Finite Group Databasebeta

← C2×#10594249 #11901578

Subgroups — C2×#10594249 #11901578

Order 1152.

2173 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64086C211yesmaximal, characteristic
2C2×C3×#879C212yesmaximal
3C2×#63982C211yesmaximal, characteristic
4C2×#63982C211yesmaximal, characteristic
5C2×#63956C212yesmaximal
6C2×#63919C211yesmaximal, characteristic, complemented
7C2×C4×#627C211yesmaximal, characteristic, complemented
8C2×#63870C211yesmaximal, characteristic, complemented
9C2×#63867C212yesmaximal, complemented
10C2×#63846C212yesmaximal, complemented
11C2×#63838C211yesmaximal, characteristic, complemented
12#10594249C2116yesmaximal, direct product factor
13C2×#5701—32nomaximal
14C23×C3⋊Dic3C2211yescharacteristic
15C22×#627C2211yescharacteristic
16C22×#627C2211yescharacteristic
17C2×C32×C22⋊C4C2211yescharacteristic
18C2×#3757C2211yescharacteristic
19C2×#3750C2211yescharacteristic
20C2×#3750C2211yescharacteristic
21C2×C4×C3⋊Dic3C2211yescharacteristic
22C2×C4×C3⋊Dic3C2211yescharacteristic
23#64086C2218yes—
24C2×C3×#217C2212yes—
25C2×C3×#217C2212yes—
26C2×C3×C4⋊Dic3C2212yes—
27C2×C3×Dic3⋊C4C2212yes—
28C2×C3×Dic3⋊C4C2212yes—
29C2×C3×C4×Dic3C2212yes—
30C3×#879C22116yes—
31#63982C2218yes—
32#63982C2218yes—
33C2×#3724C2212yes—
34C2×#3724C2211yescharacteristic
35C2×#3724C2211yescharacteristic
36C2×Dic3⋊Dic3C2212yes—
37C2×Dic3⋊Dic3C2212yes—
38C2×Dic3⋊Dic3C2212yes—
39C2×Dic3⋊Dic3C2212yes—
40#63956C22116yes—
41C2×Dic32C2212yes—
42#63919C2218yescomplemented
43C4×#627C2218yescomplemented
44#63870C2218yescomplemented
45#63867C22116yescomplemented
46#63846C22116yescomplemented
47#63838C2218yescomplemented
48C2×C3×#126—32no—
49#5701—332no—
50C2×#890—32no—
51C2×#889—32no—
52C2×#889—32no—
53C2×#889—32no—
54C2×#888—32no—
55C2×#887—32no—
56C2×#879—32no—
57C2×#879—32no—
58C2×#879D312yes—
59C2×#878—32no—
60C2×#878—32no—
61C2×#877—32no—
62C2×#877—61no—
63C2×#870—32no—
64C2×C4×Dic6—32no—
65C24×C32C2311yescharacteristic
66C22×C32×C4C2311yescharacteristic
67C22×C32×C4C2311yescharacteristic
68C22×C3⋊Dic3—24no—
69C22×C3⋊Dic3C2314yes—
70C22×C3⋊Dic3C2311yescharacteristic
71C22×C3⋊Dic3C2311yescharacteristic
72C22×C3⋊Dic3C2311yescharacteristic
73C22×C3⋊Dic3C2311yescharacteristic
74C22×C3×Dic3C2312yes—
75C22×C3×Dic3C2312yes—
76C22×C3×Dic3C2312yes—
77C22×C3×Dic3C2312yes—
78C2×#627—24no—
79C2×#627—24no—
80C2×#627C2314yes—
81C2×#627C2314yes—
82C32×C22⋊C4C2314yes—
83#3757C2314yes—
84#3750C2314yes—
85#3750C2314yes—
86C4×C3⋊Dic3C2314yes—
87C4×C3⋊Dic3C2314yes—
88C3×#217C2318yes—
89C3×#217C2318yes—
90C3×C4⋊Dic3C2318yes—
91C3×Dic3⋊C4C2318yes—
92C3×Dic3⋊C4C2318yes—
93C3×C4×Dic3C2318yes—
94#3724C2314yes—
95#3724C2318yes—
96#3724C2314yes—
97Dic3⋊Dic3C2318yes—
98Dic3⋊Dic3C2318yes—
99Dic3⋊Dic3C2318yes—
100Dic3⋊Dic3C2318yes—
101Dic32C2318yes—
102C2×#541—91noSylow
103C23×Dic3—32no—
104C23×Dic3—61no—
105C22×Dic6—32no—
106C22×Dic6—32no—
107C3×#126—316no—
108C2×C3×C4⋊C4—32no—
109C2×C3×C4⋊C4—32no—
110C2×C3×C4⋊C4—32no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—32no—
113C2×C3×C22⋊C4—32no—
114C2×C3×C22⋊C4C2×D312yes—
115C2×C3×C42—32no—
116C2×#217—32no—
117C2×#217—61no—
118C2×#217C2×D312yes—
119C2×#217C2×D312yes—
120C2×C4⋊Dic3—32no—
121C2×C4⋊Dic3—32no—
122C2×C4⋊Dic3—32no—
123C2×C4⋊Dic3—32no—
124C2×C4⋊Dic3C2×D312yes—
125C2×Dic3⋊C4—32no—
126C2×Dic3⋊C4—32no—
127C2×Dic3⋊C4—32no—
128C2×Dic3⋊C4—32no—
129C2×Dic3⋊C4—32no—
130C2×Dic3⋊C4—32no—
131C2×Dic3⋊C4—32no—
132C2×Dic3⋊C4—32no—
133C2×Dic3⋊C4—32no—
134C2×Dic3⋊C4—32no—
135C2×Dic3⋊C4—61no—
136C2×Dic3⋊C4—61no—
137C2×Dic3⋊C4C2×D312yes—
138C2×Dic3⋊C4C2×D312yes—
139C2×C4×Dic3—32no—
140C2×C4×Dic3—32no—
141C2×C4×Dic3—32no—
142C2×C4×Dic3—32no—
143C2×C4×Dic3—61no—
144C2×C4×Dic3—61no—
145C2×C4×Dic3C2×D312yes—
146#890—316no—
147#889—316no—
148#889—316no—
149#889—316no—
150#888—316no—
151#887—316no—
152#879—316no—
153#879—316no—
154#879C2×D3116yes—
155#878—316no—
156#878—316no—
157#877—316no—
158#877—68no—
159#870—316no—
160C4×Dic6—316no—
161C23×C32—22no—
162C23×C32—22no—
163C23×C32—22no—
164C23×C32C2412yes—
165C23×C32C2411yescharacteristic
166C2×C32×C4—22no—
167C2×C32×C4—22no—
168C2×C32×C4C2412yes—
169C2×C32×C4C2412yes—
170C2×C3⋊Dic3—22no—
171C2×C3⋊Dic3—28no—
172C2×C3⋊Dic3—22no—
173C2×C3⋊Dic3—22no—
174C2×C3⋊Dic3#4114yes—
175C2×C3⋊Dic3C2412yes—
176C2×C3⋊Dic3C2412yes—
177C2×C3⋊Dic3C2412yes—
178C2×C3⋊Dic3C2412yes—
179C2×C3×Dic3—24no—
180C2×C3×Dic3—24no—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3C2414yes—
184C2×C3×Dic3C2414yes—
185C2×C3×Dic3C2414yes—
186C2×C3×Dic3C2414yes—
187#627—28no—
188#627—28no—
189#541—916no—
190C2×C4⋊Q8—91no—
191C2×#126—32no—
192C2×#126—92no—
193C2×#125—92no—
194C2×#125—91no—
195C2×#125—92no—
196C2×C22⋊Q8—91no—
197C2×C22⋊Q8—91no—
198C2×C4×Q8—91no—
199C2×C4×Q8—91no—
200C2×#117—91no—
201C24×C3—21no—
202C24×C3C22×D312yes—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—32no—
206C22×C3×C4—32no—
207C22×C3×C4—32no—
208C22×C3×C4—32no—
209C22×C3×C4C22×D312yes—
210C22×C3×C4C22×D312yes—
211C22×Dic3—32no—
212C22×Dic3—38no—
213C22×Dic3—32no—
214C22×Dic3—32no—
215C22×Dic3—32no—
216C22×Dic3—64no—
217C22×Dic3—61no—
218C22×Dic3—61no—
219C22×Dic3—61no—
220C22×Dic3—61no—
221C22×Dic3—68no—
222C22×Dic3—68no—
223C22×Dic3C22×D312yes—
224C22×Dic3C22×D312yes—
225C22×Dic3C22×D312yes—
226C22×Dic3C22×D312yes—
227C2×Dic6—38no—
228C2×Dic6—38no—
229C2×Dic6—68no—
230C2×Dic6—68no—
231C3×C4⋊C4—38no—
232C3×C4⋊C4—38no—
233C3×C4⋊C4—38no—
234C3×C22⋊C4—24no—
235C3×C22⋊C4—38no—
236C3×C22⋊C4—38no—
237C3×C22⋊C4C22×D318yes—
238C3×C42—38no—
239#217—38no—
240#217—64no—
241#217C22×D318yes—
242#217C22×D318yes—
243C4⋊Dic3—38no—
244C4⋊Dic3—38no—
245C4⋊Dic3—38no—
246C4⋊Dic3—38no—
247C4⋊Dic3C22×D318yes—
248Dic3⋊C4—38no—
249Dic3⋊C4—38no—
250Dic3⋊C4—38no—
251Dic3⋊C4—38no—
252Dic3⋊C4—38no—
253Dic3⋊C4—38no—
254Dic3⋊C4—38no—
255Dic3⋊C4—38no—
256Dic3⋊C4—38no—
257Dic3⋊C4—38no—
258Dic3⋊C4—64no—
259Dic3⋊C4—64no—
260Dic3⋊C4C22×D318yes—
261Dic3⋊C4C22×D318yes—
262C4×Dic3—38no—
263C4×Dic3—38no—
264C4×Dic3—38no—
265C4×Dic3—38no—
266C4×Dic3—64no—
267C4×Dic3—64no—
268C4×Dic3C22×D318yes—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—22no—
272C22×C32—44no—
273C22×C32C2×#4112yes—
274C22×C322−1+412yes—
275C22×C322−1+412yes—
276C22×C32C2511yescharacteristic
277C32×C4—22no—
278C32×C4—22no—
279C3⋊Dic3—22no—
280C3⋊Dic3—22no—
281C3⋊Dic3—22no—
282C3⋊Dic3C2×#4114yes—
283C3×Dic3—24no—
284C3×Dic3—24no—
285C3×Dic3—24no—
286C3×Dic3—24no—
287C22×Q8—91no—
288C22×Q8—91no—
289C23×C4—91no—
290C4⋊Q8—98no—
291#126—316no—
292#126—916no—
293#125—916no—
294#125—98no—
295#125—916no—
296C22⋊Q8—98no—
297C22⋊Q8—98no—
298C4×Q8—98no—
299C4×Q8—98no—
300#117—98no—
301C2×C4⋊C4—32no—
302C2×C4⋊C4—32no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4—91no—
305C2×C4⋊C4—91no—
306C2×C4⋊C4—92no—
307C2×C4⋊C4—92no—
308C2×C4⋊C4—92no—
309C2×C4⋊C4—92no—
310C2×C4⋊C4—92no—
311C2×C4⋊C4—91no—
312C2×C4⋊C4—91no—
313C2×C22⋊C4—32no—
314C2×C22⋊C4—32no—
315C2×C22⋊C4—91no—
316C2×C22⋊C4D3211yescharacteristic
317C2×C42—32no—
318C2×C42—92no—
319C2×C42—91no—
320C2×C42—91no—
321C23×C3—24no—
322C23×C3—24no—
323C23×C3—24no—
324C23×C3—24no—
325C23×C3—22no—
326C23×C3—21no—
327C23×C3—42no—
328C23×C3—42no—
329C23×C3C23×D314yes—
330C23×C3C23×D312yes—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—34no—
336C2×C3×C4—34no—
337C2×C3×C4—34no—
338C2×C3×C4—34no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—64no—
342C2×C3×C4—64no—
343C2×C3×C4—64no—
344C2×C3×C4—64no—
345C2×C3×C4C23×D314yes—
346C2×C3×C4C23×D314yes—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—38no—
352C2×Dic3—34no—
353C2×Dic3—34no—
354C2×Dic3—34no—
355C2×Dic3—34no—
356C2×Dic3—64no—
357C2×Dic3—62no—
358C2×Dic3—62no—
359C2×Dic3—62no—
360C2×Dic3—62no—
361C2×Dic3—616no—
362C2×Dic3—616no—
363C2×Dic3—64no—
364C2×Dic3—64no—
365C2×Dic3—64no—
366C2×Dic3—64no—
367C2×Dic3—122no—
368C2×Dic3—122no—
369C2×Dic3C23×D314yes—
370C2×Dic3C23×D314yes—
371C2×Dic3C23×D314yes—
372C2×Dic3C23×D314yes—
373Dic6—616no—
374Dic6—616no—
375C2×C32—42no—
376C2×C32#54114yescomplemented
377C2×C32C22×#4111yescharacteristic
378C2×C32C2×2−1+411yescharacteristic
379C2×C32C2×2−1+411yescharacteristic
380C24C2×D3211yescharacteristic
381C2×Q8—94no—
382C2×Q8—94no—
383C2×Q8—184no—
384C2×Q8—184no—
385C22×C4—32no—
386C22×C4—32no—
387C22×C4—32no—
388C22×C4—32no—
389C22×C4—91no—
390C22×C4—94no—
391C22×C4—91no—
392C22×C4—91no—
393C22×C4—91no—
394C22×C4—184no—
395C22×C4C2×D3211yescharacteristic
396C22×C4C2×D3211yescharacteristic
397C4⋊C4—38no—
398C4⋊C4—38no—
399C4⋊C4—38no—
400C4⋊C4—94no—
401C4⋊C4—94no—
402C4⋊C4—98no—
403C4⋊C4—98no—
404C4⋊C4—98no—
405C4⋊C4—98no—
406C4⋊C4—98no—
407C4⋊C4—94no—
408C4⋊C4—94no—
409C22⋊C4—38no—
410C22⋊C4—38no—
411C22⋊C4—94no—
412C22⋊C4C2×D3214yes—
413C42—38no—
414C42—98no—
415C42—94no—
416C42—94no—
417C22×C3—24no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—24no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—21no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—48no—
428C22×C3—48no—
429C22×C3D3×#4114yes—
430C22×C3#101014yes—
431C22×C3#101014yes—
432C22×C3C24×D312yes—
433C3×C4—24no—
434C3×C4—24no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—64no—
438C3×C4—64no—
439C3×C4—64no—
440C3×C4—64no—
441Dic3—24no—
442Dic3—24no—
443Dic3—24no—
444Dic3—24no—
445Dic3—38no—
446Dic3—64no—
447Dic3—64no—
448Dic3—64no—
449Dic3—64no—
450Dic3—64no—
451Dic3—122no—
452Dic3—122no—
453C32C2×#54111yescharacteristic, complemented, Sylow
454C23—22no—
455C23—22no—
456C23—22no—
457C23C22×D3212yes—
458C23C22×D3211yescharacteristic
459Q8—188no—
460Q8—188no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—34no—
464C2×C4—34no—
465C2×C4—34no—
466C2×C4—34no—
467C2×C4—64no—
468C2×C4—64no—
469C2×C4—64no—
470C2×C4—64no—
471C2×C4—94no—
472C2×C4—92no—
473C2×C4—92no—
474C2×C4—92no—
475C2×C4—92no—
476C2×C4—188no—
477C2×C4—182no—
478C2×C4—182no—
479C2×C4—182no—
480C2×C4C22×D3212yes—
481C2×C4C22×D3212yes—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—24no—
486C2×C3—44no—
487C2×C3—44no—
488C2×C3#570118yescomplemented
489C2×C3C2×D3×#4112yes—
490C2×C3C2×#101012yes—
491C2×C3C2×#101012yes—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—44no—
496C22#6430612yes—
497C22#6430912yes—
498C22#6430912yes—
499C22C23×D3211yescharacteristic
500C4—22no—
501C4—22no—
502C4—64no—
503C4—64no—
504C4—64no—
505C4—64no—
506C4—94no—
507C4—182no—
508C4—182no—
509C4—182no—
510C3—21no—
511C3C2×#570112yescomplemented
512C2—42no—
513C2#1059424914yesdirect product factor
514C2C2×#6430611yescharacteristic
515C2C2×#6430911yescharacteristic
516C2C2×#6430911yescharacteristic
subgroupG/H
centerC23C22×D32
commutator, cosocleC22×C32C25
FrattiniC22C23×D32
FittingC2×C32×C22⋊C4C22
socleC23×C32C24
prime radicalC24×C32C23
hypercenterC2×C22⋊C4D32
hypocenterC32C2×#541
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
6. C2×#63919C2semidirect
7. C2×C4×#627C2semidirect
8. C2×#63870C2semidirect
9. C2×#63867C2semidirect
10. C2×#63846C2semidirect
11. C2×#63838C2semidirect
12. #10594249C2direct (N × H)
42. #63919C22semidirect
43. C4×#627C22semidirect
44. #63870C22semidirect
45. #63867C22semidirect
46. #63846C22semidirect
47. #63838C22semidirect
376. C2×C32#541semidirect
453. C32C2×#541semidirect
488. C2×C3#5701semidirect
511. C3C2×#5701semidirect
513. C2#10594249direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#640862
C2×C3×#8792
C2×#639822
C2×#639562
C2×#639192
C2×C4×#6272
C2×#638702
C2×#638672
C2×#638462
C2×#638382
#105942492
C2×#57013