Finite Group Databasebeta

← #29738918

Subgroups — #29738918

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#21455677C212yesmaximal
2#21454416C211yesmaximal, characteristic
3#21447332C212yesmaximal, complemented
4#21447166C212yesmaximal, complemented
5#21446604C211yesmaximal, characteristic, complemented
6#21446568C211yesmaximal, characteristic, complemented
7#21446049C212yesmaximal
8#21445982C211yesmaximal, characteristic, complemented
9#21445884C211yesmaximal, characteristic
10#21445658C211yesmaximal, characteristic
11C61×#3164C211yesmaximal, characteristic, complemented
12C2×#14519783C2211yescharacteristic
13#16840242C2212yes—
14#16840066—22no—
15#16840059—22no—
16#16840049—22no—
17#16840044—22no—
18C2×#14519713C2211yescharacteristic
19#16839805C2212yes—
20#16839805C2212yes—
21#16839801—22no—
22#16839799—21no—
23C2×#14519711C2211yescharacteristic
24#107×D61—21no—
25C2×#14519708C2212yes—
26#16839789C2212yes—
27C2×#14519703C2211yescharacteristic
28#16839761C2212yes—
29#16839756C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839627C2212yes—
32#16839531C2212yes—
33#16839530C2212yes—
34#16839526C2211yescharacteristic
35#16839521C2211yescharacteristic
36#16839521C2212yes—
37C61×#544C2212yes—
38C61×#501—22no—
39C61×#500—21no—
40C61×#499—21no—
41C61×#428C2212yes—
42C61×#427C2211yescharacteristic
43C2×#107×C61C2211yescharacteristic
44C61×#405C2211yescharacteristic
45C2×#11758222C2311yescharacteristic
46#14519783—22no—
47#14519783Q814yescomplemented
48C2×#11758194C2312yes—
49C2×#11758194C2311yescharacteristic
50C2×#11758189—21no—
51C22×#92418C2311yescharacteristic
52C2×C8×D61—21no—
53C4×Dic122—22no—
54C4×D244—22no—
55#14519737—42no—
56#14519735—22no—
57C2×C4⋊Dic61C2312yes—
58C4⋊C4×D61—22no—
59C2×C4×Dic61C2311yescharacteristic
60#14519713—22no—
61#14519711—21no—
62#14519711—21no—
63#14519710—22no—
64#14519710—22no—
65#14519708—22no—
66#14519708—22no—
67#14519708—22no—
68#14519708—22no—
69C4⋊#92418—22no—
70C4⋊#92418—22no—
71#14519703Q814yescomplemented
72#14519700—22no—
73#14519700D411yescharacteristic
74#14519700D411yescharacteristic
75#14519697—22no—
76#14519697D411yescharacteristic
77#14519697D411yescharacteristic
78C8×Dic61—22no—
79#14519684—42no—
80C2×M16×C61C2311yescharacteristic
81C22×C8×C61C2311yescharacteristic
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61—22no—
84#125×C61—22no—
85#125×C61—22no—
86C22⋊Q8×C61—42no—
87#117×C61C2311yescharacteristic
88#117×C61C2312yes—
89C2×C4⋊C4×C61C2312yes—
90#107×C61—21no—
91#107×C61—21no—
92#107×C61—22no—
93#107×C61D411yescharacteristic
94#107×C61D411yescharacteristic
95#106×C61—22no—
96#106×C61D412yes—
97#97×C61—21no—
98C4×C8×C61—21no—
99C2×Q8×C61—42no—
100C22×C4×C61—22no—
101C22×C4×C61C2411yescharacteristic
102C2×C22⋊D61—21no—
103C22×Dic61—21no—
104C22×Dic61—22no—
105#11758222—21no—
106#11758222—21no—
107#11758222C2×Q812yes—
108#11758222C2×Q812yes—
109C2×D244C2×D411yescharacteristic
110C2×C4×D61—21no—
111C2×C4×D61—42no—
112C2×Dic122C2×D411yescharacteristic
113M16×C61—21no—
114M16×C61C2×Q812yes—
115C2×C8×C61—21no—
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2×Q812yes—
119C2×C8×C61#4112yes—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—42no—
123C4⋊C4×C61—42no—
124C4⋊C4×C61—42no—
125C4⋊C4×C61C2×D411yescharacteristic
126C4⋊C4×C61C2×D411yescharacteristic
127C4⋊C4×C61C2×D412yes—
128C4⋊C4×C61C2×D412yes—
129C22⋊C4×C61—21no—
130C22⋊C4×C61—22no—
131C42×C61—21no—
132C42×C61—22no—
133#11758198—42no—
134C4⋊Dic61—22no—
135C4⋊Dic61—22no—
136#11758196—42no—
137C4×Dic61—21no—
138C4×Dic61—42no—
139C4×Dic61C2×Q812yes—
140#11758194—21no—
141#11758194—22no—
142#11758194—22no—
143#11758194C2×Q812yes—
144C2×#92418—21no—
145C2×#92418—21no—
146C2×#92418—21no—
147C2×#92418—22no—
148C2×#92418C2×D411yescharacteristic
149C2×#92418C2×D411yescharacteristic
150#11758189—24no—
151C8×D61—24no—
152C23×C61H(5,𝔽2)11yescharacteristic
153C22×D61—21no—
154Q8×C61—162no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—42no—
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—41no—
172C2×Dic61—41no—
173C2×Dic61—42no—
174C2×Dic61—82no—
175D244—21no—
176D244C4⋊Q812yes—
177C4×D61—22no—
178C4×D61—22no—
179C4×D61—162no—
180Dic122—21no—
181Dic122C4⋊Q812yes—
182C8×C61—22no—
183C8×C61C22⋊Q812yes—
184C8×C61C22⋊Q812yes—
185#92418—22no—
186#92418—21no—
187#92418—21no—
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—41no—
200C4×C61—42no—
201C4×C61—41no—
202C4×C61—82no—
203C4×C61C2×C22⋊Q811yescharacteristic
204C4×C61C2×C4⋊Q811yescharacteristic
205C4×C61#54711yescharacteristic
206C4×C61D4×Q811yescharacteristic
207Dic61—22no—
208Dic61—42no—
209Dic61—82no—
210#3164D6111yescharacteristic, complemented
211#3084—612no—
212#2913—611no—
213#2090—612no—
214#2061—612no—
215#1952—611no—
216#1945—611no—
217#1860—612no—
218#1846—611no—
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#544C2×D6112yes—
229#520—612no—
230#501—22no—
231#500—21no—
232#499—21no—
233#479—1222no—
234#478—1222no—
235#476—1222no—
236#475—1222no—
237C2×#108—611no—
238#428—612no—
239#428—612no—
240#428—1222no—
241#428C2×D6112yes—
242#427—1221no—
243#427C2×D6111yescharacteristic
244C2×#107—611no—
245C2×#107—1221no—
246C2×#107C2×D6111yescharacteristic
247C2×#106—612no—
248#424—612no—
249C2×#104—611no—
250#417—612no—
251#416—611no—
252#405C2×D6111yescharacteristic
253C2×C4×C8—611no—
254#382—612no—
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—612no—
266C2×M16—1221no—
267C2×M16C22×D6111yescharacteristic
268C22×C8—611no—
269C22×C8—1221no—
270C22×C8C22×D6111yescharacteristic
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273#125—22no—
274#125—22no—
275C22⋊Q8—42no—
276C4×Q8—1222no—
277C4×D4—1222no—
278C4×D4—2442no—
279#117—1222no—
280#117C22×D6111yescharacteristic
281#117C22×D6112yes—
282C2×C4⋊C4—612no—
283C2×C4⋊C4—1222no—
284C2×C4⋊C4C22×D6112yes—
285C2×C42—611no—
286#108—1222no—
287#107—21no—
288#107—21no—
289#107—22no—
290#107—1221no—
291#107—1221no—
292#107—1222no—
293#107—1222no—
294#107C22⋊D6111yescharacteristic
295#107C22⋊D6111yescharacteristic
296#106—22no—
297#106—1222no—
298#106—1222no—
299#106—1222no—
300#106—1222no—
301#106C22⋊D6112yes—
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—21no—
312C4×C8—21no—
313C4×C8—1222no—
314#95—2442no—
315#41—612no—
316#41—612no—
317#41—1221no—
318#41—1221no—
319C2×Q8—42no—
320C2×Q8—611no—
321C2×D4—611no—
322C2×D4—1221no—
323C22×C4—22no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—1222no—
327C22×C4—2442no—
328C22×C4C23×D6111yescharacteristic
329M16—21no—
330M16—612no—
331M16—1221no—
332M16—1224no—
333M16—1222no—
334M16—1222no—
335M16Q8×D6112yes—
336C2×C8—21no—
337C2×C8—611no—
338C2×C8—611no—
339C2×C8—1221no—
340C2×C8—1221no—
341C2×C8—1221no—
342C2×C8—1224no—
343C2×C8—1222no—
344C2×C8Q8×D6112yes—
345C2×C8#1175822612yes—
346C2×C8C2×C22⋊D6111yescharacteristic
347C2×C8C2×C22⋊D6111yescharacteristic
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—1222no—
354C4⋊C4—1222no—
355C4⋊C4—2442no—
356C4⋊C4D4×D6111yescharacteristic
357C4⋊C4D4×D6111yescharacteristic
358C4⋊C4C2×C22⋊D6112yes—
359C4⋊C4C2×C22⋊D6112yes—
360C22⋊C4—22no—
361C22⋊C4—21no—
362C22⋊C4—2442no—
363C42—22no—
364C42—21no—
365C42—612no—
366C42—1221no—
367C42—2442no—
368C23—1221no—
369C23#1451982211yescharacteristic
370Q8—162no—
371Q8—612no—
372Q8—1221no—
373D4—612no—
374D4—1222no—
375D4—1222no—
376D4—1221no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—1221no—
384C2×C4—1222no—
385C2×C4—1222no—
386C2×C4—1222no—
387C2×C4—9762no—
388C2×C4—4882no—
389C2×C4—2442no—
390C2×C4—2441no—
391C2×C4—2441no—
392C2×C4C2×D4×D6111yescharacteristic
393C2×C4C22×C22⋊D6111yescharacteristic
394C2×C4C2×Q8×D6111yescharacteristic
395C2×C4C2×Q8×D6111yescharacteristic
396C2×C4C2×#1175822611yescharacteristic
397C2×C4#1451982511yescharacteristic
398C8—22no—
399C8—1222no—
400C8—1221no—
401C8—1221no—
402C8—2442no—
403C8#1451975312yes—
404C8#1451975312yes—
405C22—21no—
406C22—21no—
407C22—1222no—
408C22—2441no—
409C22—2441no—
410C22#1684038911yescharacteristic
411C22D4×C22⋊D6111yescharacteristic
412C22#1684047311yescharacteristic
413C4—42no—
414C4—41no—
415C4—41no—
416C4—42no—
417C4—82no—
418C4—1222no—
419C4—4882no—
420C4—2442no—
421C4C2×#1451975311yescharacteristic
422C4C4⋊Q8×D6111yescharacteristic
423C4#1684041511yescharacteristic
424C4Q8×C22⋊D6111yescharacteristic
425C2—21no—
426C2—21no—
427C2—2442no—
428C2#2144951711yescharacteristic
429C2#2145760111yescharacteristic
430C2#2145760511yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3164C2
socleC22×C61D42
prime radicalC2×#11758222C23
hypercenter#3164D61
hypocenterC61#26082
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3164

Complements

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

NH ≅ G/Ntype
3. #21447332C2semidirect
4. #21447166C2semidirect
5. #21446604C2semidirect
6. #21446568C2semidirect
8. #21445982C2semidirect
11. C61×#3164C2semidirect
47. #14519783Q8semidirect
71. #14519703Q8semidirect
210. #3164D61semidirect
260. C61#26082semidirect

Maximal subgroups

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

subgroupindex
#214556772
#214544162
#214473322
#214471662
#214466042
#214465682
#214460492
#214459822
#214458842
#214456582
C61×#31642
#2608261