Finite Group Databasebeta

← #21460228

Subgroups — #21460228

Order 7808.

1002 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16840515C212yesmaximal, complemented
2#16840511C212yesmaximal, complemented
3#16840496C212yesmaximal, complemented
4#16840492C212yesmaximal, complemented
5#16840483C212yesmaximal, complemented
6#16840480C212yesmaximal, complemented
7Q8×C22⋊D61C212yesmaximal, complemented
8#16840425C212yesmaximal, complemented
9#16840380C214yesmaximal, complemented
10#16840376C214yesmaximal, complemented
11#16840335C212yesmaximal, complemented
12#16840333C211yesmaximal, characteristic, complemented
13#16840314C212yesmaximal, complemented
14#16840308C211yesmaximal, characteristic
15C61×#536C211yesmaximal, characteristic, complemented
16C2×#11758222C2211yescharacteristic
17C22×Dic122C2211yescharacteristic
18C2×Q8×D61C2212yes—
19#14519776C2212yes—
20#14519776C2214yes—
21#14519776C2214yes—
22#14519775C2211yescharacteristic
23#14519775C2212yes—
24#14519775C2214yes—
25#14519775C2212yes—
26#14519773C2211yescharacteristic
27#14519772C2214yes—
28#14519771C2214yes—
29#14519770C2214yes—
30#14519770C2214yes—
31#14519767C2214yes—
32#14519766C2211yescharacteristic
33#14519764C2211yescharacteristic
34#14519764C2212yes—
35#14519763C2214yes—
36#14519763C2214yes—
37#14519763C2214yes—
38#14519762C2214yes—
39#14519759C2214yes—
40#14519758C2212yes—
41#14519758C2212yes—
42#14519757C2212yes—
43#14519756C2212yes—
44#14519756C2214yes—
45#14519756C2212yes—
46#14519755C2214yes—
47#14519754C2214yes—
48#14519754C2214yes—
49#14519753C2212yes—
50#14519753C2214yes—
51#14519749C2212yes—
52#14519749C2212yes—
53Q8×Dic61C2212yes—
54C4×Dic122C2214yes—
55#14519742C2212yes—
56#14519742C2214yes—
57#14519742C2214yes—
58C4×C22⋊D61C2212yes—
59C4×C22⋊D61C2212yes—
60C4×C22⋊D61C2214yes—
61#14519737C2214yes—
62#14519736C2211yescharacteristic
63#14519735C2212yes—
64#14519733C2214yes—
65C2×C4×Dic61C2211yescharacteristic
66C2×#41×C61C2211yescharacteristic
67C22×Q8×C61C2211yescharacteristic
68C4⋊Q8×C61C2212yes—
69#124×C61C2212yes—
70#123×C61C2212yes—
71#123×C61C2212yes—
72C22⋊Q8×C61C2212yes—
73C22⋊Q8×C61C2212yes—
74#117×C61C2211yescharacteristic
75#41×C61—42no—
76C2×Q8×C61—42no—
77C2×Q8×C61C2312yes—
78C2×Q8×C61C2312yes—
79C2×Q8×C61C2311yescharacteristic
80C2×D4×C61C2311yescharacteristic
81C2×D4×C61C2311yescharacteristic
82C2×D4×C61C2311yescharacteristic
83C22×C4×C61C2311yescharacteristic
84C22×C4×C61C2311yescharacteristic
85C22×C4×C61C2311yescharacteristic
86C22×C4×C61C2312yes—
87C2×C22⋊D61C2312yes—
88C2×C22⋊D61C2312yes—
89C2×C22⋊D61C2312yes—
90C22×Dic61C2312yes—
91C22×Dic61C2312yes—
92Q8×D61—44no—
93#11758222—42no—
94C2×D244C2311yescharacteristic
95C2×C4×D61C2312yes—
96C2×C4×D61C2314yes—
97C2×Dic122—42no—
98C2×Dic122C2312yes—
99C2×Dic122C2314yes—
100C2×Dic122C2314yes—
101C2×Dic122C2312yes—
102C2×Dic122C2311yescharacteristic
103C4⋊C4×C61C2312yes—
104C4⋊C4×C61C2314yes—
105C4⋊C4×C61C2314yes—
106C22⋊C4×C61C2312yes—
107C22⋊C4×C61C2314yes—
108C22⋊C4×C61C2314yes—
109C42×C61C2312yes—
110C22⋊Dic61C2314yes—
111C22⋊Dic61C2312yes—
112C22⋊Dic61C2312yes—
113C22⋊Dic61C2312yes—
114C22⋊Dic61C2314yes—
115#11758198C2314yes—
116#11758198C2314yes—
117#11758198C2312yes—
118#11758198C2312yes—
119#11758198C2314yes—
120C4⋊Dic61C2312yes—
121C4⋊Dic61C2314yes—
122C4⋊Dic61C2314yes—
123#11758196C2314yes—
124#11758196C2314yes—
125#11758196C2314yes—
126#11758196C2312yes—
127#11758196C2312yes—
128#11758196C2314yes—
129#11758196C2314yes—
130#11758196C2314yes—
131#11758196C2314yes—
132C4×Dic61C2312yes—
133C4×Dic61C2311yescharacteristic
134C4×Dic61C2311yescharacteristic
135C4×Dic61C2314yes—
136C4×Dic61C2314yes—
137C4×Dic61C2314yes—
138C4×Dic61C2312yes—
139C23×C61C2411yescharacteristic
140C23×C61C2411yescharacteristic
141C23×C61C2411yescharacteristic
142C22×D61C2412yes—
143Q8×C61—24no—
144Q8×C61—42no—
145Q8×C61—41no—
146D4×C61—41no—
147D4×C61—41no—
148D4×C61—41no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—42no—
153C2×C4×C61C2411yescharacteristic
154C2×C4×C61C2414yes—
155C2×C4×C61C2411yescharacteristic
156C2×C4×C61C2414yes—
157C2×C4×C61C2411yescharacteristic
158C2×C4×C61C2411yescharacteristic
159C22⋊D61—24no—
160C22⋊D61—24no—
161C22⋊D61—42no—
162C2×Dic61—24no—
163C2×Dic61—42no—
164C2×Dic61C2414yes—
165C2×Dic61C2412yes—
166C2×Dic61C2412yes—
167C2×Dic61C2414yes—
168C2×Dic61C2412yes—
169D244—41no—
170C4×D61—24no—
171C4×D61—44no—
172Dic122—24no—
173Dic122—28no—
174Dic122—42no—
175Dic122—41no—
176Dic122—44no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—22no—
181C22×C61—22no—
182C22×C61—22no—
183C22×C61C2511yescharacteristic
184C2×D61—22no—
185C2×D61—22no—
186C2×D61—22no—
187C4×C61—21no—
188C4×C61—24no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—24no—
192C4×C61H(5,𝔽2)12yes—
193Dic61—24no—
194Dic61—22no—
195Dic61—24no—
196Dic61—22no—
197Dic612−1+414yes—
198#3550—611nomaximal, Sylow
199C2×C61—41no—
200C2×C61—41no—
201C2×C61—41no—
202C2×C61C2×H(5,𝔽2)11yescharacteristic
203C2×C61C2×2−1+412yes—
204D61—42no—
205#563—612no—
206#563—612no—
207#559—612no—
208Q82—612no—
209#556—612no—
210#556—612no—
211D4×Q8—612no—
212D4×Q8—612no—
213#542—614no—
214#542—614no—
215#536—611no—
216#536—612no—
217#536D6111yescharacteristic, complemented
218#533—612no—
219#532—611no—
220C61#355011yescharacteristic, complemented, Sylow
221C2×#41—611no—
222C2×#41C2×D6111yescharacteristic
223C22×Q8—612no—
224C22×Q8—611no—
225C22×Q8C2×D6111yescharacteristic
226C4⋊Q8—611no—
227C4⋊Q8—612no—
228C4⋊Q8—612no—
229C4⋊Q8—614no—
230C4⋊Q8—612no—
231C4⋊Q8—614no—
232C4⋊Q8—614no—
233C4⋊Q8C2×D6112yes—
234#127—611no—
235#126—614no—
236#126—614no—
237#126—614no—
238#126—614no—
239#125—614no—
240#124—611no—
241#124—611no—
242#124—612no—
243#124—614no—
244#124—614no—
245#124—614no—
246#124C2×D6112yes—
247#123—612no—
248#123—612no—
249#123—614no—
250#123—614no—
251#123C2×D6112yes—
252#123C2×D6112yes—
253C22⋊Q8—612no—
254C22⋊Q8—614no—
255C22⋊Q8—612no—
256C22⋊Q8—612no—
257C22⋊Q8—614no—
258C22⋊Q8—612no—
259C22⋊Q8—614no—
260C22⋊Q8—614no—
261C22⋊Q8—614no—
262C22⋊Q8C2×D6112yes—
263C22⋊Q8C2×D6112yes—
264#121—612no—
265#121—612no—
266C4×Q8—612no—
267C4×Q8—612no—
268C4×Q8—614no—
269C4×Q8—614no—
270C4×Q8—614no—
271C4×D4—612no—
272C4×D4—612no—
273C4×D4—614no—
274C4×D4—614no—
275#117—611no—
276#117—612no—
277#117—614no—
278#117C2×D6111yescharacteristic
279C2×C42—611no—
280#41—42no—
281#41—2442no—
282C2×Q8—42no—
283C2×Q8—614no—
284C2×Q8—612no—
285C2×Q8—614no—
286C2×Q8—612no—
287C2×Q8—611no—
288C2×Q8—2442no—
289C2×Q8—2444no—
290C2×Q8C22×D6112yes—
291C2×Q8C22×D6112yes—
292C2×Q8C22×D6111yescharacteristic
293C2×D4—612no—
294C2×D4—612no—
295C2×D4—612no—
296C2×D4—611no—
297C2×D4C22×D6111yescharacteristic
298C2×D4C22×D6111yescharacteristic
299C2×D4C22×D6111yescharacteristic
300C22×C4—612no—
301C22×C4—612no—
302C22×C4—612no—
303C22×C4—614no—
304C22×C4C22×D6111yescharacteristic
305C22×C4C22×D6111yescharacteristic
306C22×C4C22×D6111yescharacteristic
307C22×C4C22×D6112yes—
308C4⋊C4—614no—
309C4⋊C4—614no—
310C4⋊C4—614no—
311C4⋊C4—612no—
312C4⋊C4—612no—
313C4⋊C4—614no—
314C4⋊C4—614no—
315C4⋊C4—614no—
316C4⋊C4—614no—
317C4⋊C4—612no—
318C4⋊C4—614no—
319C4⋊C4—614no—
320C4⋊C4C22×D6112yes—
321C4⋊C4C22×D6114yes—
322C4⋊C4C22×D6114yes—
323C22⋊C4—614no—
324C22⋊C4—612no—
325C22⋊C4—612no—
326C22⋊C4—612no—
327C22⋊C4—614no—
328C22⋊C4—614no—
329C22⋊C4—612no—
330C22⋊C4—612no—
331C22⋊C4—614no—
332C22⋊C4—614no—
333C22⋊C4C22×D6114yes—
334C22⋊C4C22×D6112yes—
335C22⋊C4C22×D6114yes—
336C42—612no—
337C42—611no—
338C42—611no—
339C42—614no—
340C42—614no—
341C42—612no—
342C42—614no—
343C42C22×D6112yes—
344C23—612no—
345C23C23×D6111yescharacteristic
346C23C23×D6111yescharacteristic
347C23C23×D6111yescharacteristic
348Q8—24no—
349Q8—42no—
350Q8—41no—
351Q8—1228no—
352Q8—1224no—
353Q8—2442no—
354Q8—2444no—
355Q8—2441no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—1224no—
360D4—1224no—
361D4—2441no—
362D4—2442no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—42no—
367C2×C4—612no—
368C2×C4—614no—
369C2×C4—612no—
370C2×C4—612no—
371C2×C4—614no—
372C2×C4—1224no—
373C2×C4—1224no—
374C2×C4—2444no—
375C2×C4—2442no—
376C2×C4C23×D6114yes—
377C2×C4C23×D6111yescharacteristic
378C2×C4C23×D6111yescharacteristic
379C2×C4C23×D6114yes—
380C2×C4C23×D6111yescharacteristic
381C2×C4C23×D6111yescharacteristic
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—1222no—
389C22—1222no—
390C22—1222no—
391C22C24×D6111yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—24no—
396C4—24no—
397C4—614no—
398C4—1222no—
399C4—1224no—
400C4—1222no—
401C4—1224no—
402C4#1451982212yes—
403C2—41no—
404C2—41no—
405C2—41no—
406C2—2442no—
407C2C2×#1451982211yescharacteristic
408C22−1+4×D6112yes—
subgroupG/H
center, FrattiniC22C24×D61
commutator, socleC22×C61C25
FittingC61×#536C2
hypercenter#536D61
hypocenterC61#3550
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #16840515C2semidirect
2. #16840511C2semidirect
3. #16840496C2semidirect
4. #16840492C2semidirect
5. #16840483C2semidirect
6. #16840480C2semidirect
7. Q8×C22⋊D61C2semidirect
8. #16840425C2semidirect
9. #16840380C2semidirect
10. #16840376C2semidirect
11. #16840335C2semidirect
12. #16840333C2semidirect
13. #16840314C2semidirect
15. C61×#536C2semidirect
217. #536D61semidirect
220. C61#3550semidirect