Finite Group Databasebeta

← #29575425

Subgroups — #29575425

Order 15616.

532 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454842C211yesmaximal, characteristic
2C2×#16839864C211yesmaximal, characteristic
3#21444215C211yesmaximal, characteristic, complemented
4#21444209C211yesmaximal, characteristic
5#21444142C211yesmaximal, characteristic, complemented
6#21443664C211yesmaximal, characteristic, complemented
7C61×#1275C211yesmaximal, characteristic, complemented
8C22×#11758190C411yescharacteristic
9C2×C4×Dic122C2211yescharacteristic
10C2×C4×D244C2211yescharacteristic
11#16839864—22no—
12#16839864—22no—
13C8×D244—22no—
14#16839827—22no—
15#16839815C2211yescharacteristic
16#16839815C2211yescharacteristic
17C2×#14519711C411yescharacteristic
18C2×#14519702C411yescharacteristic
19C2×#14519699C411yescharacteristic
20C22⋊C8×D61—22no—
21#16839494—22no—
22#16839489—22no—
23#16839487—22no—
24C61×#431C2211yescharacteristic
25C61×#431C2211yescharacteristic
26C2×#97×C61C2211yescharacteristic
27C22×Dic122C2×C411yescharacteristic
28C22×D244C2×C411yescharacteristic
29C22×C4×D61—21no—
30C2×#11758190—22no—
31C2×#11758190—22no—
32C2×#11758190C814yescomplemented
33C2×C8×D61—24no—
34C4×Dic122—21no—
35C4×Dic122—21no—
36C4×Dic122—21no—
37C4×Dic122—21no—
38C4×D244—21no—
39C4×D244—21no—
40C4×D244D412yes—
41C4×D244D412yes—
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×C4⋊Dic61C2×C411yescharacteristic
44C2×#11758196—21no—
45C2×#11758198—21no—
46C2×C4×Dic61—21no—
47#14519711—22no—
48C4⋊#92418—21no—
49C4⋊#92418—21no—
50#14519702—22no—
51#14519699—22no—
52C22⋊#92418—21no—
53C22⋊#92418D412yes—
54#14519689—24no—
55C4×#92418—21no—
56C4×#92418—21no—
57C22×C8×C61C2×C411yescharacteristic
58C22×C8×C61C2×C411yescharacteristic
59C2×C42×C61C2311yescharacteristic
60C4⋊C8×C61—21no—
61C4⋊C8×C61—21no—
62C22⋊C8×C61—21no—
63C22⋊C8×C61—21no—
64#97×C61—21no—
65#97×C61—21no—
66C4×C8×C61—21no—
67C4×C8×C61D412yes—
68C23×D61—21no—
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61#4111yescharacteristic
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C2×D244—21no—
76C2×D244—21no—
77C2×D244—21no—
78C2×D244C22⋊C412yes—
79C2×D244C2×C812yes—
80C2×D244M1612yes—
81C2×C4×D61—22no—
82C2×C4×D61—22no—
83C2×C4×D61—82no—
84C2×Dic122—21no—
85C2×Dic122—21no—
86C2×Dic122—21no—
87C2×Dic122—21no—
88C2×Dic122—21no—
89C2×Dic122C2×C812yescomplemented
90C2×C8×C61—21no—
91C2×C8×C61—21no—
92C2×C8×C61—21no—
93C2×C8×C61—22no—
94C2×C8×C61—21no—
95C2×C8×C61—21no—
96C2×C8×C61—41no—
97C2×C8×C61C2×C812yescomplemented
98C42×C61C2×D411yescharacteristic
99C42×C61C2×D411yescharacteristic
100C42×C61C2×D411yescharacteristic
101C42×C61#4111yescharacteristic
102#11758198—22no—
103#11758198—22no—
104C4⋊Dic61—21no—
105C4⋊Dic61—21no—
106C4⋊Dic61—21no—
107C4⋊Dic61C22⋊C412yes—
108#11758196—41no—
109#11758196—41no—
110C4×Dic61—41no—
111C4×Dic61—41no—
112C2×#92418—22no—
113C2×#92418—41no—
114#11758190—28no—
115C8×D61—84no—
116C23×C61C4×D411yescharacteristic
117C22×D61—22no—
118C22×D61—22no—
119C22×D61—22no—
120C22×D61—81no—
121C22×D61—81no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61C2×C22⋊C411yescharacteristic
125C2×C4×C61C4×D411yescharacteristic
126C2×C4×C61#12011yescharacteristic
127C2×C4×C61#12111yescharacteristic
128C2×C4×C61#12111yescharacteristic
129C2×C4×C61#12311yescharacteristic
130C2×C4×C61C22×C811yescharacteristic
131C2×C4×C61C2×M1611yescharacteristic
132C2×C4×C61#13111yescharacteristic
133C2×C4×C61#13111yescharacteristic
134C2×Dic61—22no—
135C2×Dic61—22no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—81no—
140C2×Dic61—81no—
141D244—22no—
142D244—22no—
143D244—22no—
144D244C22⋊C814yes—
145C4×D61—44no—
146C4×D61—82no—
147Dic122—22no—
148Dic122—22no—
149Dic122—41no—
150Dic122—41no—
151C8×C61—22no—
152C8×C61—42no—
153C8×C61—41no—
154C8×C61—81no—
155#92418—42no—
156#92418—81no—
157#9087—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61#38611yescharacteristic
161C22×C61C8×D411yescharacteristic
162C22×C61#43511yescharacteristic
163C2×D61—24no—
164C2×D61—42no—
165C2×D61—42no—
166C2×D61—82no—
167C2×D61—82no—
168C4×C61—21no—
169C4×C61—21no—
170C4×C61—21no—
171C4×C61—21no—
172C4×C61C2×C22⋊C811yescharacteristic
173C4×C61#40811yescharacteristic
174C4×C61C8×D411yescharacteristic
175C4×C61#43511yescharacteristic
176Dic61—42no—
177Dic61—42no—
178Dic61—81no—
179#2972—611no—
180C2×#440—611no—
181#1479—611no—
182#1478—611no—
183#1461—611no—
184#1281—611no—
185#1275D6111yescharacteristic, complemented
186C2×C61—21no—
187C2×C61—21no—
188C2×C61#186911yescharacteristic
189C2×C61#190211yescharacteristic
190C2×C61#190411yescharacteristic
191D61—44no—
192C22×SD16—611no—
193C2×C4×Q8—611no—
194C2×C4×D4—611no—
195#440—1222no—
196#440—1222no—
197C8×D4—1222no—
198C8×D4—1222no—
199#431—611no—
200#431—611no—
201#431C2×D6111yescharacteristic
202#431C2×D6111yescharacteristic
203C2×#107—611no—
204C2×#103—611no—
205C2×#102—611no—
206C2×C22⋊C8—1222no—
207C2×#97C2×D6111yescharacteristic
208Q8⋊C8—1222no—
209D4⋊C8—1222no—
210#324—1222no—
211C61#908711yescharacteristic, complemented, Sylow
212C22×Q8—611no—
213C22×D4—611no—
214C23×C4—1221no—
215C2×SD16—614no—
216C2×SD16—1222no—
217C2×SD16—1222no—
218C22×C8—1224no—
219C22×C8C4×D6111yescharacteristic
220C22×C8C4×D6111yescharacteristic
221C4×Q8—1221no—
222C4×Q8—1221no—
223C4×Q8—1221no—
224C4×Q8—1221no—
225C4×D4—612no—
226C4×D4—612no—
227C4×D4—1221no—
228C4×D4—1221no—
229C2×C4⋊C4—611no—
230C2×C4⋊C4—611no—
231C2×C4⋊C4—1221no—
232C2×C22⋊C4—1221no—
233C2×C42—1221no—
234C2×C42C22×D6111yescharacteristic
235#107—1222no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—1221no—
239C4⋊C8—1221no—
240#103—1222no—
241#102—1222no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—612no—
245C22⋊C8—1224no—
246C22⋊C8—1221no—
247#97—21no—
248#97—21no—
249C4×C8—21no—
250C4×C8—1221no—
251C4×C8—1221no—
252C4×C8D24412yes—
253C24—1221no—
254C2×Q8—612no—
255C2×Q8—1221no—
256C2×Q8—1221no—
257C2×Q8—1221no—
258C2×Q8—1221no—
259C2×Q8—1221no—
260C2×D4—612no—
261C2×D4—612no—
262C2×D4—612no—
263C2×D4—1221no—
264C2×D4—1221no—
265C2×D4—1221no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4—1222no—
269C22×C4—1222no—
270C22×C4—1221no—
271C22×C4—4882no—
272C22×C4C2×C4×D6111yescharacteristic
273C22×C4D4×D6111yescharacteristic
274C22×C4#1175822611yescharacteristic
275SD16—1228no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—1222no—
284C2×C8—4884no—
285C2×C8—2441no—
286C2×C8C8×D6112yescomplemented
287C4⋊C4—612no—
288C4⋊C4—1221no—
289C4⋊C4—1221no—
290C4⋊C4—1221no—
291C4⋊C4—2441no—
292C4⋊C4—2441no—
293C22⋊C4—1222no—
294C22⋊C4—1222no—
295C42—2441no—
296C42—2441no—
297C42C2×D24411yescharacteristic
298C42#1175822211yescharacteristic
299C42D4×D6111yescharacteristic
300C42D4×D6111yescharacteristic
301C23—1222no—
302C23—1222no—
303C23—1222no—
304C23—4881no—
305C23—4881no—
306C23#1451973911yescharacteristic
307Q8—1222no—
308Q8—1222no—
309Q8—2441no—
310Q8—2441no—
311D4—614no—
312D4—1222no—
313D4—1222no—
314D4—1222no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—1222no—
318C2×C4—1222no—
319C2×C4—4882no—
320C2×C4—4881no—
321C2×C4—4881no—
322C2×C4—2444no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4C22⋊C4×D6111yescharacteristic
327C2×C4C4×D24411yescharacteristic
328C2×C4#1451974711yescharacteristic
329C2×C4#1451975011yescharacteristic
330C2×C4#1451975211yescharacteristic
331C2×C4#1451976211yescharacteristic
332C2×C4C2×C8×D6111yescharacteristic
333C2×C4M16×D6111yescharacteristic
334C2×C4#1451978311yescharacteristic
335C2×C4#1451978511yescharacteristic
336C8—22no—
337C8—42no—
338C8—41no—
339C8—81no—
340C8—4881no—
341C8—2442no—
342C22—21no—
343C22—21no—
344C22—1224no—
345C22—4882no—
346C22—4882no—
347C22—2442no—
348C22—2442no—
349C22#1683964011yescharacteristic
350C22#1683982711yescharacteristic
351C22#1683983211yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—4881no—
357C4—2442no—
358C4—2442no—
359C4C22⋊C8×D6111yescharacteristic
360C4#1683972711yescharacteristic
361C4C8×D24411yescharacteristic
362C4#1683983511yescharacteristic
363C2—21no—
364C2—21no—
365C2—2444no—
366C2#2144611011yescharacteristic
367C2#2144629311yescharacteristic
368C2#2144630711yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC2×C4×C61C22×C8
FrattiniC2×C42C22×D61
FittingC61×#1275C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1275D61
hypocenterC61#9087
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
3. #21444215C2semidirect
5. #21444142C2semidirect
6. #21443664C2semidirect
7. C61×#1275C2semidirect
32. C2×#11758190C8semidirect
89. C2×Dic122C2×C8semidirect
97. C2×C8×C61C2×C8semidirect
185. #1275D61semidirect
211. C61#9087semidirect
286. C2×C8C8×D61semidirect

Maximal subgroups

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

subgroupindex
#214548422
C2×#168398642
#214442152
#214442092
#214441422
#214436642
C61×#12752
#908761