Finite Group Databasebeta

← #21459702

Subgroups — #21459702

Order 7808.

992 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16840490C211yesmaximal, characteristic, complemented
2#16840487C212yesmaximal, complemented
3#16840486C211yesmaximal, characteristic, complemented
4#16840485C211yesmaximal, characteristic, complemented
5#16840484C212yesmaximal, complemented
6#16840453C212yesmaximal, complemented
7#16840450C212yesmaximal, complemented
8#16840444C214yesmaximal, complemented
9#16840371C214yesmaximal, complemented
10#16840337C212yesmaximal
11#16840318C212yesmaximal, complemented
12#16840316C212yesmaximal, complemented
13#16840316C211yesmaximal, characteristic, complemented
14C4⋊Q8×D61C211yesmaximal, characteristic, complemented
15#16840238C212yesmaximal, complemented
16C4×#11758226C211yesmaximal, characteristic
17C61×#557C211yesmaximal, characteristic, complemented
18C2×#11758226C2211yescharacteristic
19C2×Q8×D61C2212yes—
20#14519777C2211yescharacteristic
21#14519776C2212yes—
22#14519776C2212yes—
23#14519776C2211yescharacteristic
24#14519775C2212yes—
25#14519775C2212yes—
26#14519775C2211yescharacteristic
27#14519772C2214yes—
28#14519772C2214yes—
29#14519772C2214yes—
30#14519771C2214yes—
31#14519769C2212yes—
32#14519768C2214yes—
33#14519767C2214yes—
34#14519767C2214yes—
35#14519767C2214yes—
36#14519767C2212yes—
37#14519766C2212yes—
38#14519765C2212yes—
39#14519762C2214yes—
40#14519762C2214yes—
41#14519757C2211yescharacteristic
42#14519757C2212yes—
43#14519755C2214yes—
44#14519755C2214yes—
45#14519755C2212yes—
46#14519755C2214yes—
47#14519753C2211yescharacteristic
48#14519753C2212yes—
49Q8×Dic61C2212yes—
50Q8×Dic61C2211yescharacteristic
51C4×Dic122C2212yes—
52C4×Dic122C2212yes—
53C4×Dic122C2212yes—
54C4×Dic122C2211yescharacteristic
55C4×Dic122C2214yes—
56#14519742C2212yes—
57#14519742C2214yes—
58#14519742C2214yes—
59C4×D244C2212yes—
60C4×D244C2211yescharacteristic
61#14519739C2211yescharacteristic
62#14519739C2212yes—
63#14519735C2212yes—
64#14519735C2214yes—
65#14519735C2212yes—
66#14519735C2211yescharacteristic
67#14519735C2212yes—
68#14519734C2214yes—
69#14519734C2214yes—
70#14519734C2212yes—
71C4⋊C4×D61C2212yes—
72C4⋊C4×D61C2212yes—
73C42×D61C2211yescharacteristic
74C42×D61C2212yes—
75C4⋊Q8×C61C2211yescharacteristic
76C4⋊Q8×C61C2212yes—
77#125×C61C2212yes—
78#125×C61C2214yes—
79C4×Q8×C61C2212yes—
80C4×Q8×C61C2212yes—
81C4×Q8×C61C2211yescharacteristic
82C4×Q8×C61C2211yescharacteristic
83C2×Q8×C61C2312yes—
84C2×Q8×C61C2311yescharacteristic
85C2×Q8×C61C2311yescharacteristic
86#11758226—24no—
87Q8×D61—28no—
88C2×D244C2312yes—
89C2×D244C2311yescharacteristic
90C2×C4×D61C2314yes—
91C2×C4×D61C2312yes—
92C2×C4×D61C2311yescharacteristic
93C2×C4×D61C2311yescharacteristic
94C2×C4×D61C2312yes—
95C2×C4×D61C2312yes—
96C2×C4×D61C2311yescharacteristic
97C2×Dic122C2314yes—
98C2×Dic122C2312yes—
99C2×Dic122C2312yes—
100C2×Dic122C2311yescharacteristic
101C4⋊C4×C61C2311yescharacteristic
102C4⋊C4×C61C2311yescharacteristic
103C4⋊C4×C61C2312yes—
104C4⋊C4×C61C2314yes—
105C4⋊C4×C61C2312yes—
106C4⋊C4×C61C2312yes—
107C4⋊C4×C61C2314yes—
108C4⋊C4×C61C2312yes—
109C4⋊C4×C61C2314yes—
110C42×C61C2312yes—
111C42×C61C2314yes—
112C42×C61C2311yescharacteristic
113C42×C61C2312yes—
114#11758198C2312yes—
115#11758198C2312yes—
116#11758198C2312yes—
117#11758198C2314yes—
118#11758198C2314yes—
119#11758198C2312yes—
120#11758198C2312yes—
121#11758198C2314yes—
122#11758198C2314yes—
123C4⋊Dic61C2312yes—
124C4⋊Dic61C2312yes—
125C4⋊Dic61C2311yescharacteristic
126C4⋊Dic61C2311yescharacteristic
127C4⋊Dic61C2312yes—
128C4⋊Dic61C2314yes—
129C4⋊Dic61C2312yes—
130#11758196C2314yes—
131#11758196C2314yes—
132#11758196C2312yes—
133#11758196C2312yes—
134#11758196C2314yes—
135#11758196C2314yes—
136#11758196C2312yes—
137#11758196C2314yes—
138#11758196C2312yes—
139#11758196C2314yes—
140#11758196C2312yes—
141C4×Dic61C2311yescharacteristic
142C4×Dic61C2312yes—
143C4×Dic61C2311yescharacteristic
144C4×Dic61C2312yes—
145C4×Dic61C2311yescharacteristic
146C4×Dic61C2314yes—
147C4×Dic61C2314yes—
148C4×Dic61C2312yes—
149C4×Dic61C2314yes—
150C22×D61C2412yes—
151C22×D61C2411yescharacteristic
152Q8×C61—24no—
153Q8×C61—22no—
154Q8×C61—22no—
155C2×C4×C61C2412yes—
156C2×C4×C61C2412yes—
157C2×C4×C61C2412yes—
158C2×C4×C61C2411yescharacteristic
159C2×C4×C61C2414yes—
160C2×C4×C61C2412yes—
161C2×C4×C61C2411yescharacteristic
162C2×C4×C61C2411yescharacteristic
163C2×Dic61C2412yes—
164C2×Dic61C2412yes—
165C2×Dic61C2412yes—
166C2×Dic61C2411yescharacteristic
167C2×Dic61C2414yes—
168C2×Dic61C2411yescharacteristic
169C2×Dic61C2411yescharacteristic
170D244—24no—
171D244—22no—
172C4×D61—28no—
173C4×D61—22no—
174C4×D61—24no—
175C4×D61—24no—
176C4×D61—22no—
177C4×D61—24no—
178C4×D61#4114yes—
179Dic122—24no—
180Dic122—22no—
181Dic122—28no—
182Dic122—24no—
183C22×C61C2511yescharacteristic
184C2×D61—21no—
185C2×D61—21no—
186C2×D61—22no—
187C2×D61—22no—
188C2×D61—22no—
189C2×D61C2×#4112yes—
190C4×C61—24no—
191C4×C61—22no—
192C4×C61—22no—
193C4×C61—22no—
194C4×C61—21no—
195C4×C61—22no—
196C4×C61C2×#4112yes—
197C4×C612−1+412yes—
198Dic61—24no—
199Dic61—22no—
200Dic61—22no—
201Dic61—22no—
202Dic61—21no—
203Dic61C2×#4112yes—
204Dic612−1+412yes—
205#3509—611nomaximal, Sylow
206C2×C61C22×#4111yescharacteristic
207C2×C61C2×2−1+411yescharacteristic
208C2×C61C2×2−1+411yescharacteristic
209D61—22no—
210D61—42no—
211#557—612no—
212#557—611no—
213#557D6111yescharacteristic, complemented
214#556—611no—
215#556—612no—
216#556—611no—
217#552—612no—
218#552—614no—
219#552—612no—
220#541—614no—
221#536—612no—
222#533—612no—
223#533—611no—
224#533—612no—
225C2×C4⋊Q8—611no—
226#519—612no—
227C4×#41—611no—
228C61#350911yescharacteristic, complemented, Sylow
229C2×#41—611no—
230C22×Q8—612no—
231C4⋊Q8—612no—
232C4⋊Q8—611no—
233C4⋊Q8—612no—
234C4⋊Q8—612no—
235C4⋊Q8—612no—
236C4⋊Q8—611no—
237C4⋊Q8—611no—
238C4⋊Q8C2×D6111yescharacteristic
239C4⋊Q8C2×D6112yes—
240#126—614no—
241#126—614no—
242#126—614no—
243#126—614no—
244#125—612no—
245#125—614no—
246#125—614no—
247#125—614no—
248#125—614no—
249#125—612no—
250#125C2×D6112yes—
251#125C2×D6114yes—
252#124—612no—
253#124—612no—
254#123—614no—
255#123—614no—
256C22⋊Q8—611no—
257C22⋊Q8—612no—
258C22⋊Q8—614no—
259C22⋊Q8—614no—
260C22⋊Q8—612no—
261C22⋊Q8—611no—
262C22⋊Q8—614no—
263C22⋊Q8—612no—
264C4×Q8—611no—
265C4×Q8—612no—
266C4×Q8—614no—
267C4×Q8—612no—
268C4×Q8—614no—
269C4×Q8—614no—
270C4×Q8—612no—
271C4×Q8—612no—
272C4×Q8—612no—
273C4×Q8—611no—
274C4×Q8C2×D6111yescharacteristic
275C4×Q8C2×D6112yes—
276C4×Q8C2×D6112yes—
277C4×Q8C2×D6111yescharacteristic
278C4×D4—612no—
279C4×D4—611no—
280C4×D4—612no—
281C4×D4—611no—
282#117—611no—
283#117—612no—
284#117—614no—
285#117—612no—
286#117—612no—
287#117—612no—
288#117—614no—
289#117—614no—
290C2×C4⋊C4—612no—
291C2×C4⋊C4—612no—
292C2×C42—611no—
293C2×C42—612no—
294#41—1224no—
295C2×Q8—612no—
296C2×Q8—611no—
297C2×Q8—614no—
298C2×Q8—612no—
299C2×Q8—1228no—
300C2×Q8C22×D6112yes—
301C2×Q8C22×D6111yescharacteristic
302C2×Q8C22×D6111yescharacteristic
303C2×D4—612no—
304C2×D4—611no—
305C22×C4—611no—
306C22×C4—611no—
307C22×C4—611no—
308C22×C4—612no—
309C22×C4—612no—
310C22×C4—612no—
311C22×C4—614no—
312C4⋊C4—614no—
313C4⋊C4—614no—
314C4⋊C4—612no—
315C4⋊C4—612no—
316C4⋊C4—612no—
317C4⋊C4—612no—
318C4⋊C4—612no—
319C4⋊C4—614no—
320C4⋊C4—614no—
321C4⋊C4—614no—
322C4⋊C4—614no—
323C4⋊C4—612no—
324C4⋊C4—612no—
325C4⋊C4—611no—
326C4⋊C4—611no—
327C4⋊C4—612no—
328C4⋊C4—614no—
329C4⋊C4—612no—
330C4⋊C4C22×D6112yes—
331C4⋊C4C22×D6112yes—
332C4⋊C4C22×D6111yescharacteristic
333C4⋊C4C22×D6111yescharacteristic
334C4⋊C4C22×D6112yes—
335C4⋊C4C22×D6114yes—
336C4⋊C4C22×D6114yes—
337C4⋊C4C22×D6114yes—
338C4⋊C4C22×D6112yes—
339C22⋊C4—612no—
340C22⋊C4—612no—
341C22⋊C4—612no—
342C22⋊C4—612no—
343C22⋊C4—612no—
344C22⋊C4—614no—
345C22⋊C4—614no—
346C22⋊C4—614no—
347C22⋊C4—614no—
348C42—611no—
349C42—612no—
350C42—611no—
351C42—612no—
352C42—611no—
353C42—614no—
354C42—614no—
355C42—614no—
356C42—612no—
357C42C22×D6112yes—
358C42C22×D6111yescharacteristic
359C42C22×D6114yes—
360C42C22×D6112yes—
361C23—611no—
362C23—612no—
363Q8—24no—
364Q8—22no—
365Q8—22no—
366Q8—1224no—
367Q8—1228no—
368Q8—1224no—
369Q8—1222no—
370D4—1224no—
371D4—1222no—
372C2×C4—611no—
373C2×C4—611no—
374C2×C4—614no—
375C2×C4—612no—
376C2×C4—612no—
377C2×C4—612no—
378C2×C4—611no—
379C2×C4—614no—
380C2×C4—1222no—
381C2×C4—1224no—
382C2×C4—1224no—
383C2×C4—1228no—
384C2×C4—1222no—
385C2×C4—1224no—
386C2×C4C23×D6112yes—
387C2×C4C23×D6112yes—
388C2×C4C23×D6111yescharacteristic
389C2×C4C23×D6114yes—
390C2×C4C23×D6112yes—
391C2×C4C23×D6112yes—
392C2×C4C23×D6111yescharacteristic
393C2×C4C23×D6111yescharacteristic
394C22—612no—
395C22—1221no—
396C22—1221no—
397C22—1222no—
398C22—1222no—
399C22—1222no—
400C22C24×D6111yescharacteristic
401C4—22no—
402C4—24no—
403C4—22no—
404C4—21no—
405C4—22no—
406C4—22no—
407C4—612no—
408C4—612no—
409C4—1224no—
410C4—1222no—
411C4—1222no—
412C4—1221no—
413C4—1222no—
414C4#41×D6112yes—
415C4#1451982512yes—
416C2—1222no—
417C2—2442no—
418C2C2×#41×D6111yescharacteristic
419C22−1+4×D6111yescharacteristic
420C2C2×#1451982511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D61
commutator, socleC22×C61C25
FittingC61×#557C2
hypercenter#557D61
hypocenterC61#3509
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #16840490C2semidirect
2. #16840487C2semidirect
3. #16840486C2semidirect
4. #16840485C2semidirect
5. #16840484C2semidirect
6. #16840453C2semidirect
7. #16840450C2semidirect
8. #16840444C2semidirect
9. #16840371C2semidirect
11. #16840318C2semidirect
12. #16840316C2semidirect
13. #16840316C2semidirect
14. C4⋊Q8×D61C2semidirect
15. #16840238C2semidirect
17. C61×#557C2semidirect
213. #557D61semidirect
228. C61#3509semidirect