Finite Group Databasebeta

← C2×#21462116 #30624010

Subgroups — C2×#21462116 #30624010

Order 15616.

1198 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519838C212yesmaximal, complemented
2#21462116C218yesmaximal, direct product factor
3C2×#16840673C212yesmaximal, complemented
4C2×#16840671C212yesmaximal, complemented
5C2×#16840338C211yesmaximal, characteristic
6C22×#11758217—22no—
7C2×#14519838—22no—
8C2×#14519838—24no—
9C2×#14519838C2214yes—
10C2×#14519838C2218yescomplemented
11#16840673C2218yescomplemented
12#16840671C2218yescomplemented
13C22×#11758222C411yescharacteristic
14C23×Dic122C2211yescharacteristic
15#16840338—24no—
16#16840338C414yes—
17#16840338C2214yes—
18C2×#14519777C411yescharacteristic
19C2×#14519777C2211yescharacteristic
20C2×#14519765C411yescharacteristic
21C2×#14519765C2211yescharacteristic
22C2×#14519761—22no—
23C2×#14519757—22no—
24C2×#117×C61C411yescharacteristic
25C2×#11758217—24no—
26C2×#11758217—28no—
27C2×#11758217—42no—
28C2×#11758217—44no—
29C22×#92420—42no—
30#14519838—24no—
31#14519838—28no—
32#14519838C2318yes—
33C2×#11758222—22no—
34C2×#11758222—42no—
35C2×#11758222—42no—
36C2×#11758222C2×C414yes—
37C22×Dic122—42no—
38C22×Dic122—42no—
39C22×Dic122C2×C411yescharacteristic
40C22×Dic122C2×C411yescharacteristic
41C22×Dic122C2×C412yes—
42C22×Dic122D412yes—
43C22×Dic122D414yes—
44C22×Dic122C2311yescharacteristic
45C22×Dic122C2312yes—
46C22×D244C2×C411yescharacteristic
47C22×C22⋊D61—21no—
48C23×Dic61D412yes—
49C22×C4×D61—21no—
50#14519777—22no—
51#14519777—22no—
52#14519777C2×C414yes—
53#14519777C2×C412yes—
54#14519777C2312yes—
55#14519765—22no—
56#14519765—22no—
57#14519765C2×C414yes—
58#14519765C2×C412yes—
59#14519765C2312yes—
60#14519761—28no—
61#14519761—28no—
62#14519757—28no—
63#14519757—28no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—22no—
66C2×#11758198—22no—
67C2×#11758198—22no—
68C23×C4×C61C2×C411yescharacteristic
69#117×C61—22no—
70#117×C61C2×C414yes—
71C2×C4⋊C4×C61—21no—
72C2×C22⋊C4×C61—21no—
73C2×C42×C61C2×C411yescharacteristic
74C2×C42×C61C2×C411yescharacteristic
75C24×C61C22⋊C411yescharacteristic
76C23×D61—21no—
77C22×C4×C61—22no—
78C22×C4×C61—22no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22⋊C412yes—
83C22×C4×C61C22×C411yescharacteristic
84C22×C4×C61C22×C412yes—
85C2×C22⋊D61—24no—
86C2×C22⋊D61—42no—
87C2×C22⋊D61—42no—
88C22×Dic61—22no—
89C22×Dic61—22no—
90C22×Dic61—24no—
91C22×Dic61—21no—
92C22×Dic61C22⋊C412yes—
93C22×Dic61C22⋊C414yes—
94C22×Dic61C2×D412yes—
95C22×Dic61C2×D414yes—
96#11758222—48no—
97#11758222—48no—
98C2×D244—21no—
99C2×D244—41no—
100C2×D244—41no—
101C2×D244C22×C412yes—
102C2×C4×D61—24no—
103C2×C4×D61—42no—
104C2×C4×D61—42no—
105C2×Dic122—21no—
106C2×Dic122—41no—
107C2×Dic122—41no—
108C2×Dic122—44no—
109C2×Dic122—42no—
110C2×Dic122—44no—
111C2×Dic122—42no—
112C2×Dic122—44no—
113C2×Dic122—41no—
114C2×Dic122—44no—
115C2×Dic122—41no—
116C2×Dic122—41no—
117C2×Dic122—41no—
118C2×Dic122C22⋊C414yes—
119C2×Dic122C22⋊C412yes—
120C2×Dic122C22×C412yes—
121C2×Dic122C22×C412yes—
122C2×Dic122C22×C411yescharacteristic
123C2×Dic122C2×D414yes—
124C2×Dic122C2×D412yes—
125C2×Dic122C2411yescharacteristic
126#11758217—28no—
127#11758217—44no—
128#11758217—48no—
129C2×#92420—44no—
130C2×#92420—48no—
131C4⋊C4×C61—24no—
132C22⋊C4×C61—24no—
133C42×C61—21no—
134C42×C61—21no—
135C42×C61C22×C412yes—
136C42×C61C22×C412yes—
137#11758198—28no—
138#11758198—24no—
139#11758198—24no—
140C4⋊Dic61—28no—
141C4⋊Dic61—24no—
142C4⋊Dic61—24no—
143C23×C61—42no—
144C23×C61#9912yes—
145C23×C61#9912yes—
146C23×C61C2×C22⋊C411yescharacteristic
147C23×C61C2×C22⋊C412yes—
148C22×D61—22no—
149C22×D61—44no—
150C22×D61—41no—
151C22×D61—41no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—24no—
157C2×C4×C61—24no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61C2×C22⋊C411yescharacteristic
163C2×C4×C61C2×C22⋊C412yes—
164C2×C4×C61C23×C411yescharacteristic
165C22⋊D61—48no—
166C2×Dic61—24no—
167C2×Dic61—22no—
168C2×Dic61—22no—
169C2×Dic61—24no—
170C2×Dic61—22no—
171C2×Dic61—24no—
172C2×Dic61—24no—
173C2×Dic61—22no—
174C2×Dic61—22no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61C2×C22⋊C412yes—
178C2×Dic61C2×C22⋊C414yes—
179C2×Dic61C22×D412yes—
180D244—42no—
181D244—42no—
182C4×D61—48no—
183Dic122—42no—
184Dic122—44no—
185Dic122—42no—
186Dic122—41no—
187Dic122—42no—
188Dic122—44no—
189Dic122—41no—
190Dic122—42no—
191Dic122—42no—
192#92420—48no—
193C2×#2133—611nomaximal, Sylow
194C22×C61—22no—
195C22×C61—24no—
196C22×C61—42no—
197C22×C61—42no—
198C22×C61C2×#9911yescharacteristic
199C22×C61C2×#9914yes—
200C22×C61C2×#9911yescharacteristic
201C22×C61C22×C22⋊C411yescharacteristic
202C2×D61—44no—
203C2×D61—42no—
204C2×D61—82no—
205C2×D61—82no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—22no—
209C4×C61—42no—
210C4×C61—42no—
211Dic61—22no—
212Dic61—24no—
213Dic61—22no—
214Dic61—42no—
215C2×#536—611no—
216C22×#101—612no—
217#2133—618no—
218C2×#356—612no—
219C2×#355—612no—
220C2×C61—21no—
221C2×C61—21no—
222C2×C61—24no—
223C2×C61#213312yescomplemented
224C2×C61C22×#9911yescharacteristic
225D61—82no—
226C22×#41—611no—
227C23×Q8—611no—
228C22×M16—1222no—
229#536—614no—
230#536—614no—
231#536—1224no—
232C2×C4⋊Q8—611no—
233C2×C4⋊Q8—611no—
234C2×#124—611no—
235C2×#124—611no—
236C2×#123—1222no—
237C2×C22⋊Q8—1222no—
238C2×#117#692311yescharacteristic
239C2×#101—614no—
240C2×#101—618no—
241C2×#101—1222no—
242C2×#101—1224no—
243#356—618no—
244#355—618no—
245C61C2×#213311yescharacteristic, complemented, Sylow
246C2×#41—614no—
247C2×#41—1222no—
248C2×#41—2442no—
249C2×#41—2442no—
250C22×Q8—612no—
251C22×Q8—614no—
252C22×Q8—611no—
253C22×Q8—612no—
254C22×Q8—611no—
255C22×Q8—612no—
256C22×Q8—611no—
257C22×Q8—2442no—
258C22×Q8—2442no—
259C22×D4—611no—
260C22×D4—1221no—
261C23×C4—612no—
262C23×C4—1221no—
263C23×C4C2×#692311yescharacteristic
264C2×M16—1224no—
265C2×M16—1228no—
266C2×M16—2444no—
267C2×M16—2442no—
268C22×C8—2442no—
269C4⋊Q8—612no—
270C4⋊Q8—612no—
271C4⋊Q8—614no—
272C4⋊Q8—1222no—
273C4⋊Q8—1222no—
274#124—612no—
275#124—612no—
276#124—614no—
277#124—1222no—
278#124—1222no—
279#123—1228no—
280#123—1228no—
281C22⋊Q8—1228no—
282C22⋊Q8—1228no—
283#117—22no—
284#117C2×#692314yes—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—1222no—
287C2×C4⋊C4—1222no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—1222no—
290C2×C22⋊C4—1222no—
291C2×C42C2×#692311yescharacteristic
292C2×C42C2×#692311yescharacteristic
293#101—618no—
294#101—1228no—
295#101—1224no—
296C24—1221no—
297C24C22⋊#692311yescharacteristic
298#41—2448no—
299#41—2448no—
300C2×Q8—614no—
301C2×Q8—612no—
302C2×Q8—614no—
303C2×Q8—612no—
304C2×Q8—611no—
305C2×Q8—612no—
306C2×Q8—611no—
307C2×Q8—612no—
308C2×Q8—1221no—
309C2×Q8—2442no—
310C2×Q8—2444no—
311C2×Q8—2444no—
312C2×Q8—2441no—
313C2×Q8—2441no—
314C2×Q8—2444no—
315C2×Q8—2444no—
316C2×Q8—2442no—
317C2×Q8—2441no—
318C2×Q8—2441no—
319C2×Q8—2441no—
320C2×Q8—2441no—
321C2×D4—612no—
322C2×D4—1224no—
323C2×D4—1221no—
324C2×D4—2441no—
325C2×D4—2441no—
326C2×D4—2442no—
327C2×D4—2442no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—612no—
333C22×C4—612no—
334C22×C4—614no—
335C22×C4—614no—
336C22×C4—1222no—
337C22×C4—1224no—
338C22×C4—1222no—
339C22×C4—1221no—
340C22×C4—1224no—
341C22×C4—2442no—
342C22×C4—2442no—
343C22×C4C22⋊#692311yescharacteristic
344C22×C4C22⋊#692312yes—
345C22×C4C22×#692311yescharacteristic
346C22×C4C22×#692312yes—
347M16—1228no—
348M16—2444no—
349M16—2448no—
350C2×C8—2448no—
351C2×C8—2444no—
352C4⋊C4—24no—
353C4⋊C4—1224no—
354C4⋊C4—1228no—
355C4⋊C4—1224no—
356C22⋊C4—24no—
357C22⋊C4—1224no—
358C22⋊C4—1228no—
359C22⋊C4—1224no—
360C42—21no—
361C42—21no—
362C42C22×#692312yes—
363C42C22×#692312yes—
364C23—42no—
365C23—1222no—
366C23—2444no—
367C23—2441no—
368C23—2441no—
369C23#1451983612yes—
370C23#1451983612yes—
371C23C2×C22⋊#692311yescharacteristic
372C23C2×C22⋊#692312yes—
373Q8—2444no—
374Q8—2444no—
375Q8—2442no—
376Q8—2442no—
377Q8—2441no—
378Q8—2442no—
379Q8—2441no—
380Q8—2442no—
381Q8—2442no—
382D4—2442no—
383D4—2442no—
384D4—2448no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—21no—
388C2×C4—24no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—612no—
396C2×C4—614no—
397C2×C4—612no—
398C2×C4—1224no—
399C2×C4—1222no—
400C2×C4—1224no—
401C2×C4—1222no—
402C2×C4—1224no—
403C2×C4—1224no—
404C2×C4—1222no—
405C2×C4—1222no—
406C2×C4—1222no—
407C2×C4—2448no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4C2×C22⋊#692311yescharacteristic
411C2×C4C2×C22⋊#692312yes—
412C2×C4C23×#692311yescharacteristic
413C8—2448no—
414C22—22no—
415C22—24no—
416C22—42no—
417C22—42no—
418C22—4882no—
419C22—4882no—
420C22—2444no—
421C22—2442no—
422C22C2×#1451983611yescharacteristic
423C22C2×#1451983614yes—
424C22C2×#1451983611yescharacteristic
425C22C22×C22⋊#692311yescharacteristic
426C4—22no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—42no—
431C4—1222no—
432C4—1224no—
433C4—1222no—
434C4—2442no—
435C2—21no—
436C2—21no—
437C2—24no—
438C2—4882no—
439C2#2146211612yesdirect product factor
440C2C22×#1451983611yescharacteristic
subgroupG/H
centerC22C2×#14519836
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#117×C61C4
socleC22×C61C2×#99
prime radicalC22×#11758222C4
cosocleC2×Dic122C24
hypercenterC2×#117#6923
hypocenterC61C2×#2133
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C22×#14519838C2semidirect
2. #21462116C2direct (N × H)
3. C2×#16840673C2semidirect
4. C2×#16840671C2semidirect
10. C2×#14519838C22semidirect
11. #16840673C22semidirect
12. #16840671C22semidirect
223. C2×C61#2133semidirect
245. C61C2×#2133semidirect
439. C2#21462116direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198382
#214621162
C2×#168406732
C2×#168406712
C2×#168403382
C2×#213361