Finite Group Databasebeta

← #29665356

Subgroups — #29665356

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#21456949C211yesmaximal, characteristic
2C2×#16840057C211yesmaximal, characteristic, complemented
3#21455291C211yesmaximal, characteristic
4#21445235C214yesmaximal, complemented
5#21445201C214yesmaximal, complemented
6#21444590C211yesmaximal, characteristic, complemented
7#21444499C211yesmaximal, characteristic, complemented
8#21444466C211yesmaximal, characteristic
9C61×#2982C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11#16840312C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13#16840160C2212yes—
14#16840109C2212yes—
15#16840081—22no—
16#16840057—22no—
17#16840057C2214yes—
18#16840053—22no—
19#16839984—22no—
20#16839972C2212yes—
21#16839962C2212yes—
22C2×#14519710—21no—
23#16839786C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25#16839761—21no—
26C2×#14519701—21no—
27#16839513C2214yes—
28#16839510C2214yes—
29#16839507C2212yes—
30#16839505C2212yes—
31C2×C4×Q8×C61C2211yescharacteristic
32C8×Q8×C61—22no—
33C8×Q8×C61C2214yes—
34C61×#431—22no—
35C61×#431C2211yescharacteristic
36C61×#431C2211yescharacteristic
37C2×#11758222—21no—
38C22×Dic122—21no—
39#14519808—44no—
40C2×#11758202—22no—
41C2×#11758201—22no—
42C2×#11758194—21no—
43C22×#92418—21no—
44#14519777—21no—
45#14519777—21no—
46#14519777—21no—
47#14519777C2311yescharacteristic
48#14519777C2312yes—
49#14519774C2311yescharacteristic
50#14519765—21no—
51#14519751—21no—
52#14519751—21no—
53C4×D244—21no—
54C4×D244—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57#14519710—24no—
58#14519709—24no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61C4⋊#92418C2312yes—
62C4⋊#92418C2312yes—
63#14519701—24no—
64#14519700—22no—
65#14519699—24no—
66#14519697—22no—
67C22×Q8×C61D411yescharacteristic
68C4×Q8×C61—21no—
69C4×Q8×C61—21no—
70C4×Q8×C61D412yes—
71C4×Q8×C61C2312yes—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C42×C61—21no—
75C2×C42×C61C2311yescharacteristic
76C4⋊C8×C61—24no—
77C4⋊C8×C61D412yes—
78C4⋊C8×C61C2312yes—
79C22⋊C8×C61—21no—
80C22⋊C8×C61D412yes—
81C4×C8×C61—24no—
82C4×C8×C61—21no—
83C4×C8×C61C2312yes—
84C2×Q8×C61—21no—
85C2×Q8×C61—22no—
86C2×Q8×C61SD1614yes—
87C2×Q8×C61C2×D412yes—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C2×C22⋊D61—41no—
94C22×Dic61—41no—
95#11758222—42no—
96#11758222—42no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244—21no—
100C2×C4×D61—41no—
101C2×Dic122—21no—
102C2×Dic122—22no—
103C2×Dic122—41no—
104C2×Dic122—82no—
105C2×C8×C61—22no—
106C2×C8×C61C2×D414yes—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61C2×D412yes—
111C42×C61—22no—
112C42×C61—22no—
113C42×C61C2×D411yescharacteristic
114C42×C61C2×D411yescharacteristic
115C42×C61#4111yescharacteristic
116C42×C61C2411yescharacteristic
117#11758202—44no—
118#11758201—44no—
119#11758198—41no—
120#11758198—41no—
121C4⋊Dic61—22no—
122C4⋊Dic61—22no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61—41no—
126C4⋊Dic61—41no—
127#11758194—42no—
128C2×#92418—22no—
129C2×#92418—22no—
130C2×#92418—42no—
131C23×C61#12111yescharacteristic
132C22×D61—41no—
133Q8×C61—22no—
134Q8×C61—24no—
135Q8×C61C2×SD1614yes—
136C2×C4×C61—22no—
137C2×C4×C61—24no—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—41no—
141C2×C4×C61—81no—
142C2×C4×C61#12111yescharacteristic
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61C2×SD1612yes—
146C2×C4×C61#13512yes—
147C2×C4×C61C22×D411yescharacteristic
148C2×C4×C61C22×D411yescharacteristic
149C2×C4×C61C2×#4111yescharacteristic
150C22⋊D61—82no—
151C2×Dic61—42no—
152C2×Dic61—41no—
153C2×Dic61—82no—
154D244—41no—
155D244—41no—
156D244—42no—
157D244—81no—
158C4×D61—82no—
159Dic122—41no—
160Dic122—41no—
161Dic122—82no—
162Dic122—82no—
163C8×C61—24no—
164C8×C61—42no—
165#92418—42no—
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—24no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—42no—
180C4×C61#46512yes—
181C4×C61C22×SD1611yescharacteristic
182C4×C61C2×#13511yescharacteristic
183C4×C61#57812yes—
184Dic61—81no—
185Dic61—82no—
186#3227—611no—
187C2×#477—611no—
188#3037—611no—
189#2982D6111yescharacteristic, complemented
190#1678—614no—
191#1672—614no—
192#1570—611no—
193#1548—611no—
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⋊Q8—611no—
203C2×C4×Q8C2×D6111yescharacteristic
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—22no—
214C8×Q8C2×D6114yes—
215#431—22no—
216#431C2×D6111yescharacteristic
217#431C2×D6111yescharacteristic
218C2×#107—1221no—
219#423—611no—
220C2×C4⋊C8—611no—
221#417—1221no—
222C2×#103—1221no—
223#334—614no—
224#333—614no—
225#332—612no—
226#331—612no—
227C61#1940311yescharacteristic, complemented, Sylow
228C2×#41—1221no—
229C22×Q8—1221no—
230C22×Q8D24411yescharacteristic
231#137—2444no—
232C2×Q16—1222no—
233C2×SD16—1222no—
234C2×M16—1221no—
235C22×C8—1221no—
236C4⋊Q8—612no—
237C4⋊Q8—611no—
238C4⋊Q8—1221no—
239C4⋊Q8—1221no—
240C4⋊Q8—1221no—
241#127—611no—
242#124—1221no—
243#121—1221no—
244#121—1221no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8D24412yes—
248C4×Q8C22×D6112yes—
249C4×D4—1221no—
250C4×D4—1221no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—1221no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4D24411yescharacteristic
255C2×C42—21no—
256C2×C42C22×D6111yescharacteristic
257#107—1224no—
258#106—1224no—
259C4⋊C8—24no—
260C4⋊C8—612no—
261C4⋊C8—612no—
262C4⋊C8—1221no—
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⋊C8C22⋊D6112yes—
272C4×C8—24no—
273C4×C8—21no—
274C4×C8C22×D6112yes—
275#41—2442no—
276#41—2442no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—1221no—
280C2×Q8—1222no—
281C2×Q8—4882no—
282C2×Q8—2441no—
283C2×Q8#1175819014yes—
284C2×Q8C2×D24412yes—
285C2×D4—1221no—
286C2×D4—1221no—
287C2×D4—1221no—
288C2×D4—2441no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—2441no—
292C22×C4—2441no—
293C22×C4C2×D24411yescharacteristic
294C22×C4#1175822211yescharacteristic
295C22×C4C2×C22⋊D6111yescharacteristic
296Q16—2444no—
297SD16—2444no—
298M16—2442no—
299C2×C8—22no—
300C2×C8—1222no—
301C2×C8—1222no—
302C2×C8—2442no—
303C2×C8C2×C22⋊D6114yes—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—1222no—
308C4⋊C4—1221no—
309C4⋊C4—1222no—
310C4⋊C4—1221no—
311C4⋊C4—2441no—
312C4⋊C4—2441no—
313C4⋊C4C2×D24412yes—
314C22⋊C4—2441no—
315C22⋊C4—2441no—
316C42—22no—
317C42—22no—
318C42C2×D24411yescharacteristic
319C42#1175822211yescharacteristic
320C42C2×C22⋊D6111yescharacteristic
321C42C23×D6111yescharacteristic
322C23—2441no—
323C23#1451975111yescharacteristic
324Q8—24no—
325Q8—22no—
326Q8—4882no—
327Q8—4882no—
328Q8—2441no—
329Q8—2441no—
330Q8C2×#1175819014yes—
331D4—4881no—
332D4—4882no—
333D4—2441no—
334D4—2441no—
335D4—2442no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—4882no—
343C2×C4—4882no—
344C2×C4—2441no—
345C2×C4—2442no—
346C2×C4#1451975111yescharacteristic
347C2×C4#1451975111yescharacteristic
348C2×C4#1451975111yescharacteristic
349C2×C4C2×#1175819012yes—
350C2×C4#1451979912yes—
351C2×C4C22×C22⋊D6111yescharacteristic
352C2×C4C22×D24411yescharacteristic
353C2×C4C2×#1175822211yescharacteristic
354C8—24no—
355C8—42no—
356C8—2442no—
357C8—2442no—
358C22—21no—
359C22—21no—
360C22—4881no—
361C22—4881no—
362C22—4881no—
363C22#1683999112yes—
364C22C2×#1451975111yescharacteristic
365C4—24no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—4881no—
371C4—4882no—
372C4#1683999112yes—
373C4C22×#1175819011yescharacteristic
374C4C2×#1451979911yescharacteristic
375C4#1684057712yes—
376C2—22no—
377C2—9761no—
378C2#2145534911yescharacteristic
379C2C2×#1683999111yescharacteristic
380C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2982C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#2982D61
hypocenterC61#19403
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. C2×#16840057C2semidirect
4. #21445235C2semidirect
5. #21445201C2semidirect
6. #21444590C2semidirect
7. #21444499C2semidirect
9. C61×#2982C2semidirect
189. #2982D61semidirect
227. C61#19403semidirect

Maximal subgroups

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

subgroupindex
#214569492
C2×#168400572
#214552912
#214452352
#214452012
#214445902
#214444992
#214444662
C61×#29822
#1940361