Finite Group Databasebeta

← #29738929

Subgroups — #29738929

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455666C212yesmaximal
2#21454416C211yesmaximal, characteristic
3#21447327C212yesmaximal, complemented
4#21447161C212yesmaximal, complemented
5#21446604C211yesmaximal, characteristic, complemented
6#21446568C211yesmaximal, characteristic, complemented
7#21446042C212yesmaximal
8#21445977C211yesmaximal, characteristic, complemented
9#21445884C211yesmaximal, characteristic
10#21445658C211yesmaximal, characteristic
11C61×#3164C211yesmaximal, characteristic, complemented
12C2×#14519783C2211yescharacteristic
13#16840242C2212yes—
14#16840066—22no—
15#16840059—22no—
16#16840054—22no—
17#16840039—22no—
18C2×#14519713C2211yescharacteristic
19#16839805C2212yes—
20#16839805C2212yes—
21#16839801—22no—
22#16839799—21no—
23C2×#14519711C2211yescharacteristic
24C2×#14519710C2212yes—
25#107×D61—21no—
26#16839789C2212yes—
27C2×#14519703C2211yescharacteristic
28#16839761C2212yes—
29#16839756C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839631C2212yes—
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—
58C2×#11758196—42no—
59C4⋊C4×D61—22no—
60C2×C4×Dic61C2311yescharacteristic
61#14519713—22no—
62#14519711—21no—
63#14519711—21no—
64#14519710—22no—
65#14519710—22no—
66#14519710—22no—
67#14519710—22no—
68#14519708—22no—
69#14519708—22no—
70C4⋊#92418—22no—
71C4⋊#92418—22no—
72#14519703Q814yescomplemented
73#14519700—22no—
74#14519700D411yescharacteristic
75#14519700D411yescharacteristic
76#14519697—22no—
77#14519697D411yescharacteristic
78#14519697D411yescharacteristic
79C8×Dic61—22no—
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—
137#11758196—84no—
138C4×Dic61—21no—
139C4×Dic61—42no—
140C4×Dic61C2×Q812yes—
141#11758194—21no—
142#11758194—22no—
143#11758194—22no—
144#11758194C2×Q812yes—
145C2×#92418—21no—
146C2×#92418—21no—
147C2×#92418—21no—
148C2×#92418—22no—
149C2×#92418C2×D411yescharacteristic
150C2×#92418C2×D411yescharacteristic
151#11758189—24no—
152C8×D61—24no—
153C23×C61H(5,𝔽2)11yescharacteristic
154C22×D61—21no—
155Q8×C61—162no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—42no—
161C2×C4×C61—82no—
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C22×Q811yescharacteristic
165C2×C4×C61C22×Q811yescharacteristic
166C2×C4×C61C2×#4111yescharacteristic
167C2×C4×C612−1+411yescharacteristic
168C22⋊D61—22no—
169C22⋊D61—22no—
170C2×Dic61—22no—
171C2×Dic61—22no—
172C2×Dic61—21no—
173C2×Dic61—22no—
174C2×Dic61—42no—
175C2×Dic61—82no—
176D244—21no—
177D244C4⋊Q812yes—
178C4×D61—22no—
179C4×D61—22no—
180C4×D61—162no—
181Dic122—21no—
182Dic122C4⋊Q812yes—
183C8×C61—22no—
184C8×C61C22⋊Q812yes—
185C8×C61C22⋊Q812yes—
186#92418—22no—
187#92418—21no—
188#92418—21no—
189#92418—42no—
190#26083—611nomaximal, Sylow
191C22×C61—21no—
192C22×C61—21no—
193C22×C61#54411yescharacteristic
194C22×C61D4211yescharacteristic
195C22×C61#55411yescharacteristic
196C2×D61—22no—
197C2×D61—41no—
198C2×D61—41no—
199C4×C61—42no—
200C4×C61—41no—
201C4×C61—42no—
202C4×C61—41no—
203C4×C61—82no—
204C4×C61C2×C22⋊Q811yescharacteristic
205C4×C61C2×C4⋊Q811yescharacteristic
206C4×C61#54711yescharacteristic
207C4×C61D4×Q811yescharacteristic
208Dic61—24no—
209Dic61—22no—
210Dic61—82no—
211#3164D6111yescharacteristic, complemented
212#3083—612no—
213#2913—611no—
214#2089—612no—
215#2060—612no—
216#1952—611no—
217#1945—611no—
218#1859—612no—
219#1845—611no—
220#1816—611no—
221#1764—611no—
222C2×C61—21no—
223C2×C61—21no—
224C2×C61#244011yescharacteristic
225C2×C61#329211yescharacteristic
226C2×C61#329311yescharacteristic
227D61—42no—
228C2×#131—611no—
229#544C2×D6112yes—
230#520—612no—
231#501—22no—
232#500—21no—
233#499—21no—
234#479—1222no—
235#478—1222no—
236#477—1222no—
237#474—1222no—
238C2×#108—611no—
239#428—612no—
240#428—612no—
241#428—1222no—
242#428C2×D6112yes—
243#427—1221no—
244#427C2×D6111yescharacteristic
245C2×#107—611no—
246C2×#107—612no—
247C2×#107—1221no—
248C2×#107C2×D6111yescharacteristic
249#424—612no—
250C2×#104—611no—
251#417—612no—
252#416—611no—
253#405C2×D6111yescharacteristic
254C2×C4×C8—611no—
255#384—612no—
256#343—612no—
257#343—612no—
258#341—611no—
259#339—611no—
260#339—612no—
261C61#2608311yescharacteristic, complemented, Sylow
262C2×#41—611no—
263#131—614no—
264#131—1222no—
265C2×M16—611no—
266C2×M16—612no—
267C2×M16—1221no—
268C2×M16C22×D6111yescharacteristic
269C22×C8—611no—
270C22×C8—1221no—
271C22×C8C22×D6111yescharacteristic
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274#125—22no—
275#125—22no—
276C22⋊Q8—42no—
277C4×Q8—1222no—
278C4×D4—1222no—
279C4×D4—2442no—
280#117—1222no—
281#117C22×D6111yescharacteristic
282#117C22×D6112yes—
283C2×C4⋊C4—612no—
284C2×C4⋊C4—1222no—
285C2×C4⋊C4—2442no—
286C2×C4⋊C4C22×D6112yes—
287C2×C42—611no—
288#108—1222no—
289#107—21no—
290#107—21no—
291#107—22no—
292#107—1222no—
293#107—1222no—
294#107—1221no—
295#107—1221no—
296#107—1222no—
297#107—1222no—
298#107C22⋊D6111yescharacteristic
299#107C22⋊D6111yescharacteristic
300#106—22no—
301#106—1222no—
302#106—1222no—
303#106C22⋊D6112yes—
304C4⋊C8—1222no—
305C4⋊C8—1222no—
306#104—614no—
307#103—611no—
308#103—611no—
309#103—1222no—
310#102—611no—
311#102—611no—
312#102—1222no—
313#97—21no—
314C4×C8—21no—
315C4×C8—1222no—
316#41—612no—
317#41—612no—
318#41—1221no—
319#41—1221no—
320C2×Q8—42no—
321C2×Q8—611no—
322C2×D4—611no—
323C2×D4—1221no—
324C22×C4—22no—
325C22×C4—1221no—
326C22×C4—1221no—
327C22×C4—1222no—
328C22×C4—2442no—
329C22×C4C23×D6111yescharacteristic
330M16—21no—
331M16—612no—
332M16—1221no—
333M16—1224no—
334M16—1222no—
335M16—1222no—
336M16Q8×D6112yes—
337C2×C8—21no—
338C2×C8—611no—
339C2×C8—611no—
340C2×C8—1221no—
341C2×C8—1221no—
342C2×C8—1221no—
343C2×C8—1224no—
344C2×C8—1222no—
345C2×C8Q8×D6112yes—
346C2×C8#1175822612yes—
347C2×C8C2×C22⋊D6111yescharacteristic
348C2×C8C2×C22⋊D6111yescharacteristic
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—1222no—
355C4⋊C4—1222no—
356C4⋊C4—4884no—
357C4⋊C4—2442no—
358C4⋊C4D4×D6111yescharacteristic
359C4⋊C4D4×D6111yescharacteristic
360C4⋊C4C2×C22⋊D6112yes—
361C4⋊C4C2×C22⋊D6112yes—
362C22⋊C4—22no—
363C22⋊C4—21no—
364C22⋊C4—2442no—
365C42—22no—
366C42—21no—
367C42—612no—
368C42—1221no—
369C42—2442no—
370C23—1221no—
371C23#1451982211yescharacteristic
372Q8—162no—
373Q8—612no—
374Q8—1221no—
375D4—612no—
376D4—1222no—
377D4—1222no—
378D4—1221no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—42no—
384C2×C4—82no—
385C2×C4—1222no—
386C2×C4—1222no—
387C2×C4—1222no—
388C2×C4—1221no—
389C2×C4—1222no—
390C2×C4—1222no—
391C2×C4—9762no—
392C2×C4—4882no—
393C2×C4—2442no—
394C2×C4C2×D4×D6111yescharacteristic
395C2×C4C22×C22⋊D6111yescharacteristic
396C2×C4C2×Q8×D6111yescharacteristic
397C2×C4C2×Q8×D6111yescharacteristic
398C2×C4C2×#1175822611yescharacteristic
399C2×C4#1451982511yescharacteristic
400C8—22no—
401C8—1222no—
402C8—1221no—
403C8—1221no—
404C8—2442no—
405C8#1451975312yes—
406C8#1451975312yes—
407C22—21no—
408C22—21no—
409C22—1222no—
410C22—2441no—
411C22—2441no—
412C22#1684038911yescharacteristic
413C22D4×C22⋊D6111yescharacteristic
414C22#1684047311yescharacteristic
415C4—42no—
416C4—41no—
417C4—42no—
418C4—41no—
419C4—82no—
420C4—1224no—
421C4—1222no—
422C4—4882no—
423C4C2×#1451975311yescharacteristic
424C4C4⋊Q8×D6111yescharacteristic
425C4#1684041511yescharacteristic
426C4Q8×C22⋊D6111yescharacteristic
427C2—21no—
428C2—21no—
429C2—2442no—
430C2#2144951711yescharacteristic
431C2#2145761611yescharacteristic
432C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3164C2
socleC22×C61D42
prime radicalC2×#11758222C23
hypercenter#3164D61
hypocenterC61#26083
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. #21447327C2semidirect
4. #21447161C2semidirect
5. #21446604C2semidirect
6. #21446568C2semidirect
8. #21445977C2semidirect
11. C61×#3164C2semidirect
47. #14519783Q8semidirect
72. #14519703Q8semidirect
211. #3164D61semidirect
261. C61#26083semidirect

Maximal subgroups

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

subgroupindex
#214556662
#214544162
#214473272
#214471612
#214466042
#214465682
#214460422
#214459772
#214458842
#214456582
C61×#31642
#2608361