Finite Group Databasebeta

← C2×#21444990 #29655764

Subgroups — C2×#21444990 #29655764

Order 15616.

946 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840485C211yesmaximal, characteristic
2C2×#16840112C211yesmaximal, characteristic, complemented
3C2×#16840049C211yesmaximal, characteristic
4#21444990C218yesmaximal, direct product factor
5C2×#16839507C211yesmaximal, characteristic, complemented
6C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
7C2×#16839489C211yesmaximal, characteristic
8C2×C61×#477C211yesmaximal, characteristic, complemented
9#16840485—22no—
10#16840485—22no—
11#16840485—22no—
12#16840485C2214yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519772—22no—
15C2×#14519765—21no—
16C2×#14519753—21no—
17C2×C4×Dic122—21no—
18C2×C4×D244C2211yescharacteristic
19C2×#14519735—21no—
20#16840112—22no—
21#16840112C2214yescomplemented
22#16840049—22no—
23#16840049C2214yes—
24C2×#14519708—21no—
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701—22no—
27C2×#14519697—21no—
28C2×C4×#92418C2211yescharacteristic
29#16839507C2214yescomplemented
30Q8⋊#92418C2214yescomplemented
31#16839489C2214yes—
32C2×C4⋊Q8×C61C2211yescharacteristic
33C2×C4×Q8×C61C2211yescharacteristic
34C61×#477—22no—
35C61×#477C2214yescomplemented
36C2×#107×C61—21no—
37C2×C4⋊C8×C61C2211yescharacteristic
38C2×#103×C61—21no—
39C22×Dic122—21no—
40C22×D244D411yescharacteristic
41C22×C4×D61—21no—
42C22×#92418—21no—
43C22×#92418—21no—
44#14519777—21no—
45#14519777—21no—
46#14519777D412yes—
47#14519777C2312yes—
48#14519772—28no—
49#14519772—28no—
50#14519765—24no—
51#14519765—24no—
52#14519753—24no—
53#14519753—24no—
54C4×Dic122—24no—
55C4×Dic122—24no—
56C4×D244—21no—
57C4×D244—21no—
58C4×D244D412yes—
59C4×D244C2312yes—
60#14519735—24no—
61#14519735—24no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61D411yescharacteristic
64C2×C4⋊Dic61D412yes—
65C2×#11758196—22no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C2×#11758198—21no—
69C2×C4×Dic61—22no—
70#14519708—24no—
71C4⋊#92418—21no—
72C4⋊#92418C2312yes—
73#14519701—28no—
74#14519697—24no—
75C4×#92418—21no—
76C4×#92418C2312yes—
77C22×Q8×C61—21no—
78C22×Q8×C61D411yescharacteristic
79C22×C8×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61C2312yes—
84C4×Q8×C61—21no—
85C4×Q8×C61—21no—
86C4×Q8×C61D412yes—
87C4×Q8×C61C2312yes—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61D411yescharacteristic
92C2×C42×C61—21no—
93C2×C42×C61C2311yescharacteristic
94#107×C61—24no—
95C4⋊C8×C61—21no—
96C4⋊C8×C61C2312yes—
97#103×C61—24no—
98C23×D61—21no—
99C2×Q8×C61—21no—
100C2×Q8×C61—22no—
101C2×Q8×C61—22no—
102C2×Q8×C61—22no—
103C2×Q8×C61—44no—
104C2×Q8×C61—41no—
105C2×Q8×C61C2×D412yes—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—41no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×Dic61—21no—
113C22×Dic61—22no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244—21no—
117C2×D244C2×D412yes—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—42no—
121C2×C4×D61—42no—
122C2×Dic122—22no—
123C2×Dic122—22no—
124C2×Dic122—42no—
125C2×Dic122—42no—
126C2×C8×C61—22no—
127C2×C8×C61—42no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61C2×D412yes—
136C42×C61—22no—
137C42×C61—22no—
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61C2411yescharacteristic
142#11758198—28no—
143#11758198—28no—
144#11758198—22no—
145#11758198—22no—
146C4⋊Dic61—22no—
147C4⋊Dic61—21no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61C2×D414yes—
151C4⋊Dic61C2×D412yes—
152#11758196—28no—
153C4×Dic61—28no—
154C2×#92418—22no—
155C2×#92418—22no—
156C2×#92418—42no—
157C2×#92418—42no—
158C23×C61#12011yescharacteristic
159C22×D61—22no—
160C22×D61—42no—
161C22×D61—42no—
162C22×D61—42no—
163Q8×C61—22no—
164Q8×C61—22no—
165Q8×C61—44no—
166Q8×C61—42no—
167Q8×C61—82no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—42no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—82no—
175C2×C4×C61#12011yescharacteristic
176C2×C4×C61#12011yescharacteristic
177C2×C4×C61#12011yescharacteristic
178C2×C4×C61#13512yes—
179C2×C4×C61#13512yes—
180C2×C4×C61#13712yes—
181C2×C4×C61#13712yes—
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C22×D411yescharacteristic
185C2×Dic61—22no—
186C2×Dic61—24no—
187C2×Dic61—44no—
188C2×Dic61—42no—
189D244—22no—
190D244—22no—
191D244—42no—
192C4×D61—42no—
193C4×D61—42no—
194C4×D61—44no—
195Dic122—42no—
196Dic122—42no—
197Dic122—44no—
198C8×C61—42no—
199#92418—42no—
200#92418—42no—
201C2×#1638—611nomaximal, Sylow
202C22×C61#45212yes—
203C22×C61#45212yes—
204C22×C61#45412yes—
205C22×C61C2×#12011yescharacteristic
206C2×D61—42no—
207C2×D61—42no—
208C2×D61—42no—
209C2×D61—84no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—42no—
213C4×C61—42no—
214C4×C61—82no—
215C4×C61#45212yes—
216C4×C61#45212yes—
217C4×C61C2×#13511yescharacteristic
218C4×C61C2×#13511yescharacteristic
219C4×C61C2×#13711yescharacteristic
220C4×C61C2×#13711yescharacteristic
221Dic61—44no—
222Dic61—42no—
223C2×#556—611no—
224C2×#487—611no—
225C2×#477D6111yescharacteristic, complemented
226C2×#476—611no—
227#1638—618no—
228C2×#332—611no—
229C2×Q8⋊C8—611no—
230C2×D4⋊C8—611no—
231C2×C61#163814yescomplemented
232C2×C61C2×#45211yescharacteristic
233C2×C61C2×#45211yescharacteristic
234C2×C61C2×#45411yescharacteristic
235D61—82no—
236#556—614no—
237#556—1222no—
238#556—1222no—
239#556—1222no—
240C2×C4⋊Q8—611no—
241C2×C4⋊Q8C2×D6111yescharacteristic
242C2×#126—1222no—
243C2×#124—1221no—
244C2×C22⋊Q8—1221no—
245C2×C4×Q8—1221no—
246C2×C4×Q8C2×D6111yescharacteristic
247C2×C4×D4—611no—
248C2×#117—1221no—
249#487—614no—
250#487—1222no—
251#477—22no—
252#477C2×D6114yescomplemented
253#476—614no—
254#476—1222no—
255C2×#107—21no—
256C2×#106—1221no—
257C2×C4⋊C8—611no—
258C2×C4⋊C8C2×D6111yescharacteristic
259C2×#103—21no—
260C2×#103—1222no—
261C2×#102—1221no—
262C2×C4×C8—611no—
263#332—614no—
264Q8⋊C8—614no—
265D4⋊C8—614no—
266C61C2×#163811yescharacteristic, complemented, Sylow
267C22×Q8—21no—
268C22×Q8—1221no—
269C22×Q8C22⋊D6111yescharacteristic
270C22×D4—611no—
271C23×C4—1221no—
272C22×C8—21no—
273C22×C8—1221no—
274C22×C8—1221no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—612no—
279C4⋊Q8—612no—
280C4⋊Q8—1221no—
281C4⋊Q8—1221no—
282C4⋊Q8C22×D6112yes—
283#126—1228no—
284#126—1228no—
285#124—1224no—
286#124—1224no—
287C22⋊Q8—1224no—
288C22⋊Q8—1224no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8—1224no—
292C4×Q8—1224no—
293C4×Q8C22⋊D6112yes—
294C4×Q8C22×D6112yes—
295C4×D4—612no—
296C4×D4—612no—
297C4×D4—1221no—
298C4×D4—1221no—
299#117—1224no—
300#117—1224no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—611no—
305C2×C4⋊C4—612no—
306C2×C4⋊C4—1222no—
307C2×C4⋊C4—1221no—
308C2×C4⋊C4C22⋊D6111yescharacteristic
309C2×C22⋊C4—1222no—
310C2×C22⋊C4—1222no—
311C2×C22⋊C4—1221no—
312C2×C42—21no—
313C2×C42—1222no—
314C2×C42C22×D6111yescharacteristic
315#107—24no—
316#106—1224no—
317C4⋊C8—21no—
318C4⋊C8—612no—
319C4⋊C8—1221no—
320C4⋊C8C22×D6112yes—
321#103—24no—
322#103—1228no—
323#102—1224no—
324C4×C8—612no—
325C4×C8—1221no—
326C24—1221no—
327C2×Q8—21no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—44no—
332C2×Q8—41no—
333C2×Q8—1222no—
334C2×Q8—1222no—
335C2×Q8—2442no—
336C2×Q8—2442no—
337C2×Q8C2×C22⋊D6112yes—
338C2×D4—612no—
339C2×D4—1222no—
340C2×D4—1221no—
341C2×D4—1222no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—41no—
345C22×C4—1221no—
346C22×C4—1222no—
347C22×C4—1222no—
348C22×C4—1222no—
349C22×C4—2442no—
350C22×C4—2442no—
351C22×C4D4×D6111yescharacteristic
352C22×C4D4×D6111yescharacteristic
353C22×C4C2×C22⋊D6111yescharacteristic
354C2×C8—22no—
355C2×C8—42no—
356C2×C8—1222no—
357C2×C8—1222no—
358C2×C8—2442no—
359C2×C8—2442no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—614no—
368C4⋊C4—612no—
369C4⋊C4—1228no—
370C4⋊C4—1222no—
371C4⋊C4—1221no—
372C4⋊C4—1222no—
373C4⋊C4—1222no—
374C4⋊C4C2×C22⋊D6112yes—
375C22⋊C4—1228no—
376C22⋊C4—1228no—
377C22⋊C4—1222no—
378C22⋊C4—1222no—
379C42—22no—
380C42—22no—
381C42—1228no—
382C42D4×D6111yescharacteristic
383C42D4×D6111yescharacteristic
384C42C2×C22⋊D6111yescharacteristic
385C42C23×D6111yescharacteristic
386C23—1222no—
387C23—2442no—
388C23—2442no—
389C23—2442no—
390C23#1451974511yescharacteristic
391Q8—22no—
392Q8—22no—
393Q8—44no—
394Q8—42no—
395Q8—82no—
396Q8—2442no—
397Q8—2442no—
398Q8—2444no—
399D4—1222no—
400D4—1222no—
401D4—2442no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—82no—
409C2×C4—1222no—
410C2×C4—1224no—
411C2×C4—2442no—
412C2×C4—2444no—
413C2×C4—2442no—
414C2×C4—2442no—
415C2×C4—2444no—
416C2×C4#1451974511yescharacteristic
417C2×C4#1451974511yescharacteristic
418C2×C4#1451974511yescharacteristic
419C2×C4#1451979712yes—
420C2×C4#1451980012yes—
421C2×C4#1451980812yes—
422C2×C4#1451980912yes—
423C2×C4C2×D4×D6111yescharacteristic
424C2×C4C2×D4×D6111yescharacteristic
425C2×C4C22×C22⋊D6111yescharacteristic
426C8—42no—
427C8—2442no—
428C8—2442no—
429C22—4884no—
430C22—2442no—
431C22—2442no—
432C22—2442no—
433C22#1683992212yes—
434C22#1683992312yes—
435C22#1683993312yes—
436C22C2×#1451974511yescharacteristic
437C4—21no—
438C4—21no—
439C4—42no—
440C4—42no—
441C4—82no—
442C4—2442no—
443C4—2444no—
444C4#1683992212yes—
445C4#1683992312yes—
446C4C2×#1451979711yescharacteristic
447C4C2×#1451980011yescharacteristic
448C4C2×#1451980811yescharacteristic
449C4C2×#1451980911yescharacteristic
450C2—4882no—
451C2#2144499014yesdirect product factor
452C2C2×#1683992211yescharacteristic
453C2C2×#1683992311yescharacteristic
454C2C2×#1683993311yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#477C2
socleC23×C61#120
prime radicalC22×D244D4
hypercenterC2×#477D61
hypocenterC61C2×#1638
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
2. C2×#16840112C2semidirect
4. #21444990C2direct (N × H)
5. C2×#16839507C2semidirect
6. C2×Q8⋊#92418C2semidirect
8. C2×C61×#477C2semidirect
21. #16840112C22semidirect
29. #16839507C22semidirect
30. Q8⋊#92418C22semidirect
35. C61×#477C22semidirect
225. C2×#477D61semidirect
231. C2×C61#1638semidirect
252. #477C2×D61semidirect
266. C61C2×#1638semidirect
451. C2#21444990direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168404852
C2×#168401122
C2×#168400492
#214449902
C2×#168395072
C2×Q8⋊#924182
C2×#168394892
C2×C61×#4772
C2×#163861