Finite Group Databasebeta

← #29663443

Subgroups — #29663443

Order 15616.

562 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456998C212yesmaximal
2#21454794C211yesmaximal, characteristic
3#21445344C211yesmaximal, characteristic, complemented
4#21445339C211yesmaximal, characteristic, complemented
5#21445310C211yesmaximal, characteristic, complemented
6#21445305C211yesmaximal, characteristic, complemented
7#21445274C212yesmaximal, complemented
8#21445267C212yesmaximal, complemented
9#21444567C211yesmaximal, characteristic, complemented
10#21444529C212yesmaximal, complemented
11C61×#3170C211yesmaximal, characteristic, complemented
12#16840296C2211yescharacteristic
13#16840161—22no—
14#16840119—22no—
15#16840103—22no—
16#16840086—22no—
17#16840054C2212yes—
18#16840049C2212yes—
19#16840044C2212yes—
20#16840039C2212yes—
21#16839892C2211yescharacteristic
22C8×Dic122C2211yescharacteristic
23#16839841C2211yescharacteristic
24#16839833—21no—
25C8×D244C2211yescharacteristic
26C8×C22⋊D61—21no—
27#16839821—21no—
28#16839819C2211yescharacteristic
29#16839815C2212yes—
30#16839716—21no—
31#16839515C2212yes—
32#16839513C2211yescharacteristic
33#16839513C2211yescharacteristic
34#16839511C2212yes—
35#16839511C2212yes—
36#16839508C2212yes—
37#16839508C2212yes—
38C61×#533C2212yes—
39C61×#500C2211yescharacteristic
40C61×#494C2211yescharacteristic
41C61×#487C2212yes—
42C61×#467—21no—
43C61×#467—21no—
44C4×Q16×C61—21no—
45C4×Q16×C61—21no—
46C2×C4×C8×C61C2211yescharacteristic
47C2×#11758189—21no—
48C2×C8×D61—21no—
49#14519771—21no—
50#14519769D411yescharacteristic
51#14519765D411yescharacteristic
52#14519760—21no—
53#14519756D411yescharacteristic
54#14519751D411yescharacteristic
55C4×Dic122C2311yescharacteristic
56C4×D244C2311yescharacteristic
57C4×C22⋊D61—21no—
58#14519734—21no—
59#14519710—22no—
60#14519710—22no—
61#14519708—22no—
62#14519708—22no—
63C4⋊#92418C2311yescharacteristic
64C4⋊#92418C2311yescharacteristic
65C4⋊#92418C2312yes—
66C4⋊#92418C2312yes—
67#14519706—21no—
68#14519706—21no—
69#14519700—22no—
70#14519700—22no—
71#14519697—22no—
72#14519697—22no—
73C22⋊#92418—22no—
74C22⋊#92418—21no—
75#14519689—21no—
76#14519689—21no—
77#14519688—21no—
78#14519687—21no—
79C4×#92418—22no—
80C8×Dic61—21no—
81C2×Q16×C61—21no—
82C2×Q16×C61—21no—
83C22×C8×C61D411yescharacteristic
84C22×C8×C61D411yescharacteristic
85C4⋊Q8×C61C2312yes—
86C4⋊Q8×C61C2312yes—
87#125×C61—22no—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—22no—
90C4×Q8×C61—22no—
91C4×Q8×C61—22no—
92#117×C61—22no—
93C2×C42×C61C2311yescharacteristic
94#107×C61—21no—
95#107×C61—21no—
96#103×C61—22no—
97#103×C61—22no—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C2×C22⋊D61—21no—
108C2×D244C2×D411yescharacteristic
109C2×C4×D61—21no—
110C2×Dic122C2×D411yescharacteristic
111Q16×C61—42no—
112Q16×C61—42no—
113C2×C8×C61—22no—
114C2×C8×C61—41no—
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2×D411yescharacteristic
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—42no—
124C4⋊C4×C61—42no—
125C22⋊C4×C61—42no—
126C42×C61—42no—
127C42×C61C2×D411yescharacteristic
128C42×C61C2×D411yescharacteristic
129C42×C61#4111yescharacteristic
130C42×C61C2411yescharacteristic
131C22⋊Dic61—21no—
132#11758198—21no—
133#11758198—21no—
134#11758198—21no—
135#11758198—21no—
136C4⋊Dic61C2×D411yescharacteristic
137C4⋊Dic61C2×D411yescharacteristic
138#11758196—21no—
139#11758196—21no—
140#11758196—21no—
141#11758196—21no—
142C4×Dic61—21no—
143C2×#92418—21no—
144C2×#92418—22no—
145C2×#92418—22no—
146C2×#92418—21no—
147#11758189—42no—
148C8×D61—42no—
149C23×C61#12111yescharacteristic
150C22×D61—21no—
151Q8×C61—42no—
152Q8×C61—42no—
153Q8×C61—82no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—22no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C22⋊D61—42no—
167C2×Dic61—21no—
168C2×Dic61—21no—
169C2×Dic61—21no—
170D244—22no—
171C4×D61—41no—
172C4×D61—41no—
173Dic122—22no—
174C8×C61—21no—
175C8×C61—21no—
176C8×C61—21no—
177C8×C61#13512yes—
178#92418—41no—
179#92418—41no—
180#92418—42no—
181#92418—42no—
182#19226—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61C2×#12111yescharacteristic
187C2×D61—41no—
188C2×D61—41no—
189C2×D61—41no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—82no—
195C4×C61—82no—
196C4×C61C2×#13511yescharacteristic
197C4×C61#57511yescharacteristic
198C4×C61#57812yes—
199Dic61—41no—
200Dic61—41no—
201Dic61—41no—
202#3233—612no—
203#3170D6111yescharacteristic, complemented
204#2966—611no—
205#1696—611no—
206#1695—611no—
207#1690—611no—
208#1689—611no—
209#1684—612no—
210#1683—612no—
211#1564—611no—
212#1554—612no—
213C2×C61—41no—
214C2×C61#303511yescharacteristic
215C2×C61#305911yescharacteristic
216C2×C61#306711yescharacteristic
217D61—81no—
218#533C2×D6112yes—
219#529—611no—
220#500C2×D6111yescharacteristic
221#499—1222no—
222#494C2×D6111yescharacteristic
223#488—1222no—
224#487C2×D6112yes—
225#485—1222no—
226#482—1222no—
227#477—612no—
228#476—612no—
229#475—612no—
230#474—612no—
231#467—21no—
232#467—21no—
233#446—611no—
234C8×Q8—611no—
235C4×Q16—21no—
236C4×Q16—21no—
237#436—611no—
238#435—1221no—
239C8×D4—611no—
240C8×D4—1221no—
241#433—1221no—
242#432—611no—
243#431—612no—
244#405—1221no—
245C2×C4×C8C2×D6111yescharacteristic
246#335—612no—
247#334—611no—
248#334—611no—
249#333—612no—
250#333—612no—
251#332—612no—
252#332—612no—
253C61#1922611yescharacteristic, complemented, Sylow
254C2×Q16—21no—
255C2×Q16—21no—
256C2×M16—1221no—
257C22×C8—1221no—
258C22×C8C22⋊D6111yescharacteristic
259C22×C8C22⋊D6111yescharacteristic
260C4⋊Q8C22×D6112yes—
261C4⋊Q8C22×D6112yes—
262#126—1221no—
263#125—22no—
264#125—611no—
265#124—611no—
266#123—1221no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269C22⋊Q8—611no—
270#121—611no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—611no—
274C4×D4—611no—
275C4×D4—1221no—
276#117—22no—
277#117—1221no—
278C2×C42C22×D6111yescharacteristic
279#107—21no—
280#107—21no—
281#107—1222no—
282#107—1222no—
283#106—1222no—
284#106—1222no—
285C4⋊C8—611no—
286C4⋊C8—611no—
287C4⋊C8—612no—
288C4⋊C8—612no—
289C4⋊C8—1221no—
290C4⋊C8—1221no—
291#103—22no—
292#103—22no—
293#103—1222no—
294#103—1222no—
295#102—1222no—
296#102—1222no—
297C22⋊C8—1222no—
298C22⋊C8—1221no—
299C22⋊C8—1221no—
300C22⋊C8—1221no—
301#97—1221no—
302#97—1221no—
303C4×C8—1221no—
304C4×C8—1222no—
305C4×C8C22⋊D6111yescharacteristic
306C4×C8C22⋊D6111yescharacteristic
307C4×C8C22×D6111yescharacteristic
308C4×C8C22×D6111yescharacteristic
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—611no—
312C2×D4—611no—
313C2×D4—1221no—
314C22×C4—1221no—
315C22×C4D4×D6111yescharacteristic
316C22×C4#1175822411yescharacteristic
317C22×C4C2×C22⋊D6111yescharacteristic
318Q16—42no—
319Q16—42no—
320M16—2442no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—1222no—
324C2×C8—1222no—
325C2×C8—1221no—
326C2×C8—1221no—
327C2×C8—2442no—
328C2×C8C2×C22⋊D6111yescharacteristic
329C2×C8C2×C22⋊D6111yescharacteristic
330C2×C8C2×C22⋊D6111yescharacteristic
331C2×C8C2×C22⋊D6111yescharacteristic
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—611no—
339C4⋊C4—611no—
340C4⋊C4—1221no—
341C4⋊C4—1221no—
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C22⋊C4—42no—
345C22⋊C4—1221no—
346C22⋊C4—1221no—
347C22⋊C4—1221no—
348C22⋊C4—1221no—
349C22⋊C4—1221no—
350C42—42no—
351C42—1221no—
352C42D4×D6111yescharacteristic
353C42#1175822411yescharacteristic
354C42C2×C22⋊D6111yescharacteristic
355C42C23×D6111yescharacteristic
356C23—1221no—
357C23#1451974811yescharacteristic
358Q8—42no—
359Q8—42no—
360Q8—82no—
361Q8—1222no—
362D4—1222no—
363D4—2442no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—1221no—
371C2×C4—1221no—
372C2×C4—1221no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4#1451974811yescharacteristic
376C2×C4#1451974811yescharacteristic
377C2×C4#1451974811yescharacteristic
378C2×C4C2×D4×D6111yescharacteristic
379C2×C4C22×C22⋊D6111yescharacteristic
380C2×C4C2×#1175822411yescharacteristic
381C8—21no—
382C8—21no—
383C8—21no—
384C8—2441no—
385C8—2441no—
386C8—2442no—
387C8—2442no—
388C8#1451979712yes—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—2441no—
393C22—2441no—
394C22—2441no—
395C22C2×#1451974811yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—82no—
401C4—82no—
402C4—2441no—
403C4—2441no—
404C4—2441no—
405C4C2×#1451979711yescharacteristic
406C4#1684056011yescharacteristic
407C4#1684057512yes—
408C2—41no—
409C2—4881no—
410C2#2145526811yescharacteristic
411C2#2145547011yescharacteristic
412C2#2145553511yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3170C2
socleC22×C61C2×#121
prime radical#14519751D4
hypercenter#3170D61
hypocenterC61#19226
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
3. #21445344C2semidirect
4. #21445339C2semidirect
5. #21445310C2semidirect
6. #21445305C2semidirect
7. #21445274C2semidirect
8. #21445267C2semidirect
9. #21444567C2semidirect
10. #21444529C2semidirect
11. C61×#3170C2semidirect
203. #3170D61semidirect
253. C61#19226semidirect

Maximal subgroups

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

subgroupindex
#214569982
#214547942
#214453442
#214453392
#214453102
#214453052
#214452742
#214452672
#214445672
#214445292
C61×#31702
#1922661