Finite Group Databasebeta

← C2×#72229320 #72276253

Subgroups — C2×#72229320 #72276253

Order 5184.

1588 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 1588 conjugacy classes of subgroups — too many to draw. Only its 60 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×#13048167C211yesmaximal, characteristic, complemented
2#72229320C212yesmaximal, direct product factor
3C2×#11721950C311yesmaximal, characteristic, complemented
4C2×C3×#63515—31nomaximal
5#13048167C2211yescharacteristic, complemented
6C2×((C22⋊C27)⋊C2)⋊C3—43nomaximal
7C25×C3×C9D311yescharacteristic
8C22×C3×#611—33no—
9C2×C22⋊(C22⋊C27)—21no—
10C2×C22⋊(C22⋊C27)C2×C311yescharacteristic, complemented
11#11721950C2×C312yescomplemented
12C3×#63515—36no—
13C3×#63515—32no—
14C2×C3×#3676—33no—
15C2×#66353—43no—
16C2×#66353—81no—
17((C22⋊C27)⋊C2)⋊C3—46no—
18C2×C3×#953—91no—
19C2×#63515—31no—
20C24×C3×C9—33no—
21C24×C3×C9—33no—
22C24×C3×C9—61no—
23C24×C3×C9—61no—
24C24×C3×C9C2×D311yescharacteristic
25C23×C3×D9—31no—
26C2×C3×#611—36no—
27C2×C3×#611—36no—
28C2×C3×#611—63no—
29C2×C3×#611—63no—
30C2×C3×#611—63no—
31C2×C3×#611—63no—
32C22×C3×Dic9—33no—
33C22⋊(C22⋊C27)—21no—
34C22⋊(C22⋊C27)C22×C311yescharacteristic, complemented
35C2×(C22⋊C27)⋊C2—43no—
36C3×#3676—36no—
37C3×#3676—36no—
38C2×D27⋊C3—161no—
39#66353—43no—
40#66353—81no—
41C25×C32D911yescharacteristic
42C22×C3×#67—93no—
43C25×C9—21no—
44C25×C9C3×D311yescharacteristic
45C3×#953—96no—
46C3×#953—92no—
47C2×C3×#217—93no—
48C22×#611—33no—
49#63515—36no—
50#63515—32no—
51C2×#3676—33no—
52C23×C3×C9—21no—
53C23×C3×C9—31no—
54C23×C3×C9—33no—
55C23×C3×C9—33no—
56C23×C3×C9—33no—
57C23×C3×C9—33no—
58C23×C3×C9—33no—
59C23×C3×C9—61no—
60C23×C3×C9—61no—
61C23×C3×C9—66no—
62C23×C3×C9—63no—
63C23×C3×C9—63no—
64C23×C3×C9—63no—
65C23×C3×C9S413yes—
66C22×C3×D9—32no—
67C22×C3×D9—63no—
68C22×C3×D9—63no—
69C3×#611—66no—
70C3×#611—66no—
71C3×#611—66no—
72C3×#611—66no—
73C2×C3×Dic9—36no—
74C2×C3×Dic9—63no—
75C2×C3×Dic9—63no—
76C2×C22⋊C27—43no—
77C2×C22⋊C27—81no—
78C2×C22⋊C27—82no—
79C2×C22⋊C27—83no—
80(C22⋊C27)⋊C2—46no—
81C2×C3×#120—271no—
82C2×#953—91no—
83C2×C27⋊C3—161no—
84D27⋊C3—162no—
85C24×C32—33no—
86C24×C32—33no—
87C24×C32—61no—
88C24×C32—61no—
89C24×C32C2×D911yescharacteristic
90C23×C3×D3—91no—
91C2×C3×#67—96no—
92C2×C3×#67—96no—
93C2×C3×#67—183no—
94C2×C3×#67—183no—
95C2×C3×#67—183no—
96C2×C3×#67—183no—
97C22×C3×Dic3—93no—
98C24×C9—21no—
99C24×C9—33no—
100C24×C9—33no—
101C24×C9—61no—
102C24×C9—63no—
103C24×C9—62no—
104C24×C9—61no—
105C24×C9—63no—
106C24×C9—62no—
107C24×C9C2×C3×D311yescharacteristic
108C23×D9—31no—
109C3×#217—96no—
110C3×#217—96no—
111C2×#611—36no—
112C2×#611—36no—
113C2×#611—63no—
114C2×#611—63no—
115C2×#611—63no—
116C2×#611—63no—
117C22×Dic9—33no—
118#3676—36no—
119#3676—36no—
120C22×C3×C9—21no—
121C22×C3×C9—31no—
122C22×C3×C9—33no—
123C22×C3×C9—33no—
124C22×C3×C9—33no—
125C22×C3×C9—33no—
126C22×C3×C9—33no—
127C22×C3×C9—63no—
128C22×C3×C9—66no—
129C22×C3×C9—63no—
130C22×C3×C9—63no—
131C22×C3×C9—61no—
132C22×C3×C9—61no—
133C22×C3×C9C2×S413yes—
134C2×C3×D9—66no—
135C2×C3×D9—123no—
136C2×C3×D9—121no—
137C3×Dic9—66no—
138C2×D27—161no—
139C22⋊C27—43no—
140C22⋊C27—81no—
141C22⋊C27—82no—
142C22⋊C27—83no—
143C25×C3—31no—
144C25×C3C3×D911yescharacteristic
145C22×C3×D4—273no—
146C22×#67—93no—
147C3×#120—276no—
148C3×#120—272no—
149C2×C3×C22⋊C4—273no—
150#953—96no—
151#953—92no—
152C2×#217—93no—
153C27⋊C3—161noSylow
154C23×C32—21no—
155C23×C32—31no—
156C23×C32—33no—
157C23×C32—33no—
158C23×C32—33no—
159C23×C32—33no—
160C23×C32—33no—
161C23×C32—61no—
162C23×C32—61no—
163C23×C32—66no—
164C23×C32—63no—
165C23×C32—63no—
166C23×C32—63no—
167C23×C32(C22⋊C9)⋊C213yes—
168C22×C3×D3—92no—
169C22×C3×D3—183no—
170C22×C3×D3—183no—
171C3×#67—186no—
172C3×#67—186no—
173C3×#67—186no—
174C3×#67—186no—
175C2×C3×Dic3—96no—
176C2×C3×Dic3—183no—
177C2×C3×Dic3—183no—
178C23×C9—21no—
179C23×C9—22no—
180C23×C9—23no—
181C23×C9—31no—
182C23×C9—33no—
183C23×C9—33no—
184C23×C9—33no—
185C23×C9—33no—
186C23×C9—33no—
187C23×C9—63no—
188C23×C9—63no—
189C23×C9—61no—
190C23×C9—63no—
191C23×C9—63no—
192C23×C9—63no—
193C23×C9—61no—
194C23×C9—62no—
195C23×C9—61no—
196C23×C9—62no—
197C23×C9—66no—
198C23×C9—63no—
199C23×C9—63no—
200C23×C9—63no—
201C23×C9—66no—
202C23×C9—612no—
203C23×C9—66no—
204C23×C9—66no—
205C23×C9C3×S413yes—
206C22×D9—32no—
207C22×D9—63no—
208C22×D9—63no—
209#611—66no—
210#611—66no—
211#611—66no—
212#611—66no—
213C2×Dic9—36no—
214C2×Dic9—63no—
215C2×Dic9—63no—
216C2×#120—271noSylow
217C2×C3×C9—33no—
218C2×C3×C9—33no—
219C2×C3×C9—61no—
220C2×C3×C9—61no—
221C2×C3×C9#102011yescharacteristic
222C3×D9—122no—
223C2×C27—161no—
224C2×C27—321no—
225D27—162no—
226C24×C3—33no—
227C24×C3—33no—
228C24×C3—33no—
229C24×C3—33no—
230C24×C3—31no—
231C24×C3—63no—
232C24×C3—63no—
233C24×C3—61no—
234C24×C3—61no—
235C24×C3—63no—
236C24×C3—63no—
237C24×C3—271no—
238C24×C3C2×C3×D911yescharacteristic
239C23×D3—91no—
240C2×C3×D4—276no—
241C2×C3×D4—276no—
242C2×C3×D4—543no—
243C2×C3×D4—543no—
244C2×C3×D4—543no—
245C2×C3×D4—543no—
246C22×C3×C4—273no—
247C2×#67—96no—
248C2×#67—96no—
249C2×#67—183no—
250C2×#67—183no—
251C2×#67—183no—
252C2×#67—183no—
253C22×Dic3—93no—
254C3×C22⋊C4—276no—
255C3×C22⋊C4—276no—
256#217—96no—
257#217—96no—
258C22×C32—21no—
259C22×C32—33no—
260C22×C32—31no—
261C22×C32—33no—
262C22×C32—33no—
263C22×C32—33no—
264C22×C32—33no—
265C22×C32—63no—
266C22×C32—66no—
267C22×C32—63no—
268C22×C32—63no—
269C22×C32—61no—
270C22×C32—61no—
271C22×C32C2×(C22⋊C9)⋊C213yes—
272C2×C3×D3—186no—
273C2×C3×D3—363no—
274C2×C3×D3—361no—
275C3×Dic3—186no—
276C22×C9—21no—
277C22×C9—22no—
278C22×C9—23no—
279C22×C9—33no—
280C22×C9—31no—
281C22×C9—33no—
282C22×C9—33no—
283C22×C9—33no—
284C22×C9—33no—
285C22×C9—63no—
286C22×C9—63no—
287C22×C9—61no—
288C22×C9—63no—
289C22×C9—63no—
290C22×C9—63no—
291C22×C9—62no—
292C22×C9—63no—
293C22×C9—66no—
294C22×C9—63no—
295C22×C9—63no—
296C22×C9—62no—
297C22×C9—61no—
298C22×C9—66no—
299C22×C9—612no—
300C22×C9—66no—
301C22×C9—66no—
302C22×C9—61no—
303C22×C9C2×C3×S413yes—
304C2×D9—66no—
305C2×D9—123no—
306C2×D9—121no—
307Dic9—66no—
308C25D27⋊C311yescharacteristic, complemented
309C22×D4—273no—
310#120—276no—
311#120—272no—
312C2×C22⋊C4—273no—
313C3×C9C2×#102011yescharacteristic
314C27—161no—
315C27—321no—
316C23×C3—21no—
317C23×C3—31no—
318C23×C3—31no—
319C23×C3—33no—
320C23×C3—33no—
321C23×C3—33no—
322C23×C3—33no—
323C23×C3—33no—
324C23×C3—33no—
325C23×C3—33no—
326C23×C3—33no—
327C23×C3—33no—
328C23×C3—33no—
329C23×C3—33no—
330C23×C3—61no—
331C23×C3—63no—
332C23×C3—63no—
333C23×C3—61no—
334C23×C3—63no—
335C23×C3—63no—
336C23×C3—63no—
337C23×C3—61no—
338C23×C3—61no—
339C23×C3—66no—
340C23×C3—63no—
341C23×C3—63no—
342C23×C3—63no—
343C23×C3—63no—
344C23×C3—66no—
345C23×C3—66no—
346C23×C3—66no—
347C23×C3—66no—
348C23×C3—63no—
349C23×C3—63no—
350C23×C3—63no—
351C23×C3—66no—
352C23×C3—63no—
353C23×C3—66no—
354C23×C3—272no—
355C23×C3—543no—
356C23×C3—543no—
357C23×C3C3×(C22⋊C9)⋊C213yes—
358C22×D3—92no—
359C22×D3—183no—
360C22×D3—183no—
361C3×D4—546no—
362C3×D4—546no—
363C3×D4—546no—
364C3×D4—546no—
365C2×C3×C4—276no—
366C2×C3×C4—543no—
367C2×C3×C4—543no—
368#67—186no—
369#67—186no—
370#67—186no—
371#67—186no—
372C2×Dic3—96no—
373C2×Dic3—183no—
374C2×Dic3—183no—
375C2×C32—33no—
376C2×C32—33no—
377C2×C32—61no—
378C2×C32—61no—
379C2×C32#6418811yescharacteristic
380C3×D3—362no—
381C2×C9—21no—
382C2×C9—33no—
383C2×C9—33no—
384C2×C9—63no—
385C2×C9—63no—
386C2×C9—62no—
387C2×C9—62no—
388C2×C9—61no—
389C2×C9—61no—
390C2×C9C3×#102011yescharacteristic
391D9—122no—
392C24—33no—
393C24—33no—
394C24—61no—
395C24—61no—
396C24—271no—
397C24C2×D27⋊C311yescharacteristic, complemented
398C2×D4—276no—
399C2×D4—276no—
400C2×D4—543no—
401C2×D4—543no—
402C2×D4—543no—
403C2×D4—543no—
404C22×C4—273no—
405C22⋊C4—276no—
406C22⋊C4—276no—
407C22×C3—21no—
408C22×C3—33no—
409C22×C3—31no—
410C22×C3—33no—
411C22×C3—31no—
412C22×C3—33no—
413C22×C3—33no—
414C22×C3—33no—
415C22×C3—33no—
416C22×C3—33no—
417C22×C3—33no—
418C22×C3—33no—
419C22×C3—33no—
420C22×C3—33no—
421C22×C3—61no—
422C22×C3—63no—
423C22×C3—63no—
424C22×C3—61no—
425C22×C3—63no—
426C22×C3—63no—
427C22×C3—63no—
428C22×C3—63no—
429C22×C3—66no—
430C22×C3—63no—
431C22×C3—63no—
432C22×C3—66no—
433C22×C3—66no—
434C22×C3—66no—
435C22×C3—66no—
436C22×C3—63no—
437C22×C3—63no—
438C22×C3—61no—
439C22×C3—63no—
440C22×C3—66no—
441C22×C3—63no—
442C22×C3—66no—
443C22×C3—63no—
444C22×C3—61no—
445C22×C3—546no—
446C22×C3—1083no—
447C22×C3—1081no—
448C22×C3C2×C3×(C22⋊C9)⋊C213yes—
449C2×D3—186no—
450C2×D3—363no—
451C2×D3—361no—
452C3×C4—546no—
453Dic3—186no—
454C32C2×#6418811yescharacteristic
455C9—21no—
456C9C2×C3×#102011yescharacteristic
457C23—21no—
458C23—31no—
459C23—33no—
460C23—33no—
461C23—33no—
462C23—33no—
463C23—33no—
464C23—61no—
465C23—61no—
466C23—66no—
467C23—63no—
468C23—63no—
469C23—63no—
470C23—272no—
471C23—543no—
472C23—543no—
473C23((C22⋊C27)⋊C2)⋊C313yescomplemented
474D4—546no—
475D4—546no—
476D4—546no—
477D4—546no—
478C2×C4—276no—
479C2×C4—543no—
480C2×C4—543no—
481C2×C3—33no—
482C2×C3—33no—
483C2×C3—33no—
484C2×C3—33no—
485C2×C3—31no—
486C2×C3—63no—
487C2×C3—63no—
488C2×C3—63no—
489C2×C3—63no—
490C2×C3—61no—
491C2×C3—61no—
492C2×C3—1082no—
493C2×C3C3×#6418811yescharacteristic
494D3—362no—
495C22—21no—
496C22—33no—
497C22—31no—
498C22—33no—
499C22—33no—
500C22—33no—
501C22—33no—
502C22—63no—
503C22—66no—
504C22—63no—
505C22—63no—
506C22—61no—
507C22—61no—
508C22—546no—
509C22—1083no—
510C22—1081no—
511C22C2×((C22⋊C27)⋊C2)⋊C313yescomplemented
512C4—546no—
513C3—31no—
514C3C2×C3×#6418811yescharacteristic
515C2—33no—
516C2—33no—
517C2—61no—
518C2—61no—
519C2—1082no—
520C2#7222932011yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2#72229320
commutator, cosocle, hypocenterC22⋊(C22⋊C27)C22×C3
FrattiniC9C2×C3×#1020
FittingC25×C3×C9D3
socleC25×C3C3×D9
LCSG ▷ C22⋊(C22⋊C27)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×#13048167C2semidirect
2. #72229320C2direct (N × H)
3. C2×#11721950C3semidirect
5. #13048167C22semidirect
10. C2×C22⋊(C22⋊C27)C2×C3semidirect
11. #11721950C2×C3semidirect
34. C22⋊(C22⋊C27)C22×C3semidirect
308. C25D27⋊C3semidirect
397. C24C2×D27⋊C3semidirect
473. C23((C22⋊C27)⋊C2)⋊C3semidirect
511. C22C2×((C22⋊C27)⋊C2)⋊C3semidirect
520. C2#72229320direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#130481672
#722293202
C2×#117219503
C2×C3×#635153
C2×((C22⋊C27)⋊C2)⋊C34