Finite Group Databasebeta

← C2×#21447512 #29832854

Subgroups — C2×#21447512 #29832854

Order 15616.

1198 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840337C211yesmaximal, characteristic
2C2×#16839745C212yesmaximal
3#21447512C218yesmaximal, direct product factor
4C2×#16839561C211yesmaximal, characteristic, complemented
5C2×#16839560C212yesmaximal, complemented
6C2×C61×#356C211yesmaximal, characteristic, complemented
7C22×#11758226C2211yescharacteristic
8C22×Q8×D61C411yescharacteristic
9C2×M16×D61—22no—
10#16840337—24no—
11#16840337C414yes—
12#16840337C2214yes—
13C2×#14519775C2211yescharacteristic
14C2×#14519766C411yescharacteristic
15C2×#14519765C411yescharacteristic
16C2×#14519762—22no—
17C2×#14519755—22no—
18C2×#14519734C411yescharacteristic
19#16839745—24no—
20#16839745C2218yes—
21C2×#14519696C2212yes—
22C2×#14519693—22no—
23#16839561C2214yescomplemented
24#16839560C2218yescomplemented
25C2×C4⋊Q8×C61C2211yescharacteristic
26C2×#101×C61C2212yes—
27C61×#356C2214yescomplemented
28C2×#11758226—42no—
29C2×#11758226—42no—
30C2×#11758226C2×C412yes—
31C2×#11758226D414yes—
32C2×#11758226C2312yes—
33C22×Dic122—21no—
34C22×Dic122C2×C411yescharacteristic
35C2×Q8×D61—22no—
36C2×Q8×D61—42no—
37C2×Q8×D61—42no—
38C2×Q8×D61C2×C414yes—
39C22×D244C2×C411yescharacteristic
40C22×D244D412yes—
41C22×C4×D61—21no—
42C22×C4×D61C2×C411yescharacteristic
43C22×C4×D61D412yes—
44C2×#11758194—22no—
45C2×#11758189—42no—
46M16×D61—28no—
47M16×D61—44no—
48C2×C8×D61—42no—
49#14519775—22no—
50#14519775C2×C412yes—
51#14519775C2312yes—
52#14519766—22no—
53#14519766C2×C414yes—
54#14519765—22no—
55#14519765C2×C414yes—
56#14519762—28no—
57#14519762—28no—
58#14519755—28no—
59#14519755—28no—
60#14519734—22no—
61#14519734C2×C414yes—
62C2×#11758196—22no—
63C2×#11758196—21no—
64C2×#11758198—21no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×C4×Dic61C2×C411yescharacteristic
68#14519696—22no—
69#14519696C2314yes—
70#14519693—28no—
71C22×Q8×C61C2×C411yescharacteristic
72C22×Q8×C61C2311yescharacteristic
73C2×M16×C61—22no—
74C4⋊Q8×C61—22no—
75C4⋊Q8×C61C2×C412yes—
76C4⋊Q8×C61C2312yes—
77C2×C4⋊C4×C61—22no—
78C2×C42×C61C2×C411yescharacteristic
79#101×C61—22no—
80#101×C61C2314yes—
81C23×D61C22⋊C411yescharacteristic
82C23×D61C22⋊C412yes—
83C2×Q8×C61—21no—
84C2×Q8×C61—41no—
85C2×Q8×C61—41no—
86C2×Q8×C61—41no—
87C2×Q8×C61—41no—
88C2×Q8×C61C22×C412yes—
89C2×Q8×C61C22×C411yescharacteristic
90C2×Q8×C61C2×D412yes—
91C2×Q8×C61C2411yescharacteristic
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61C22×C411yescharacteristic
95C22×C4×C61C2×D412yes—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61C22⋊C411yescharacteristic
99#11758226—44no—
100#11758226—44no—
101#11758226—44no—
102#11758226—44no—
103Q8×D61—48no—
104Q8×D61—48no—
105C2×D244—41no—
106C2×D244—41no—
107C2×D244—42no—
108C2×D244—42no—
109C2×D244C22⋊C412yes—
110C2×D244C22⋊C414yes—
111C2×D244C22×C412yes—
112C2×D244C2×D414yes—
113C2×C4×D61—22no—
114C2×C4×D61—22no—
115C2×C4×D61—22no—
116C2×C4×D61—24no—
117C2×C4×D61—22no—
118C2×C4×D61—24no—
119C2×C4×D61—42no—
120C2×C4×D61—42no—
121C2×C4×D61C22⋊C412yes—
122C2×C4×D61C22⋊C414yes—
123C2×C4×D61C22×C412yes—
124C2×C4×D61C2×D414yes—
125C2×Dic122—21no—
126C2×Dic122—24no—
127C2×Dic122—42no—
128C2×Dic122—42no—
129C2×Dic122—41no—
130C2×Dic122—41no—
131C2×Dic122C22×C412yes—
132M16×C61—24no—
133M16×C61—42no—
134C2×C8×C61—42no—
135C4⋊C4×C61—24no—
136C4⋊C4×C61—24no—
137C42×C61—21no—
138C42×C61C22×C412yes—
139#11758198—24no—
140#11758198—28no—
141#11758198—28no—
142#11758196—24no—
143#11758196—24no—
144#11758196—24no—
145C4×Dic61—21no—
146C4×Dic61C22×C412yes—
147#11758194—24no—
148#11758194—42no—
149C2×#92418—42no—
150#11758189—48no—
151C8×D61—48no—
152C23×C61C2×C22⋊C411yescharacteristic
153C22×D61—21no—
154C22×D61—21no—
155C22×D61—22no—
156C22×D61—22no—
157C22×D61—42no—
158C22×D61—44no—
159C22×D61C2×C22⋊C414yes—
160C22×D61C2×C22⋊C412yes—
161Q8×C61—41no—
162Q8×C61—42no—
163Q8×C61—41no—
164Q8×C61—42no—
165Q8×C61—42no—
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—42no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61C2×C22⋊C411yescharacteristic
176C2×C4×C61C2×C22⋊C412yes—
177C2×C4×C61C23×C411yescharacteristic
178C2×C4×C61C22×D412yes—
179C2×Dic61—22no—
180C2×Dic61—22no—
181C2×Dic61—42no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61#9912yes—
185C2×Dic61#9912yes—
186C2×Dic61C2×C22⋊C412yes—
187D244—42no—
188D244—44no—
189D244—42no—
190D244—44no—
191C4×D61—24no—
192C4×D61—24no—
193C4×D61—24no—
194C4×D61—24no—
195C4×D61—24no—
196C4×D61—24no—
197C4×D61—48no—
198Dic122—42no—
199Dic122—48no—
200Dic122—42no—
201C8×C61—44no—
202#92418—44no—
203C2×#2135—611nomaximal, Sylow
204C22×C61—22no—
205C22×C61C2×#9911yescharacteristic
206C22×C61C2×#9911yescharacteristic
207C22×C61C22×C22⋊C411yescharacteristic
208C2×D61—22no—
209C2×D61—24no—
210C2×D61—42no—
211C2×D61—42no—
212C2×D61—44no—
213C2×D61—42no—
214C2×D61—44no—
215C2×D61—82no—
216C4×C61—22no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—22no—
220C4×C61—42no—
221C4×C61—42no—
222Dic61—42no—
223Dic61—42no—
224Dic61C2×#9914yes—
225C2×#536—611no—
226C2×#413—612no—
227#2135—618no—
228C2×#356—612no—
229C2×#356—611no—
230C2×#356D6111yescharacteristic, complemented
231C2×C61—21no—
232C2×C61—21no—
233C2×C61#213512yescomplemented
234C2×C61C22×#9911yescharacteristic
235D61—42no—
236D61—44no—
237C22×#41—611no—
238C23×Q8—611no—
239C22×M16—1222no—
240#536—614no—
241#536—614no—
242#536—1224no—
243C2×C4⋊Q8—611no—
244C2×C4⋊Q8C2×D6111yescharacteristic
245C2×#124—611no—
246C2×#124—611no—
247C2×#123—1222no—
248C2×C22⋊Q8—1222no—
249C2×#117—611no—
250#413—618no—
251#413—1224no—
252C2×#101—612no—
253C2×#101C2×D6112yes—
254C2×#100—1222no—
255#356—614no—
256#356—618no—
257#356C2×D6114yescomplemented
258C61C2×#213511yescharacteristic, complemented, Sylow
259C2×#41—612no—
260C2×#41—614no—
261C2×#41—612no—
262C2×#41—2442no—
263C2×#41—2442no—
264C22×Q8—614no—
265C22×Q8—611no—
266C22×Q8—1221no—
267C22×Q8—1222no—
268C22×Q8—2442no—
269C22×Q8—2442no—
270C22×Q8C4×D6111yescharacteristic
271C22×Q8C22×D6111yescharacteristic
272C22×D4—611no—
273C22×D4—612no—
274C23×C4—611no—
275C23×C4—612no—
276C23×C4—1221no—
277C2×M16—22no—
278C2×M16—1228no—
279C2×M16—1222no—
280C2×M16—2442no—
281C2×M16—2444no—
282C22×C8—2442no—
283C4⋊Q8—22no—
284C4⋊Q8—612no—
285C4⋊Q8—612no—
286C4⋊Q8—1222no—
287C4⋊Q8C4×D6112yes—
288C4⋊Q8C22×D6112yes—
289#124—614no—
290#124—614no—
291#124—1222no—
292#124—1222no—
293#123—1228no—
294#123—1228no—
295C22⋊Q8—1228no—
296C22⋊Q8—1228no—
297#117—614no—
298#117—1222no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—1221no—
301C2×C4⋊C4—1222no—
302C2×C22⋊C4—1221no—
303C2×C22⋊C4—1222no—
304C2×C22⋊C4—1222no—
305C2×C42—611no—
306C2×C42C4×D6111yescharacteristic
307#101—22no—
308#101—614no—
309#101—1222no—
310#101C22×D6114yes—
311#100—1228no—
312C24—612no—
313C24—611no—
314#41—2444no—
315#41—2444no—
316#41—2444no—
317#41—2444no—
318C2×Q8—21no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—612no—
324C2×Q8—1224no—
325C2×Q8—1221no—
326C2×Q8—2441no—
327C2×Q8—2441no—
328C2×Q8—2442no—
329C2×Q8—2442no—
330C2×Q8—2448no—
331C2×Q8—2448no—
332C2×Q8C2×C4×D6112yes—
333C2×Q8C2×C4×D6111yescharacteristic
334C2×Q8D4×D6112yes—
335C2×Q8C23×D6111yescharacteristic
336C2×D4—612no—
337C2×D4—614no—
338C2×D4—614no—
339C2×D4—612no—
340C2×D4—2442no—
341C2×D4—2441no—
342C2×D4—2442no—
343C2×D4—2441no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—611no—
347C22×C4—612no—
348C22×C4—614no—
349C22×C4—612no—
350C22×C4—614no—
351C22×C4—1222no—
352C22×C4—1224no—
353C22×C4—1222no—
354C22×C4—1222no—
355C22×C4—1222no—
356C22×C4—1224no—
357C22×C4—1221no—
358C22×C4—1221no—
359C22×C4—2442no—
360C22×C4—2442no—
361C22×C4C2×C4×D6111yescharacteristic
362C22×C4D4×D6112yes—
363M16—24no—
364M16—42no—
365M16—1224no—
366M16—2448no—
367M16—2442no—
368C2×C8—42no—
369C2×C8—2448no—
370C2×C8—2442no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—1224no—
374C4⋊C4—1224no—
375C4⋊C4—1224no—
376C22⋊C4—1224no—
377C22⋊C4—1228no—
378C22⋊C4—1228no—
379C42—21no—
380C42—612no—
381C42—1221no—
382C42C2×C4×D6112yes—
383C23—614no—
384C23—612no—
385C23—1222no—
386C23—1221no—
387C23—1221no—
388C23—1222no—
389C23—2444no—
390C23—2442no—
391C23C22⋊C4×D6111yescharacteristic
392Q8—41no—
393Q8—42no—
394Q8—41no—
395Q8—42no—
396Q8—42no—
397Q8—2442no—
398Q8—2442no—
399Q8—2448no—
400D4—2444no—
401D4—2442no—
402D4—2444no—
403D4—2442no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—612no—
414C2×C4—612no—
415C2×C4—612no—
416C2×C4—1224no—
417C2×C4—1224no—
418C2×C4—1224no—
419C2×C4—1224no—
420C2×C4—1224no—
421C2×C4—1224no—
422C2×C4—1222no—
423C2×C4—1222no—
424C2×C4—2448no—
425C2×C4—2442no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4C22⋊C4×D6111yescharacteristic
429C2×C4C22⋊C4×D6112yes—
430C2×C4C22×C4×D6111yescharacteristic
431C2×C4C2×D4×D6112yes—
432C8—44no—
433C8—2444no—
434C22—22no—
435C22—1222no—
436C22—1224no—
437C22—4882no—
438C22—2444no—
439C22—2442no—
440C22—2442no—
441C22—2444no—
442C22—2442no—
443C22#99×D6111yescharacteristic
444C22#99×D6111yescharacteristic
445C22C2×C22⋊C4×D6111yescharacteristic
446C4—22no—
447C4—21no—
448C4—22no—
449C4—21no—
450C4—42no—
451C4—42no—
452C4—614no—
453C4—2442no—
454C4—2442no—
455C2—21no—
456C2—21no—
457C2—2444no—
458C2—2442no—
459C2#2144751212yesdirect product factor
460C2C2×#99×D6111yescharacteristic
subgroupG/H
centerC22#99×D61
commutatorC2×C4×C61C23×C4
FrattiniC2×Q8C23×D61
FittingC2×C61×#356C2
socleC22×C61C2×#99
prime radicalC22×#11758226C22
cosocleC2×Q8×C61C24
hypercenterC2×#356D61
hypocenterC61C2×#2135
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
3. #21447512C2direct (N × H)
4. C2×#16839561C2semidirect
5. C2×#16839560C2semidirect
6. C2×C61×#356C2semidirect
23. #16839561C22semidirect
24. #16839560C22semidirect
27. C61×#356C22semidirect
230. C2×#356D61semidirect
233. C2×C61#2135semidirect
257. #356C2×D61semidirect
258. C61C2×#2135semidirect
459. C2#21447512direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403372
C2×#168397452
#214475122
C2×#168395612
C2×#168395602
C2×C61×#3562
C2×#213561