Finite Group Databasebeta

← C2×#21444793 #29652197

Subgroups — C2×#21444793 #29652197

Order 15616.

1200 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839917C212yesmaximal
2C2×#16839682C211yesmaximal, characteristic
3#21444793C218yesmaximal, direct product factor
4C2×#16839499C212yesmaximal
5C2×#16839488C211yesmaximal, characteristic
6C2×C61×#544C211yesmaximal, characteristic
7C23×Dic122C2211yescharacteristic
8C22×#11758202—22no—
9C22×#11758196—21no—
10#16839917—24no—
11#16839917C2218yes—
12C2×#14519700—22no—
13C2×C22⋊#92418C2212yes—
14#16839682—22no—
15#16839682C2214yes—
16C2×#14519684—21no—
17C2×#14519684C2211yescharacteristic
18#16839499C2218yes—
19#16839488C2214yes—
20C61×#544—24no—
21C61×#544—22no—
22C61×#544C2214yes—
23C2×C4⋊Q8×C61—22no—
24C2×#125×C61—22no—
25C2×C22⋊Q8×C61—21no—
26C2×C22⋊Q8×C61C2212yes—
27C2×#117×C61—21no—
28C22×C4⋊C4×C61C2211yescharacteristic
29C22×Dic122—22no—
30C22×Dic122—22no—
31C22×Dic122—21no—
32C22×Dic122—44no—
33C22×Dic122C2312yes—
34C23×Dic61—21no—
35C23×Dic61—21no—
36C2×#11758202—28no—
37C2×#11758202—48no—
38C22×#92418—22no—
39C2×#11758196—24no—
40C2×#11758196—44no—
41#14519700—28no—
42C22⋊#92418—22no—
43C22⋊#92418C2314yes—
44#14519684—21no—
45#14519684—24no—
46#14519684C2312yes—
47C22×Q8×C61—21no—
48C22×Q8×C61D412yes—
49C23×C4×C61D411yescharacteristic
50C23×C4×C61D411yescharacteristic
51C23×C4×C61C2311yescharacteristic
52C4⋊Q8×C61—28no—
53C4⋊Q8×C61—28no—
54#125×C61—28no—
55#125×C61—28no—
56C22⋊Q8×C61—24no—
57C22⋊Q8×C61—22no—
58C22⋊Q8×C61—22no—
59C22⋊Q8×C61—24no—
60C22⋊Q8×C61D414yes—
61C22⋊Q8×C61C2314yes—
62#117×C61—24no—
63#117×C61—24no—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61—22no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—22no—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61—22no—
71C2×C4⋊C4×C61D412yes—
72C2×C4⋊C4×C61D412yes—
73C2×C4⋊C4×C61C2312yes—
74C2×C22⋊C4×C61—22no—
75C2×C42×C61—22no—
76C24×C61C2×D411yescharacteristic
77C2×Q8×C61—24no—
78C2×Q8×C61—22no—
79C2×Q8×C61—44no—
80C2×Q8×C61—82no—
81C2×Q8×C61C2×D414yes—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—22no—
85C22×C4×C61—22no—
86C22×C4×C61—22no—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—42no—
90C22×C4×C61—42no—
91C22×C4×C61Q1614yescomplemented
92C22×C4×C61C2×D412yes—
93C22×C4×C61C2×D412yes—
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2411yescharacteristic
98C22×Dic61—24no—
99C22×Dic61—24no—
100C22×Dic61—22no—
101C22×Dic61—22no—
102C22×Dic61—42no—
103C22×Dic61—42no—
104C22×Dic61—41no—
105C22×Dic61—41no—
106C22×Dic61—42no—
107C2×Dic122—28no—
108C2×Dic122—24no—
109C2×Dic122—44no—
110C2×Dic122—48no—
111C2×Dic122—44no—
112C2×Dic122—44no—
113C2×Dic122—42no—
114C4⋊C4×C61—24no—
115C4⋊C4×C61—24no—
116C4⋊C4×C61—24no—
117C4⋊C4×C61—28no—
118C4⋊C4×C61—24no—
119C4⋊C4×C61—24no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—24no—
122C4⋊C4×C61—24no—
123C4⋊C4×C61C2×D414yes—
124C22⋊C4×C61—24no—
125C22⋊C4×C61—24no—
126C42×C61—28no—
127#11758202—416no—
128#11758196—48no—
129C2×#92418—24no—
130C2×#92418—44no—
131C23×C61—22no—
132C23×C61—22no—
133C23×C61#12011yescharacteristic
134C23×C61#12011yescharacteristic
135C23×C61C2×Q1612yes—
136C23×C61#13712yes—
137C23×C61C22×D411yescharacteristic
138Q8×C61—44no—
139Q8×C61—82no—
140Q8×C61—84no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—24no—
144C2×C4×C61—24no—
145C2×C4×C61—24no—
146C2×C4×C61—24no—
147C2×C4×C61—24no—
148C2×C4×C61—22no—
149C2×C4×C61—42no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—44no—
153C2×C4×C61—42no—
154C2×C4×C61—44no—
155C2×C4×C61—42no—
156C2×C4×C61—81no—
157C2×C4×C61#12012yes—
158C2×C4×C61C2×Q1614yescomplemented
159C2×C4×C61C22×D412yes—
160C2×Dic61—24no—
161C2×Dic61—24no—
162C2×Dic61—42no—
163C2×Dic61—42no—
164C2×Dic61—44no—
165C2×Dic61—48no—
166C2×Dic61—42no—
167C2×Dic61—42no—
168C2×Dic61—44no—
169Dic122—28no—
170Dic122—44no—
171Dic122—48no—
172#92418—44no—
173C2×#1605—611nomaximal, Sylow
174C22×C61—24no—
175C22×C61—24no—
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61#45112yes—
181C22×C61#45112yes—
182C22×C61#45612yes—
183C22×C61B(4,𝔽2)12yes—
184C22×C61C2×#12011yescharacteristic
185C22×C61C22×Q1611yescharacteristic
186C22×C61C2×#13711yescharacteristic
187C4×C61—24no—
188C4×C61—22no—
189C4×C61—22no—
190C4×C61—44no—
191C4×C61—42no—
192C4×C61—44no—
193Dic61—44no—
194Dic61—44no—
195C2×#544D6111yescharacteristic
196C2×#451—612no—
197C2×#395—611no—
198#1605—618no—
199C2×#328—612no—
200C2×#324—611no—
201C2×C61—21no—
202C2×C61—21no—
203C2×C61—22no—
204C2×C61#160514yescomplemented
205C2×C61C2×#45111yescharacteristic
206C2×C61C2×#45611yescharacteristic
207C2×C61C2×B(4,𝔽2)11yescharacteristic
208C23×Q8—611no—
209C22×Q16—1222no—
210#544—24no—
211#544—22no—
212#544C2×D6114yes—
213C2×C4⋊Q8—22no—
214C2×#125—22no—
215C2×C22⋊Q8—21no—
216C2×C22⋊Q8C2×D6112yes—
217C2×#117—21no—
218C22×C4⋊C4—1221no—
219C22×C4⋊C4C2×D6111yescharacteristic
220#451—618no—
221#451—1224no—
222C2×#103—1222no—
223C2×C22⋊C8—612no—
224#395—614no—
225#395—1222no—
226C2×#95—611no—
227C2×#95—1221no—
228#328—618no—
229#324—614no—
230C61C2×#160511yescharacteristic, complemented, Sylow
231C22×Q8—21no—
232C22×Q8—612no—
233C22×Q8—1221no—
234C22×Q8—1222no—
235C22×Q8—1222no—
236C22×Q8—2444no—
237C22×Q8C22⋊D6112yes—
238C23×C4—1221no—
239C23×C4—1221no—
240C23×C4C22⋊D6111yescharacteristic
241C23×C4C22⋊D6111yescharacteristic
242C23×C4C22×D6111yescharacteristic
243C2×Q16—1228no—
244C2×Q16—2448no—
245C22×C8—1222no—
246C4⋊Q8—28no—
247C4⋊Q8—28no—
248#125—28no—
249#125—28no—
250C22⋊Q8—24no—
251C22⋊Q8—24no—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254C22⋊Q8C22⋊D6114yes—
255C22⋊Q8C22×D6114yes—
256#117—24no—
257#117—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—1224no—
266C2×C4⋊C4—2444no—
267C2×C4⋊C4C22⋊D6112yes—
268C2×C4⋊C4C22⋊D6112yes—
269C2×C4⋊C4C22×D6112yes—
270C2×C22⋊C4—22no—
271C2×C42—22no—
272#103—1228no—
273C22⋊C8—614no—
274C22⋊C8—1222no—
275#95—612no—
276#95—1221no—
277#95—1224no—
278C24C2×C22⋊D6111yescharacteristic
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—44no—
282C2×Q8—82no—
283C2×Q8—1228no—
284C2×Q8—1224no—
285C2×Q8—2444no—
286C2×Q8—2444no—
287C2×Q8—2444no—
288C2×Q8—2448no—
289C2×Q8—2442no—
290C2×Q8C2×C22⋊D6114yes—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—1224no—
301C22×C4—1224no—
302C22×C4—1222no—
303C22×C4—1222no—
304C22×C4—2442no—
305C22×C4—2442no—
306C22×C4—2442no—
307C22×C4—2441no—
308C22×C4—2441no—
309C22×C4#1175820214yescomplemented
310C22×C4C2×C22⋊D6112yes—
311C22×C4C2×C22⋊D6112yes—
312C22×C4C2×C22⋊D6112yes—
313C22×C4C2×C22⋊D6112yes—
314C22×C4C2×C22⋊D6111yescharacteristic
315C22×C4C23×D6111yescharacteristic
316Q16—24416no—
317C2×C8—1224no—
318C2×C8—2444no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—2448no—
329C4⋊C4C2×C22⋊D6114yes—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C42—28no—
333C23—22no—
334C23—22no—
335C23#1451974611yescharacteristic
336C23#1451974611yescharacteristic
337C23C2×#1175820212yes—
338C23#1451980812yes—
339C23C22×C22⋊D6111yescharacteristic
340Q8—44no—
341Q8—82no—
342Q8—84no—
343Q8—1228no—
344Q8—2444no—
345Q8—2448no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—1224no—
363C2×C4—1224no—
364C2×C4—2442no—
365C2×C4—2448no—
366C2×C4—2442no—
367C2×C4—2444no—
368C2×C4—2444no—
369C2×C4—2442no—
370C2×C4—2442no—
371C2×C4#1451974612yes—
372C2×C4C2×#1175820214yescomplemented
373C2×C4C22×C22⋊D6112yes—
374C8—2444no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—24no—
380C22—24no—
381C22#1683991812yes—
382C22#1683991812yes—
383C22#1683994512yes—
384C22#1683994912yes—
385C22C2×#1451974611yescharacteristic
386C22C22×#1175820211yescharacteristic
387C22C2×#1451980811yescharacteristic
388C4—22no—
389C4—22no—
390C4—24no—
391C4—44no—
392C4—42no—
393C4—44no—
394C4—2444no—
395C4—2444no—
396C2—21no—
397C2—21no—
398C2—22no—
399C2#2144479314yesdirect product factor
400C2C2×#1683991811yescharacteristic
401C2C2×#1683994511yescharacteristic
402C2C2×#1683994911yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#544C2
socleC23×C61#120
prime radicalC24×C61C2×D4
hypercenterC2×#544D61
hypocenterC61C2×#1605
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #21444793C2direct (N × H)
91. C22×C4×C61Q16semidirect
158. C2×C4×C61C2×Q16semidirect
204. C2×C61#1605semidirect
230. C61C2×#1605semidirect
309. C22×C4#11758202semidirect
372. C2×C4C2×#11758202semidirect
399. C2#21444793direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168399172
C2×#168396822
#214447932
C2×#168394992
C2×#168394882
C2×C61×#5442
C2×#160561