Finite Group Databasebeta

← #29663158

Subgroups — #29663158

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456940C211yesmaximal, characteristic
2#21456744C211yesmaximal, characteristic
3#21454532C211yesmaximal, characteristic
4#21445363C214yesmaximal, complemented
5#21445239C214yesmaximal, complemented
6#21444567C211yesmaximal, characteristic, complemented
7#21444504C211yesmaximal, characteristic, complemented
8#21444491C211yesmaximal, characteristic, complemented
9C2×C61×#487C211yesmaximal, characteristic, complemented
10C2×#14519783—21no—
11#16840312C2211yescharacteristic
12#16840165—21no—
13#16840136—21no—
14#16840112—21no—
15#16840110—21no—
16#16840075—22no—
17#16840039C2214yes—
18#16839964C2214yes—
19#16839901—22no—
20#16839841—21no—
21#16839841C2212yes—
22C8×D244—21no—
23C8×D244C2212yes—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839728—21no—
28#16839515C2212yes—
29#16839513C2212yes—
30#16839508C2214yes—
31#16839508C2214yes—
32C2×C4⋊Q8×C61C2211yescharacteristic
33C2×C4⋊Q8×C61C2211yescharacteristic
34C61×#487—21no—
35C61×#487—21no—
36C61×#487C2212yes—
37C61×#487C2212yes—
38C2×#103×C61—21no—
39C2×#103×C61—21no—
40C2×C4×C8×C61C2211yescharacteristic
41C2×#11758222—21no—
42C2×#11758201—24no—
43#14519783—44no—
44C2×#11758194—21no—
45C2×#11758189—22no—
46C2×C8×D61—22no—
47#14519777D411yescharacteristic
48#14519774D411yescharacteristic
49#14519765—21no—
50#14519751—21no—
51#14519751D412yes—
52C4×D244—21no—
53C4×D244C2312yes—
54#14519710—24no—
55C4⋊#92418—21no—
56C4⋊#92418C2312yes—
57C4⋊#92418C2312yes—
58C4⋊#92418C2312yes—
59#14519701—24no—
60#14519697—24no—
61C22⋊#92418—21no—
62C22⋊#92418—21no—
63#14519689—22no—
64#14519689—22no—
65C4×#92418—21no—
66C4×#92418—21no—
67C22×Q8×C61—21no—
68C22×Q8×C61—21no—
69C22×C8×C61D411yescharacteristic
70C22×C8×C61D411yescharacteristic
71C4⋊Q8×C61—21no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—21no—
77C4⋊Q8×C61C2312yes—
78C4⋊Q8×C61C2312yes—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—21no—
83C2×C42×C61C2311yescharacteristic
84#103×C61—22no—
85#103×C61—22no—
86#103×C61—22no—
87#103×C61—22no—
88C4×C8×C61D411yescharacteristic
89C4×C8×C61D411yescharacteristic
90C4×C8×C61C2311yescharacteristic
91C4×C8×C61C2311yescharacteristic
92C2×Q8×C61—22no—
93C2×Q8×C61—22no—
94C2×Q8×C61—41no—
95C2×Q8×C61—41no—
96C2×Q8×C61—42no—
97C2×Q8×C61—42no—
98C2×Q8×C61—82no—
99C22×C4×C61—41no—
100C22×C4×C61—41no—
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61#4111yescharacteristic
104C2×C22⋊D61—22no—
105#11758222—42no—
106#11758222—42no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244C2×D412yes—
110C2×C4×D61—22no—
111C2×Dic122—21no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2×D411yescharacteristic
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—41no—
123C4⋊C4×C61—41no—
124C4⋊C4×C61—41no—
125C4⋊C4×C61—41no—
126C42×C61C2×D411yescharacteristic
127C42×C61C2×D411yescharacteristic
128C42×C61#4111yescharacteristic
129C42×C61C2411yescharacteristic
130#11758201—84no—
131#11758198—22no—
132#11758198—22no—
133C4⋊Dic61—21no—
134C4⋊Dic61C2×D412yes—
135#11758194—42no—
136C2×#92418—22no—
137C2×#92418—22no—
138C2×#92418—22no—
139#11758189—44no—
140C8×D61—44no—
141C23×C61#12111yescharacteristic
142C22×D61—22no—
143Q8×C61—42no—
144Q8×C61—42no—
145Q8×C61—44no—
146Q8×C61—82no—
147Q8×C61—82no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—82no—
153C2×C4×C61—82no—
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#13512yes—
158C2×C4×C61AGL(1,ℤ/8)12yes—
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C2×#4111yescharacteristic
162C22⋊D61—44no—
163C2×Dic61—22no—
164D244—24no—
165D244—42no—
166D244—42no—
167C4×D61—42no—
168C4×D61—42no—
169Dic122—42no—
170C8×C61—21no—
171C8×C61—21no—
172C8×C61—21no—
173C8×C61—21no—
174#92418—42no—
175#92418—42no—
176#92418—42no—
177#19197—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61#46312yes—
181C22×C61C2×#12111yescharacteristic
182C2×D61—42no—
183C2×D61—42no—
184C2×D61—42no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—82no—
188C4×C61—82no—
189C4×C61#46312yes—
190C4×C61#47312yes—
191C4×C61C2×#13511yescharacteristic
192C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
193C4×C61#57712yes—
194Dic61—42no—
195#3226—611no—
196#3206—611no—
197C2×#487D6111yescharacteristic, complemented
198#2931—611no—
199#1699—614no—
200#1679—614no—
201#1564—611no—
202#1549—611no—
203#1546—611no—
204C2×C61—22no—
205C2×C61C2×#46311yescharacteristic
206C2×C61#305711yescharacteristic
207C2×C61C2×#47311yescharacteristic
208D61—82no—
209C2×#131—1221no—
210#532—611no—
211C2×C4⋊Q8C2×D6111yescharacteristic
212C2×C4⋊Q8C2×D6111yescharacteristic
213#500—1221no—
214#492—1221no—
215#487—21no—
216#487—21no—
217#487—1221no—
218#487C2×D6112yes—
219#487C2×D6112yes—
220#486—1221no—
221#480—1222no—
222#474—614no—
223#461—614no—
224#448—1222no—
225#436—612no—
226#436—1221no—
227C8×D4—612no—
228C8×D4—1221no—
229#431—611no—
230#431—611no—
231#423—611no—
232C2×#103—21no—
233C2×#103—21no—
234#408—1221no—
235C2×C4×C8C2×D6111yescharacteristic
236#335—612no—
237#334—612no—
238#332—614no—
239#332—614no—
240C61#1919711yescharacteristic, complemented, Sylow
241C2×#41—1221no—
242C22×Q8—21no—
243C22×Q8—21no—
244C2×SD16—1224no—
245#131—2444no—
246C2×M16—1221no—
247C2×M16—1222no—
248C22×C8—1222no—
249C22×C8C22⋊D6111yescharacteristic
250C22×C8C22⋊D6111yescharacteristic
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—611no—
258C4⋊Q8C22×D6112yes—
259C4⋊Q8C22×D6112yes—
260#127—611no—
261#124—1221no—
262#121—612no—
263#121—1221no—
264C4×D4—612no—
265C4×D4—1221no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C42C22×D6111yescharacteristic
271#107—1224no—
272C4⋊C8—612no—
273C4⋊C8—612no—
274C4⋊C8—612no—
275C4⋊C8—1221no—
276#103—22no—
277#103—22no—
278#103—22no—
279#103—22no—
280#103—1224no—
281#102—1224no—
282C22⋊C8—1221no—
283C22⋊C8—1221no—
284C22⋊C8—1222no—
285C22⋊C8—1222no—
286C4×C8—1221no—
287C4×C8—1221no—
288C4×C8C22⋊D6111yescharacteristic
289C4×C8C22⋊D6111yescharacteristic
290C4×C8C22×D6111yescharacteristic
291C4×C8C22×D6111yescharacteristic
292#41—2442no—
293#41—2442no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—82no—
301C2×Q8—1221no—
302C2×D4—612no—
303C2×D4—1222no—
304C2×D4—1221no—
305C2×D4—1221no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—1222no—
309C22×C4D4×D6111yescharacteristic
310C22×C4#1175822411yescharacteristic
311C22×C4C2×C22⋊D6111yescharacteristic
312SD16—4884no—
313M16—2444no—
314M16—2442no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—1222no—
318C2×C8—1222no—
319C2×C8—1222no—
320C2×C8—2444no—
321C2×C8C2×C22⋊D6111yescharacteristic
322C2×C8C2×C22⋊D6111yescharacteristic
323C2×C8C2×C22⋊D6111yescharacteristic
324C2×C8C2×C22⋊D6111yescharacteristic
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—612no—
334C4⋊C4—1221no—
335C22⋊C4—1222no—
336C22⋊C4—1222no—
337C42D4×D6111yescharacteristic
338C42#1175822411yescharacteristic
339C42C2×C22⋊D6111yescharacteristic
340C42C23×D6111yescharacteristic
341C23—1222no—
342C23#1451974811yescharacteristic
343Q8—42no—
344Q8—42no—
345Q8—44no—
346Q8—82no—
347Q8—82no—
348Q8—2442no—
349D4—1224no—
350D4—2444no—
351D4—2442no—
352D4—2442no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—1222no—
360C2×C4—2442no—
361C2×C4—2442no—
362C2×C4#1451974811yescharacteristic
363C2×C4#1451974811yescharacteristic
364C2×C4#1451974811yescharacteristic
365C2×C4#1451979712yes—
366C2×C4#1451980212yes—
367C2×C4C2×D4×D6111yescharacteristic
368C2×C4C22×C22⋊D6111yescharacteristic
369C2×C4C2×#1175822411yescharacteristic
370C8—21no—
371C8—21no—
372C8—21no—
373C8—21no—
374C8—2442no—
375C8—2442no—
376C8—2442no—
377C22—21no—
378C22—21no—
379C22—2442no—
380C22—2442no—
381C22—2442no—
382C22#1683997412yes—
383C22C2×#1451974811yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—82no—
388C4—2442no—
389C4#1683997412yes—
390C4#1684003412yes—
391C4C2×#1451979711yescharacteristic
392C4C2×#1451980211yescharacteristic
393C4#1684056812yes—
394C2—22no—
395C2—4882no—
396C2C2×#1683997411yescharacteristic
397C2#2145545411yescharacteristic
398C2C2×#1684003411yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#487C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenterC2×#487D61
hypocenterC61#19197
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. #21445363C2semidirect
5. #21445239C2semidirect
6. #21444567C2semidirect
7. #21444504C2semidirect
8. #21444491C2semidirect
9. C2×C61×#487C2semidirect
197. C2×#487D61semidirect
240. C61#19197semidirect

Maximal subgroups

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

subgroupindex
#214569402
#214567442
#214545322
#214453632
#214452392
#214445672
#214445042
#214444912
C2×C61×#4872
#1919761