Finite Group Databasebeta

← C2×#13077226 #15617470

Subgroups — C2×#13077226 #15617470

Order 2816.

2586 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10623276C212yesmaximal
2C22×#66960C211yesmaximal, characteristic
3#13077226C2116yesmaximal, direct product factor
4C2×#10622738C211yesmaximal, characteristic, complemented
5C2×#10622506C214yesmaximal, complemented
6C22×#66886C212yesmaximal, complemented
7C2×D11×#108C214yesmaximal, complemented
8C2×C11×#430C211yesmaximal, characteristic, complemented
9C22×#4339C2211yescharacteristic
10C22×C11×M16C2212yes—
11C22×C11×M16C2211yescharacteristic
12C22×#4318C2212yes—
13C23×C11⋊C8C2211yescharacteristic
14#10623276—28no—
15#10623276C22116yes—
16C2×#66960—24no—
17C2×#66960C418yes—
18C2×#66960C418yes—
19C2×#66960C2218yes—
20C2×M16×D11C2218yes—
21C2×#66954—24no—
22C22×#4313C2212yes—
23C22×C8×D11C2212yes—
24C11×#430C2218yescomplemented
25C2×C11×#108C2214yes—
26#10622738C2218yescomplemented
27C2×#66886C2214yes—
28C2×#66886C2214yes—
29C2×#66886C22116yescomplemented
30C2×#66883C2214yes—
31#10622506C22132yescomplemented
32D11×#108C22132yescomplemented
33C23×C4×C11C2311yescharacteristic
34C22×C22⋊D11Q812yes—
35C2×#4339C2×C414yes—
36C2×#4339C2×C414yes—
37C2×#4339D414yes—
38C2×#4339D414yes—
39C2×#4339Q814yes—
40C2×#4339C2314yes—
41C22×D44Q811yescharacteristic
42C22×C4×D11C2312yes—
43C22×Dic22Q811yescharacteristic
44C2×C11×M16—22no—
45C2×C11×M16—24no—
46C2×C11×M16C2×C414yes—
47C2×C11×M16C2×C414yes—
48C2×C11×M16C2314yes—
49C2×C11×M16C2318yes—
50C2×C11×M16C2314yes—
51C22×C8×C11—22no—
52C22×C8×C11C2312yes—
53C2×#4318—22no—
54C2×#4318—24no—
55C2×#4318C2314yes—
56C2×#4318C2318yes—
57C22×C11⋊C8—22no—
58C22×C11⋊C8C2×C414yes—
59C22×C11⋊C8C2×C414yes—
60C22×C11⋊C8C2311yescharacteristic
61C22×C11⋊C8C2311yescharacteristic
62C22×C11⋊C8C2314yes—
63#66960—232no—
64#66960C2×C4164yes—
65M16×D11—216no—
66M16×D11C23132yes—
67#66954—232no—
68C2×#4313—28no—
69C2×#4313C2×C418yes—
70C2×#4313C2×C418yes—
71C2×#4313C2318yes—
72C2×C8×D11—28no—
73C2×C8×D11C2×C418yes—
74C2×C8×D11C2×C418yes—
75C2×C8×D11C2318yes—
76C11×#108C23116yes—
77#66886C23116yes—
78#66886C23116yes—
79#66883C23116yes—
80C2×#2916—111nomaximal, Sylow
81C24×C11C2×Q811yescharacteristic
82C23×D11C2×Q812yes—
83C22×C4×C11C22×C412yes—
84C22×C4×C11C22×C412yes—
85C22×C4×C11C2×D412yes—
86C22×C4×C11C2×D412yes—
87C22×C4×C11C2×Q811yescharacteristic
88C22×C4×C11C2×Q812yes—
89C22×C4×C11C2411yescharacteristic
90C22×C4×C11C2412yes—
91C2×C22⋊D11C4⋊C418yes—
92C2×C22⋊D11C4⋊C418yes—
93C2×C22⋊D11C2×Q818yes—
94C22×Dic11C2×Q812yes—
95#4339C4⋊C4116yes—
96#4339C4⋊C4116yes—
97#4339C22×C4116yes—
98#4339C2×D4116yes—
99C2×D44C4⋊C414yes—
100C2×D44C4⋊C414yes—
101C2×D44C2×Q814yes—
102C2×C4×D11C22×C414yes—
103C2×C4×D11C22×C414yes—
104C2×C4×D11C2×D414yes—
105C2×C4×D11C2×D414yes—
106C2×C4×D11C2×Q814yes—
107C2×C4×D11C2414yes—
108C2×Dic22C4⋊C414yes—
109C2×Dic22C4⋊C414yes—
110C2×Dic22C2×Q814yes—
111C11×M16—24no—
112C11×M16—28no—
113C11×M16C22×C4116yes—
114C11×M16C2418yes—
115C2×C8×C11—24no—
116C2×C8×C11—28no—
117C2×C8×C11C22×C414yes—
118C2×C8×C11C22×C414yes—
119C2×C8×C11C2414yes—
120#4318—28no—
121#4318—24no—
122#4318C2418yes—
123C2×C11⋊C8—24no—
124C2×C11⋊C8—28no—
125C2×C11⋊C8C22×C412yes—
126C2×C11⋊C8C22×C412yes—
127C2×C11⋊C8C22×C412yes—
128C2×C11⋊C8C22×C412yes—
129C2×C11⋊C8C22×C4116yes—
130C2×C11⋊C8C2412yes—
131C2×C11⋊C8C2412yes—
132#4313—232no—
133#4313C22×C4132yes—
134C8×D11—232no—
135C8×D11C22×C4132yes—
136C2×#568—112no—
137C22×#131—111no—
138#2916—1116no—
139C2×#430—114no—
140C2×#430—111no—
141C2×#430D1111yescharacteristic, complemented
142C22×#108—114no—
143C22×#108—112no—
144C23×C11—42no—
145C23×C11C2×C4⋊C412yes—
146C23×C11C2×C4⋊C412yes—
147C23×C11C22×Q811yescharacteristic
148C23×C11C22×Q812yes—
149C22×D11—44no—
150C22×D11C2×C4⋊C414yes—
151C22×D11C2×C4⋊C414yes—
152C22×D11C22×Q814yes—
153C2×C4×C11C2×C4⋊C412yes—
154C2×C4×C11C2×C4⋊C412yes—
155C2×C4×C11C2×C4⋊C414yes—
156C2×C4×C11C2×C4⋊C414yes—
157C2×C4×C11C23×C411yescharacteristic
158C2×C4×C11C23×C411yescharacteristic
159C2×C4×C11C23×C414yes—
160C2×C4×C11C22×D414yes—
161C2×C4×C11C22×D411yescharacteristic
162C2×C4×C11C22×D411yescharacteristic
163C2×C4×C11C22×Q812yes—
164C2×C4×C11C22×Q811yescharacteristic
165C2×C4×C11C2511yescharacteristic
166C22⋊D11C2×C4⋊C4132yes—
167C2×Dic11C2×C4⋊C414yes—
168C2×Dic11C2×C4⋊C414yes—
169C2×Dic11C22×Q814yes—
170D44C2×C4⋊C4116yes—
171C4×D11C2×C4⋊C418yes—
172C4×D11C2×C4⋊C418yes—
173C4×D11C23×C418yes—
174C4×D11C22×D418yes—
175Dic22C2×C4⋊C4116yes—
176C8×C11—28no—
177C8×C11C23×C418yes—
178C11⋊C8—28no—
179C11⋊C8C23×C414yes—
180C11⋊C8C23×C414yes—
181C22×#41—111no—
182#568—1116no—
183#568—228no—
184C2×#131—118no—
185C2×#131—118no—
186C2×#131—118no—
187C2×#131—224no—
188C2×#131—224no—
189C22×M16—112no—
190C22×M16—112no—
191C22×M16—118no—
192C22×M16C2×D1112yes—
193C22×M16C2×D1111yescharacteristic
194C23×C8—111no—
195C23×C8—112no—
196#430—118no—
197#430—1132no—
198#430C2×D1118yescomplemented
199C2×#108—114no—
200C2×#108—114no—
201C2×#108—1116no—
202C2×#108—114no—
203C2×#108—1132no—
204C2×#108C2×D1114yes—
205C22×C11—22no—
206C22×C11—24no—
207C22×C11—42no—
208C22×C11—42no—
209C22×C11C22×C4⋊C411yescharacteristic
210C22×C11C22×C4⋊C414yes—
211C22×C11C22×C4⋊C411yescharacteristic
212C22×C11C23×Q811yescharacteristic
213C2×D11—28no—
214C2×D11—44no—
215C2×D11—44no—
216C2×D11C22×C4⋊C418yes—
217C4×C11C22×C4⋊C414yes—
218C4×C11C22×C4⋊C411yescharacteristic
219C4×C11C22×C4⋊C411yescharacteristic
220C4×C11C24×C411yescharacteristic
221C4×C11C23×D411yescharacteristic
222Dic11C22×C4⋊C418yes—
223C2×#41—114no—
224C2×#41—114no—
225C2×#41—114no—
226C2×#41—114no—
227C2×#41—114no—
228C2×#41—114no—
229C22×Q8—111no—
230C22×D4—112no—
231C22×D4—111no—
232C23×C4—112no—
233C23×C4C22×D1111yescharacteristic
234#131—1164no—
235#131—2232no—
236#131—2232no—
237C2×M16—22no—
238C2×M16—24no—
239C2×M16—118no—
240C2×M16—118no—
241C2×M16—118no—
242C2×M16—114no—
243C2×M16—118no—
244C2×M16—1132no—
245C2×M16—228no—
246C2×M16—2216no—
247C2×M16—222no—
248C2×M16—224no—
249C2×M16C4×D1114yes—
250C2×M16C4×D1114yes—
251C2×M16C22×D1114yes—
252C2×M16C22×D1118yes—
253C2×M16C22×D1114yes—
254C22×C8—22no—
255C22×C8—114no—
256C22×C8—114no—
257C22×C8—111no—
258C22×C8—111no—
259C22×C8—114no—
260C22×C8—118no—
261C22×C8—118no—
262C22×C8—118no—
263C22×C8—228no—
264C22×C8—222no—
265C22×C8C22×D1112yes—
266#108—1116no—
267#108—1116no—
268#108—1116no—
269#108C22×D11116yes—
270C2×C11—21no—
271C2×C11—21no—
272C2×C11—24no—
273C2×C11#291612yescomplemented
274C2×C11C23×C4⋊C411yescharacteristic
275D11—28no—
276C24—112no—
277C24Q8×D1111yescharacteristic
278#41—1116no—
279#41—1116no—
280#41—1116no—
281#41—1116no—
282C2×Q8—114no—
283C2×Q8—114no—
284C2×Q8—114no—
285C2×D4—118no—
286C2×D4—118no—
287C2×D4—118no—
288C2×D4—114no—
289C2×D4—114no—
290C2×D4—114no—
291C22×C4—112no—
292C22×C4—114no—
293C22×C4—114no—
294C22×C4—114no—
295C22×C4—114no—
296C22×C4—114no—
297C22×C4—114no—
298C22×C4C2×C4×D1112yes—
299C22×C4C2×C4×D1112yes—
300C22×C4D4×D1112yes—
301C22×C4D4×D1112yes—
302C22×C4Q8×D1111yescharacteristic
303C22×C4Q8×D1112yes—
304C22×C4C23×D1112yes—
305C22×C4C23×D1111yescharacteristic
306M16—24no—
307M16—28no—
308M16—1132no—
309M16—118no—
310M16—2232no—
311M16—224no—
312M16—228no—
313M16C2×C4×D11116yes—
314M16C23×D1118yes—
315C2×C8—28no—
316C2×C8—24no—
317C2×C8—112no—
318C2×C8—112no—
319C2×C8—112no—
320C2×C8—112no—
321C2×C8—1116no—
322C2×C8—112no—
323C2×C8—112no—
324C2×C8—1132no—
325C2×C8—2232no—
326C2×C8—228no—
327C2×C8—224no—
328C2×C8C2×C4×D1114yes—
329C2×C8C2×C4×D1114yes—
330C2×C8C23×D1114yes—
331C11C2×#291611yescharacteristic, complemented, Sylow
332C23—42no—
333C23—114no—
334C23—114no—
335C23—114no—
336C23—444no—
337C23C4⋊C4×D1112yes—
338C23C4⋊C4×D1112yes—
339C23C2×Q8×D1111yescharacteristic
340C23C2×Q8×D1112yes—
341Q8—1116no—
342D4—1132no—
343D4—1116no—
344C2×C4—114no—
345C2×C4—114no—
346C2×C4—114no—
347C2×C4—118no—
348C2×C4—118no—
349C2×C4—118no—
350C2×C4—118no—
351C2×C4C4⋊C4×D1112yes—
352C2×C4C4⋊C4×D1112yes—
353C2×C4C4⋊C4×D1114yes—
354C2×C4C4⋊C4×D1114yes—
355C2×C4C22×C4×D1111yescharacteristic
356C2×C4C22×C4×D1111yescharacteristic
357C2×C4C22×C4×D1114yes—
358C2×C4C2×D4×D1114yes—
359C2×C4C2×D4×D1111yescharacteristic
360C2×C4C2×D4×D1111yescharacteristic
361C2×C4C2×Q8×D1111yescharacteristic
362C2×C4C2×Q8×D1112yes—
363C2×C4C24×D1111yescharacteristic
364C8—28no—
365C8—114no—
366C8—114no—
367C8—228no—
368C8C22×C4×D1118yes—
369C22—22no—
370C22—24no—
371C22—42no—
372C22—42no—
373C22—118no—
374C22—228no—
375C22—444no—
376C22—444no—
377C22C2×C4⋊C4×D1111yescharacteristic
378C22C2×C4⋊C4×D1114yes—
379C22C2×C4⋊C4×D1111yescharacteristic
380C22C22×Q8×D1111yescharacteristic
381C4—118no—
382C4C2×C4⋊C4×D1111yescharacteristic
383C4C2×C4⋊C4×D1111yescharacteristic
384C4C2×C4⋊C4×D1114yes—
385C4C23×C4×D1111yescharacteristic
386C4C22×D4×D1111yescharacteristic
387C2—21no—
388C2—21no—
389C2—24no—
390C2—228no—
391C2#1307722612yesdirect product factor
392C2C22×C4⋊C4×D1111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D11
commutatorC4×C11C24×C4
FrattiniC2×C4C24×D11
FittingC2×C11×#430C2
socleC22×C11C22×C4⋊C4
prime radicalC22×#4339C22
cosocleC2×C4×C11C25
hypercenterC2×#430D11
hypocenterC11C2×#2916
LCSG ▷ C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#430

Complements

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

NH ≅ G/Ntype
3. #13077226C2direct (N × H)
4. C2×#10622738C2semidirect
5. C2×#10622506C2semidirect
6. C22×#66886C2semidirect
7. C2×D11×#108C2semidirect
8. C2×C11×#430C2semidirect
24. C11×#430C22semidirect
26. #10622738C22semidirect
29. C2×#66886C22semidirect
31. #10622506C22semidirect
32. D11×#108C22semidirect
141. C2×#430D11semidirect
198. #430C2×D11semidirect
273. C2×C11#2916semidirect
331. C11C2×#2916semidirect
391. C2#13077226direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#106232762
C22×#669602
#130772262
C2×#106227382
C2×#106225062
C22×#668862
C2×D11×#1082
C2×C11×#4302
C2×#291611