—61no—
448C3×C4—61no—
449C3×C4—92no—
450C3×C4—92no—
451C3×C4—92no—
452C3×C4C4×D3211yescharacteristic
453C3×C4C22×D3211yescharacteristic
454Dic3—32no—
455Dic3—32no—
456Dic3—32no—
457Dic3—32no—
458Dic3—62no—
459Dic3—92no—
460Dic3—92no—
461C32—21no—
462C32C22×ΓL(1,𝔽25)11yescharacteristic, complemented
463C32C2×D3×M1611yescharacteristic, complemented
464C23—91no—
465C23—91no—
466C23—91no—
467C23—274no—
468C23—544no—
469C2×C4—91no—
470C2×C4—91no—
471C2×C4—91no—
472C2×C4—92no—
473C2×C4—92no—
474C2×C4—92no—
475C2×C4—92no—
476C2×C4—92no—
477C2×C4—92no—
478C2×C4—274no—
479C2×C4—274no—
480C2×C4—274no—
481C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
482C8—92no—
483C8—92no—
484C8—92no—
485C8C2×((C3⋊D3)⋊C3)⋊C212yescomplemented
486C2×C3—32no—
487C2×C3—32no—
488C2×C3—31no—
489C2×C3—31no—
490C2×C3—62no—
491C2×C3—61no—
492C2×C3—94no—
493C2×C3—182no—
494C2×C3—182no—
495C2×C3D3×ΓL(1,𝔽25)12yes—
496C2×C3C2×C4×D3211yescharacteristic
497D3—34no—
498D3—62no—
499D3—62no—
500D3—62no—
501D3—94no—
502D3—122no—
503D3—182no—
504C22—94no—
505C22—92no—
506C22—92no—
507C22—92no—
508C22—182no—
509C22—182no—
510C22—548no—
511C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
512C4—92no—
513C4—92no—
514C4—92no—
515C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
516C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
517C3—31no—
518C3—31no—
519C3—61no—
520C3C2×D3×ΓL(1,𝔽25)11yescharacteristic
521C2—94no—
522C2—182no—
523C2—182no—
524C2#1172083512yesdirect product factor
525C2C2×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 radicalC22×((C3⋊D3)⋊C3)⋊C2C4
cosocleC4×H(3,𝔽3)C24
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×M16
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×#88435C2semidirect
6. C2×#88430C2semidirect
7. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
8. #11720835C2direct (N × H)
20. C2×C8×H(3,𝔽3)C22semidirect
21. C2×#6180C22semidirect
22. #88527C22semidirect
23. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
24. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
25. #3743⋊C3C22semidirect
26. C8×(C3⋊D3)⋊C3C22semidirect
27. #88435C22semidirect
28. #88430C22semidirect
29. ((C3⋊D3)⋊C3)⋊C8C22semidirect
32. C2×#3743D3semidirect
33. C2×C8×C3⋊D3D3semidirect
65. C8×H(3,𝔽3)C23semidirect
66. #6180C23semidirect
67. (C3⋊(C3⋊C8))⋊C3C23semidirect
68. (C3⋊(C3⋊C8))⋊C3C23semidirect
92. C2×C32×C8C2×D3semidirect
93. C2×C32×C8C2×D3semidirect
94. C2×C3⋊(C3⋊C8)C2×D3semidirect
95. C2×C3⋊(C3⋊C8)C2×D3semidirect
96. #3743C2×D3semidirect
97. C8×C3⋊D3C2×D3semidirect
118. C2×(C3⋊D3)⋊C3M16semidirect
194. C32×C8C22×D3semidirect
195. C32×C8C22×D3semidirect
196. C3⋊(C3⋊C8)C22×D3semidirect
197. C3⋊(C3⋊C8)C22×D3semidirect
204. C2×H(3,𝔽3)C2×M16semidirect
208. (C3⋊D3)⋊C3C2×M16semidirect
274. C2×C3⋊D3ΓL(1,𝔽25)semidirect
310. H(3,𝔽3)C22×M16semidirect
386. C2×C32C2×ΓL(1,𝔽25)semidirect
387. C2×C32D3×M16semidirect
391. C3⋊D3C2×ΓL(1,𝔽25)semidirect
415. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
462. C32C22×ΓL(1,𝔽25)semidirect
463. C32C2×D3×M16semidirect
485. C8C2×((C3⋊D3)⋊C3)⋊C2semidirect
524. C2#11720835direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×((C3⋊D3)⋊C3)⋊C22
C2×#885272
C2×#3743⋊C32
C2×C8×(C3⋊D3)⋊C32
C2×#884352
C2×#884302
C2×((C3⋊D3)⋊C3)⋊C82
#117208352
C2×#637913
C2×D3×ΓL(1,𝔽25)3