Finite Group Databasebeta

← #29665355

Subgroups — #29665355

Order 15616.

644 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456947C211yesmaximal, characteristic
2C2×#16840054C211yesmaximal, characteristic, complemented
3#21455293C211yesmaximal, characteristic
4#21454899C211yesmaximal, characteristic, complemented
5#21445238C214yesmaximal, complemented
6#21445204C214yesmaximal, complemented
7#21444591C211yesmaximal, characteristic, complemented
8#21444466C211yesmaximal, characteristic
9C61×#1548C211yesmaximal, characteristic, complemented
10C2×#14519810—21no—
11#16840312C2211yescharacteristic
12C2×C4×Dic122C2211yescharacteristic
13#16840158C2212yes—
14#16840110C2212yes—
15#16840080—22no—
16#16840054—22no—
17#16840054C2214yes—
18#16840049—22no—
19#16839985—22no—
20#16839973C2212yes—
21#16839963C2212yes—
22C8×Dic122—22no—
23#16839886C2214yes—
24#16839815C2211yescharacteristic
25#16839812—22no—
26C2×#14519710—21no—
27C2×C4⋊#92418C2211yescharacteristic
28#16839764—21no—
29C2×#14519700—21no—
30#16839512C2214yes—
31#16839507C2212yes—
32#16839505C2212yes—
33C2×C4⋊Q8×C61C2211yescharacteristic
34C61×#431C2211yescharacteristic
35C61×#423C2211yescharacteristic
36C61×#333C2214yes—
37C2×#11758222—21no—
38C22×Dic122D411yescharacteristic
39#14519810—44no—
40C2×Dic244—22no—
41C2×#11758190—22no—
42C22×#92418—21no—
43#14519777C2311yescharacteristic
44#14519774C2311yescharacteristic
45#14519765—21no—
46#14519751—21no—
47#14519751—21no—
48C4×Dic122—21no—
49C4×Dic122—21no—
50C4×Dic122D412yes—
51C4×Dic122C2312yes—
52C4×D244—21no—
53C4×D244—21no—
54C2×C4⋊Dic61D411yescharacteristic
55C2×#11758196—21no—
56C2×C4×Dic61—21no—
57#14519710—24no—
58#14519708—24no—
59C4⋊#92418—21no—
60C4⋊#92418D412yes—
61C4⋊#92418C2312yes—
62#14519706—24no—
63#14519702—22no—
64#14519700—24no—
65#14519699—22no—
66#14519697—24no—
67C22⋊#92418D412yes—
68C4×#92418C2312yes—
69C8×Dic61—24no—
70C22×Q8×C61—21no—
71C2×M16×C61—21no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61C2312yes—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C42×C61C2311yescharacteristic
79C4⋊C8×C61—21no—
80C4⋊C8×C61C2312yes—
81C4⋊C8×C61C2312yes—
82C22⋊C8×C61—21no—
83C4×C8×C61—21no—
84C2×Q8×C61—22no—
85C2×Q8×C61—41no—
86C2×Q8×C61—82no—
87C22×C4×C61—41no—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61#4111yescharacteristic
91C2×C22⋊D61—41no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94#11758222—42no—
95#11758222—42no—
96C2×D244—21no—
97C2×D244—21no—
98C2×D244—21no—
99C2×C4×D61—41no—
100C2×Dic122—21no—
101C2×Dic122—22no—
102C2×Dic122—21no—
103C2×Dic122SD1614yes—
104C2×Dic122C2×D412yes—
105M16×C61—42no—
106C2×C8×C61—22no—
107C2×C8×C61—22no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C42×C61C2×D411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C42×C61#4111yescharacteristic
115C42×C61C2411yescharacteristic
116#11758198—41no—
117#11758198—41no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61C2×D412yes—
122#11758196—22no—
123#11758196—22no—
124C4×Dic61—22no—
125C4×Dic61—22no—
126C2×#92418—22no—
127C2×#92418—42no—
128C2×#92418C2×D414yes—
129Dic244—44no—
130#11758190—44no—
131C23×C61#12111yescharacteristic
132C22×D61—41no—
133Q8×C61—82no—
134Q8×C61—82no—
135C2×C4×C61—22no—
136C2×C4×C61—41no—
137C2×C4×C61—42no—
138C2×C4×C61—82no—
139C2×C4×C61#12111yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61C2×SD1612yes—
143C2×C4×C61#13512yes—
144C2×C4×C61C22×D411yescharacteristic
145C2×C4×C61C22×D411yescharacteristic
146C2×C4×C61C2×#4111yescharacteristic
147C22⋊D61—82no—
148C2×Dic61—22no—
149C2×Dic61—24no—
150C2×Dic61—22no—
151C2×Dic61—41no—
152C2×Dic61—81no—
153D244—41no—
154D244—41no—
155D244—42no—
156D244—81no—
157C4×D61—82no—
158Dic122—22no—
159Dic122—24no—
160Dic122—41no—
161Dic122—41no—
162Dic122C2×SD1614yes—
163C8×C61—42no—
164C8×C61—42no—
165#92418—24no—
166#92418—42no—
167#19403—611nomaximal, Sylow
168C22×C61—21no—
169C22×C61—21no—
170C22×C61#46512yes—
171C22×C61C2×#12111yescharacteristic
172C2×D61—81no—
173C2×D61—81no—
174C2×D61—81no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—82no—
179C4×C61#46512yes—
180C4×C61C22×SD1611yescharacteristic
181C4×C61C2×#13511yescharacteristic
182C4×C61#57812yes—
183Dic61—24no—
184Dic61—42no—
185Dic61—81no—
186#3227—611no—
187C2×#477—611no—
188#3037—611no—
189#2982—611no—
190#1678—614no—
191#1672—614no—
192#1570—611no—
193#1548D6111yescharacteristic, complemented
194#1540—611no—
195C2×C61—22no—
196C2×C61#304411yescharacteristic
197C2×C61C2×#46511yescharacteristic
198C2×C61#306811yescharacteristic
199D61—161no—
200C2×#137—1221no—
201#532—611no—
202C2×C4⋊Q8C2×D6111yescharacteristic
203C2×C4×Q8—611no—
204#498—612no—
205#486—612no—
206#481—1222no—
207#477—614no—
208#477—1222no—
209#476—1222no—
210#464—1222no—
211#462—612no—
212#460—612no—
213C8×Q8—614no—
214C8×Q8—1222no—
215#431—611no—
216#431—1222no—
217#431C2×D6111yescharacteristic
218C2×#107—1221no—
219#423C2×D6111yescharacteristic
220C2×C4⋊C8—611no—
221#417—1221no—
222C2×#103—1221no—
223#334—614no—
224#333C2×D6114yes—
225#332—612no—
226#331—612no—
227C61#1940311yescharacteristic, complemented, Sylow
228C2×#41—1221no—
229C22×Q8—21no—
230C22×Q8—611no—
231#137—2444no—
232C2×Q16—1222no—
233C2×SD16—1222no—
234C2×M16—21no—
235C22×C8—1221no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—611no—
240C4⋊Q8C22×D6112yes—
241#127—611no—
242#124—1221no—
243#121—1221no—
244#121—1221no—
245C4×Q8—612no—
246C4×Q8—612no—
247C4×Q8—1221no—
248C4×Q8—1221no—
249C4×D4—1221no—
250C4×D4—1221no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—611no—
254C2×C4⋊C4—1221no—
255C2×C42—1221no—
256C2×C42C22×D6111yescharacteristic
257#107—1224no—
258#106—1224no—
259C4⋊C8—21no—
260C4⋊C8—612no—
261C4⋊C8—612no—
262C4⋊C8—1224no—
263C4⋊C8—1221no—
264C4⋊C8C22×D6112yes—
265C4⋊C8C22×D6112yes—
266#103—1224no—
267#103—1222no—
268#102—1224no—
269#102—1222no—
270C22⋊C8—21no—
271C22⋊C8—612no—
272C4×C8—21no—
273C4×C8—612no—
274C4×C8—1224no—
275#41—2442no—
276#41—2442no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—82no—
280C2×Q8—612no—
281C2×Q8—614no—
282C2×Q8—1221no—
283C2×Q8—1222no—
284C2×Q8—1221no—
285C2×D4—1221no—
286C2×D4—1221no—
287C2×D4—1221no—
288C2×D4—2441no—
289C22×C4—41no—
290C22×C4—1221no—
291C22×C4—1221no—
292C22×C4—2441no—
293C22×C4#1175822211yescharacteristic
294C22×C4D4×D6111yescharacteristic
295C22×C4D4×D6111yescharacteristic
296Q16—2444no—
297SD16—2444no—
298M16—42no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—614no—
302C2×C8—1222no—
303C2×C8—2442no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—612no—
309C4⋊C4—1222no—
310C4⋊C4—1221no—
311C4⋊C4—1222no—
312C4⋊C4—1221no—
313C4⋊C4—1221no—
314C22⋊C4—2441no—
315C22⋊C4—2441no—
316C42—1222no—
317C42—1222no—
318C42#1175822211yescharacteristic
319C42D4×D6111yescharacteristic
320C42D4×D6111yescharacteristic
321C42C23×D6111yescharacteristic
322C23—2441no—
323C23#1451975011yescharacteristic
324Q8—82no—
325Q8—82no—
326Q8—614no—
327Q8—1224no—
328Q8—1222no—
329Q8—2441no—
330Q8—2441no—
331D4—4881no—
332D4—4882no—
333D4—2441no—
334D4—2441no—
335D4—2442no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—82no—
340C2×C4—1224no—
341C2×C4—1222no—
342C2×C4—1222no—
343C2×C4—4882no—
344C2×C4—4881no—
345C2×C4—2441no—
346C2×C4#1451975011yescharacteristic
347C2×C4#1451975011yescharacteristic
348C2×C4#1451975011yescharacteristic
349C2×C4SD16×D6112yes—
350C2×C4#1451980012yes—
351C2×C4C2×D4×D6111yescharacteristic
352C2×C4C2×D4×D6111yescharacteristic
353C2×C4C2×#1175822211yescharacteristic
354C8—42no—
355C8—42no—
356C8—1224no—
357C8—2442no—
358C22—21no—
359C22—21no—
360C22—4881no—
361C22—4881no—
362C22—4881no—
363C22#1683999412yes—
364C22C2×#1451975011yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—82no—
369C4—1224no—
370C4—4881no—
371C4—2442no—
372C4#1683999412yes—
373C4C2×SD16×D6111yescharacteristic
374C4C2×#1451980011yescharacteristic
375C4#1684057812yes—
376C2—22no—
377C2—9761no—
378C2#2145534811yescharacteristic
379C2C2×#1683999411yescharacteristic
380C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1548C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1548D61
hypocenterC61#19403
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
2. C2×#16840054C2semidirect
4. #21454899C2semidirect
5. #21445238C2semidirect
6. #21445204C2semidirect
7. #21444591C2semidirect
9. C61×#1548C2semidirect
193. #1548D61semidirect
227. C61#19403semidirect

Maximal subgroups

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

subgroupindex
#214569472
C2×#168400542
#214552932
#214548992
#214452382
#214452042
#214445912
#214444662
C61×#15482
#1940361