Finite Group Databasebeta

← #29738925

Subgroups — #29738925

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456355C211yesmaximal, characteristic, complemented
2#21455686C212yesmaximal
3#21454415C211yesmaximal, characteristic
4#21447330C212yesmaximal, complemented
5#21447169C212yesmaximal, complemented
6#21446602C211yesmaximal, characteristic, complemented
7#21446567C211yesmaximal, characteristic, complemented
8#21446051C212yesmaximal
9#21445884C211yesmaximal, characteristic
10#21445656C211yesmaximal, characteristic
11C61×#1846C211yesmaximal, characteristic, complemented
12C2×#14519783C2211yescharacteristic
13#16840393C2212yes—
14#16840243C2212yes—
15#16840170—22no—
16#16840166—21no—
17#16840163—21no—
18#16840072—22no—
19#16840061—22no—
20#16840051—22no—
21#16840046—22no—
22C2×#14519713C2211yescharacteristic
23#16839803—22no—
24#16839802C2212yes—
25#16839802C2212yes—
26#16839802C2212yes—
27#16839800C2211yescharacteristic
28#16839799—21no—
29#107×D61—21no—
30#16839788C2212yes—
31C2×#14519705C2211yescharacteristic
32#16839761C2212yes—
33#16839759C2211yescharacteristic
34#16839717C2211yescharacteristic
35#16839531C2212yes—
36#16839531C2212yes—
37#16839527C2211yescharacteristic
38#16839522C2211yescharacteristic
39#16839522C2212yes—
40C2×#107×C61C2211yescharacteristic
41C2×#107×C61C2211yescharacteristic
42C2×#106×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#382C2212yes—
45C2×#11758222C2311yescharacteristic
46#14519783—22no—
47#14519783Q814yescomplemented
48C2×#11758194C2312yes—
49C2×#11758194C2311yescharacteristic
50C2×#11758194C2311yescharacteristic
51C2×#11758189—21no—
52C2×C8×D61—21no—
53#14519776—22no—
54#14519776—22no—
55#14519768—22no—
56#14519768—22no—
57#14519756—42no—
58#14519742—22no—
59#14519739—22no—
60C4×C22⋊D61—42no—
61#14519736C2311yescharacteristic
62#14519736C2312yes—
63#14519735—22no—
64C4⋊C4×D61—22no—
65#14519713—22no—
66#14519711D411yescharacteristic
67#14519711D411yescharacteristic
68#14519710—22no—
69#14519710—22no—
70#14519710—22no—
71#14519709D412yes—
72#14519708—22no—
73#14519708—22no—
74#14519708—22no—
75#14519706—22no—
76#14519706—22no—
77#14519705Q814yescomplemented
78#14519702D411yescharacteristic
79#14519702D411yescharacteristic
80#14519700—22no—
81#14519699D411yescharacteristic
82#14519699D411yescharacteristic
83#14519697—22no—
84#14519687—21no—
85C8×Dic61—21no—
86C22×C8×C61C2311yescharacteristic
87C22×C8×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2312yes—
89C2×C4⋊C4×C61C2312yes—
90C2×C42×C61C2311yescharacteristic
91#107×C61—21no—
92#107×C61—21no—
93#107×C61—21no—
94#107×C61—21no—
95#106×C61—22no—
96#106×C61—22no—
97C4×C8×C61—22no—
98#95×C61—42no—
99C22×C4×C61—21no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61C2411yescharacteristic
103C2×C22⋊D61—21no—
104#11758222—21no—
105#11758222—21no—
106#11758222C2×Q812yes—
107#11758222C2×Q812yes—
108C2×D244C2×D411yescharacteristic
109C2×C4×D61—21no—
110C2×C4×D61—42no—
111C2×Dic122—42no—
112C2×Dic122C2×D411yescharacteristic
113C2×C8×C61—21no—
114C2×C8×C61—21no—
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2×Q812yes—
120C2×C8×C61#4112yes—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C42×C61—21no—
126C42×C61C2×Q812yes—
127C22⋊Dic61—22no—
128C22⋊Dic61—21no—
129#11758198—42no—
130C4⋊Dic61—42no—
131C4⋊Dic61C2×D411yescharacteristic
132C4⋊Dic61C2×D411yescharacteristic
133C4⋊Dic61C2×D412yes—
134C4⋊Dic61C2×D412yes—
135#11758196—42no—
136#11758196—42no—
137#11758196—42no—
138C4×Dic61—21no—
139C4×Dic61—22no—
140C4×Dic61—42no—
141#11758194—21no—
142#11758194—21no—
143#11758194—22no—
144#11758194—22no—
145#11758194C2×Q812yes—
146#11758194C2×Q812yes—
147C2×#92418—21no—
148C2×#92418—21no—
149C2×#92418—22no—
150#11758189—24no—
151C8×D61—24no—
152C23×C61H(5,𝔽2)11yescharacteristic
153C22×D61—21no—
154C2×C4×C61—22no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—82no—
160C2×C4×C61—82no—
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C22×Q811yescharacteristic
164C2×C4×C61C22×Q811yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C2×C4×C612−1+411yescharacteristic
167C22⋊D61—22no—
168C22⋊D61—22no—
169C2×Dic61—21no—
170C2×Dic61—22no—
171C2×Dic61—21no—
172C2×Dic61—22no—
173C2×Dic61—21no—
174D244—21no—
175D244C4⋊Q812yes—
176C4×D61—22no—
177C4×D61—22no—
178C4×D61—162no—
179Dic122—21no—
180Dic122—162no—
181Dic122C4⋊Q812yes—
182C8×C61—21no—
183C8×C61—21no—
184C8×C61C22⋊Q812yes—
185C8×C61C22⋊Q812yes—
186#92418—22no—
187#92418—22no—
188#92418—42no—
189#26082—611nomaximal, Sylow
190C22×C61—21no—
191C22×C61—21no—
192C22×C61#54411yescharacteristic
193C22×C61D4211yescharacteristic
194C22×C61#55411yescharacteristic
195C2×D61—22no—
196C2×D61—41no—
197C2×D61—41no—
198C4×C61—42no—
199C4×C61—82no—
200C4×C61—82no—
201C4×C61C2×C22⋊Q811yescharacteristic
202C4×C61C2×C4⋊Q811yescharacteristic
203C4×C61#54711yescharacteristic
204C4×C61D4×Q811yescharacteristic
205Dic61—22no—
206Dic61—41no—
207Dic61—42no—
208Dic61—42no—
209Dic61—41no—
210#3164—611no—
211#3084—612no—
212#2913—611no—
213#2090—612no—
214#2061—612no—
215#1952—611no—
216#1945—611no—
217#1860—612no—
218#1846D6111yescharacteristic, complemented
219#1816—611no—
220#1764—611no—
221C2×C61—21no—
222C2×C61—21no—
223C2×C61#244011yescharacteristic
224C2×C61#329011yescharacteristic
225C2×C61#329011yescharacteristic
226D61—42no—
227C2×#131—611no—
228#544—612no—
229#520—612no—
230#501—1222no—
231#500—1221no—
232#499—1221no—
233#479—1222no—
234#478—1222no—
235#476—1222no—
236#475—1222no—
237C2×#108—611no—
238#428—612no—
239#428—612no—
240#428—612no—
241#428—1222no—
242#427—611no—
243#427—1221no—
244C2×#107—1221no—
245C2×#107C2×D6111yescharacteristic
246C2×#107C2×D6111yescharacteristic
247C2×#106C2×D6112yes—
248#424—612no—
249C2×#104—611no—
250#417—612no—
251#416—611no—
252#405—611no—
253C2×C4×C8C2×D6111yescharacteristic
254#382C2×D6112yes—
255#343—612no—
256#343—612no—
257#341—611no—
258#339—611no—
259#339—612no—
260C61#2608211yescharacteristic, complemented, Sylow
261C2×#41—611no—
262#131—614no—
263#131—1222no—
264C2×M16—611no—
265C2×M16—611no—
266C2×M16—612no—
267C2×M16—1221no—
268C22×C8—1221no—
269C22×C8C22×D6111yescharacteristic
270C22×C8C22×D6111yescharacteristic
271C4⋊Q8—1222no—
272C4⋊Q8—1222no—
273#125—1222no—
274#125—1222no—
275C22⋊Q8—2442no—
276C4×Q8—1222no—
277C4×D4—1222no—
278C4×D4—2442no—
279#117—612no—
280#117—611no—
281#117—1222no—
282C2×C4⋊C4—1222no—
283C2×C4⋊C4C22×D6112yes—
284C2×C4⋊C4C22×D6112yes—
285C2×C42C22×D6111yescharacteristic
286#108—1222no—
287#107—21no—
288#107—21no—
289#107—21no—
290#107—21no—
291#107—611no—
292#107—611no—
293#107—1222no—
294#107—1222no—
295#107—1222no—
296#106—22no—
297#106—22no—
298#106—612no—
299#106—1222no—
300#106—1222no—
301#106—1222no—
302C4⋊C8—1222no—
303C4⋊C8—1222no—
304#104—614no—
305#103—611no—
306#103—611no—
307#103—1222no—
308#102—611no—
309#102—611no—
310#102—1222no—
311#97—1221no—
312C4×C8—22no—
313C4×C8—1221no—
314#95—42no—
315#41—612no—
316#41—612no—
317#41—1221no—
318#41—1221no—
319C2×Q8—611no—
320C2×Q8—2442no—
321C2×D4—611no—
322C2×D4—1221no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—1221no—
327C22×C4—2442no—
328C22×C4C23×D6111yescharacteristic
329M16—612no—
330M16—612no—
331M16—1221no—
332M16—1221no—
333M16—1224no—
334M16—1222no—
335M16—1222no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—1221no—
339C2×C8—1221no—
340C2×C8—1224no—
341C2×C8—1222no—
342C2×C8#1175822212yes—
343C2×C8D4×D6111yescharacteristic
344C2×C8D4×D6111yescharacteristic
345C2×C8D4×D6111yescharacteristic
346C2×C8D4×D6111yescharacteristic
347C2×C8Q8×D6112yes—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—611no—
353C4⋊C4—611no—
354C4⋊C4—612no—
355C4⋊C4—612no—
356C4⋊C4—2442no—
357C4⋊C4—2442no—
358C4⋊C4—2442no—
359C4⋊C4—2442no—
360C22⋊C4—1222no—
361C22⋊C4—1221no—
362C22⋊C4—2442no—
363C42—21no—
364C42—1221no—
365C42—1222no—
366C42—2442no—
367C42Q8×D6112yes—
368C23—1221no—
369C23#1451982211yescharacteristic
370Q8—612no—
371Q8—1221no—
372Q8—9762no—
373D4—612no—
374D4—1222no—
375D4—1222no—
376D4—1221no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—1221no—
385C2×C4—1222no—
386C2×C4—1222no—
387C2×C4—1222no—
388C2×C4—1221no—
389C2×C4—1222no—
390C2×C4—1221no—
391C2×C4—9762no—
392C2×C4C2×D4×D6111yescharacteristic
393C2×C4C2×D4×D6111yescharacteristic
394C2×C4C2×Q8×D6111yescharacteristic
395C2×C4C2×Q8×D6111yescharacteristic
396C2×C4C2×#1175822211yescharacteristic
397C2×C4#1451982511yescharacteristic
398C8—21no—
399C8—21no—
400C8—1222no—
401C8—1222no—
402C8—2442no—
403C8#1451975512yes—
404C8#1451975512yes—
405C22—21no—
406C22—21no—
407C22—1222no—
408C22—2441no—
409C22—2441no—
410C22#1684038911yescharacteristic
411C22#1684039711yescharacteristic
412C22#1684046711yescharacteristic
413C4—42no—
414C4—82no—
415C4—82no—
416C4—1222no—
417C4—2441no—
418C4—2442no—
419C4—2442no—
420C4—2441no—
421C4C2×#1451975511yescharacteristic
422C4C4⋊Q8×D6111yescharacteristic
423C4#1684041311yescharacteristic
424C4#1684042711yescharacteristic
425C2—21no—
426C2—21no—
427C2—2442no—
428C2#2144952411yescharacteristic
429C2#2145760411yescharacteristic
430C2#2145760411yescharacteristic
subgroupG/H
centerC22#16840397
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1846C2
socleC22×C61D42
prime radicalC2×#11758222C23
hypercenter#1846D61
hypocenterC61#26082
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1846

Complements

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

NH ≅ G/Ntype
1. #21456355C2semidirect
4. #21447330C2semidirect
5. #21447169C2semidirect
6. #21446602C2semidirect
7. #21446567C2semidirect
11. C61×#1846C2semidirect
47. #14519783Q8semidirect
77. #14519705Q8semidirect
218. #1846D61semidirect
260. C61#26082semidirect

Maximal subgroups

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

subgroupindex
#214563552
#214556862
#214544152
#214473302
#214471692
#214466022
#214465672
#214460512
#214458842
#214456562
C61×#18462
#2608261