Finite Group Databasebeta

← #29596850

Subgroups — #29596850

Order 15616.

460 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448907C211yesmaximal, characteristic
2C2×#16839560C211yesmaximal, characteristic
3#21446212C211yesmaximal, characteristic
4#21446034C211yesmaximal, characteristic
5#21443946C211yesmaximal, characteristic
6#21443945C211yesmaximal, characteristic
7C61×#1857C211yesmaximal, characteristic
8C2×#14519775C2211yescharacteristic
9#16839805—21no—
10#16839770—21no—
11C2×#14519696C2211yescharacteristic
12C2×#14519696C2211yescharacteristic
13#16839669C2211yescharacteristic
14#16839666—21no—
15#16839659—21no—
16#16839657—21no—
17C4×#11758196—21no—
18#16839613—21no—
19#16839613—21no—
20#16839560C414yescomplemented
21#16839521—21no—
22#16839521—21no—
23C61×#519C2211yescharacteristic
24C61×#428—21no—
25C2×#101×C61C2211yescharacteristic
26C61×#391C2211yescharacteristic
27C61×#339—21no—
28C22×Dic122D411yescharacteristic
29C2×#11758194—21no—
30C2×#11758194—21no—
31#14519775—21no—
32#14519775—21no—
33#14519775C2×C412yes—
34#14519775Q812yes—
35C2×C4⋊Dic61D411yescharacteristic
36C2×#11758196—21no—
37C2×#11758196—21no—
38C2×#11758196—21no—
39C2×#11758196—21no—
40C2×#11758196—21no—
41C2×C4×Dic61—21no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—21no—
44C2×C4×Dic61D411yescharacteristic
45C2×C4×Dic61D411yescharacteristic
46#14519710—41no—
47#14519708—41no—
48#14519700—41no—
49#14519700—41no—
50#14519696—21no—
51#14519696—21no—
52#14519696—21no—
53#14519696C2×C412yes—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58C22×Q8×C61C2311yescharacteristic
59C2×M16×C61—21no—
60C4×Q8×C61—41no—
61C4×Q8×C61—41no—
62#117×C61—21no—
63#117×C61—21no—
64#117×C61—21no—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C42×C61D411yescharacteristic
67#107×C61—41no—
68#106×C61—41no—
69#101×C61—21no—
70#101×C61C2×C412yes—
71#95×C61—21no—
72#95×C61—21no—
73C2×Q8×C61—41no—
74C2×Q8×C61—41no—
75C2×Q8×C61C22×C411yescharacteristic
76C2×Q8×C61C2×Q811yescharacteristic
77C2×Q8×C61#4111yescharacteristic
78C2×Q8×C61#4111yescharacteristic
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C2×Dic122—21no—
89C2×Dic122—41no—
90C2×Dic122—41no—
91C2×Dic122C4⋊C412yes—
92M16×C61—22no—
93M16×C61—41no—
94C2×C8×C61—41no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—41no—
102C22⋊C4×C61—41no—
103C42×C61—41no—
104C42×C61—41no—
105C42×C61—41no—
106C42×C61—41no—
107C4⋊Dic61—41no—
108#11758196—24no—
109#11758196—22no—
110#11758196—22no—
111#11758196—22no—
112#11758196—22no—
113#11758196—42no—
114#11758196—81no—
115C4×Dic61—21no—
116C4×Dic61—42no—
117C4×Dic61—42no—
118C4×Dic61—41no—
119C4×Dic61—81no—
120C4×Dic61C4⋊C412yes—
121#11758194—22no—
122#11758194—41no—
123#11758194—41no—
124#11758194—41no—
125C2×#92418—41no—
126C2×#92418—41no—
127C23×C61#12011yescharacteristic
128Q8×C61—41no—
129Q8×C61—41no—
130Q8×C61—41no—
131Q8×C61—41no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—22no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61C2×C4⋊C411yescharacteristic
147C2×C4×C61C4×D411yescharacteristic
148C2×C4×C61C4×D411yescharacteristic
149C2×C4×C61C22⋊Q811yescharacteristic
150C2×C4×C61C22⋊Q811yescharacteristic
151C2×C4×C61#12311yescharacteristic
152C2×Dic61—22no—
153C2×Dic61—22no—
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—22no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164Dic122—42no—
165Dic122—81no—
166C8×C61—42no—
167#92418—42no—
168#92418—81no—
169#12973—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61#38511yescharacteristic
173C22×C61B(4,𝔽2)11yescharacteristic
174C22×C61#45811yescharacteristic
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—42no—
182C4×C61—81no—
183C4×C61—81no—
184C4×C61—81no—
185Dic61—24no—
186Dic61—42no—
187Dic61—42no—
188Dic61—81no—
189#2369—611no—
190C2×#356—611no—
191#1886—611no—
192#1857—611no—
193#1857D6111yescharacteristic
194#1394—611no—
195#1394—611no—
196C2×C61—21no—
197C2×C61—21no—
198C2×C61#186711yescharacteristic
199C2×C61#220411yescharacteristic
200C2×C61#220511yescharacteristic
201C2×C4⋊Q8—611no—
202#519C2×D6111yescharacteristic
203#428—21no—
204#428—1221no—
205#419—1221no—
206C2×#101—611no—
207C2×#101—611no—
208C2×#101C2×D6111yescharacteristic
209#391—611no—
210#391—1221no—
211#391C2×D6111yescharacteristic
212#389—1221no—
213#389—1221no—
214C4×C4⋊C4—1221no—
215C4×C4⋊C4—1221no—
216C4×C4⋊C4—1221no—
217#356—614no—
218#339—21no—
219#339—1221no—
220#339—1221no—
221C61#1297311yescharacteristic, complemented, Sylow
222C22×Q8—611no—
223C22×Q8C22×D6111yescharacteristic
224C2×M16—21no—
225C2×M16—1221no—
226C2×M16—1221no—
227C4⋊Q8—612no—
228C4⋊Q8—612no—
229C4⋊Q8—1221no—
230C4⋊Q8—1221no—
231C4×Q8—41no—
232C4×Q8—41no—
233#117—21no—
234#117—21no—
235#117—21no—
236C2×C4⋊C4—611no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—1221no—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4—1221no—
242C2×C4⋊C4C22⋊D6111yescharacteristic
243C2×C42—611no—
244C2×C42—611no—
245C2×C42—1221no—
246C2×C42—1221no—
247C2×C42—1221no—
248C2×C42C22⋊D6111yescharacteristic
249#107—41no—
250#107—2441no—
251#106—41no—
252#106—2441no—
253#103—2441no—
254#103—2441no—
255#101—21no—
256#101—612no—
257#101—1221no—
258#101—1221no—
259#101—1221no—
260#101C4×D6112yes—
261#95—21no—
262#95—21no—
263#95—1221no—
264#95—1221no—
265#95—1221no—
266#95—1221no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—612no—
270C2×Q8—1221no—
271C2×Q8—2441no—
272C2×Q8—2441no—
273C2×Q8C2×C4×D6111yescharacteristic
274C2×Q8#1175822211yescharacteristic
275C2×Q8Q8×D6111yescharacteristic
276C2×Q8#1175822611yescharacteristic
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4—1221no—
282C22×C4—1221no—
283C22×C4D4×D6111yescharacteristic
284C22×C4D4×D6111yescharacteristic
285C22×C4C2×C22⋊D6111yescharacteristic
286M16—22no—
287M16—41no—
288M16—1222no—
289M16—2441no—
290M16—2441no—
291M16—2441no—
292C2×C8—41no—
293C2×C8—2441no—
294C2×C8—2441no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—1224no—
303C4⋊C4—1222no—
304C4⋊C4—1222no—
305C4⋊C4—1222no—
306C4⋊C4—1222no—
307C4⋊C4—4881no—
308C4⋊C4—2441no—
309C4⋊C4—2442no—
310C22⋊C4—41no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42—612no—
316C42—1221no—
317C42—4881no—
318C42—2441no—
319C42—2442no—
320C42—2442no—
321C23#1451974511yescharacteristic
322Q8—41no—
323Q8—41no—
324Q8—41no—
325Q8—41no—
326Q8—4881no—
327Q8—2442no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—1222no—
343C2×C4—1222no—
344C2×C4—1222no—
345C2×C4—1222no—
346C2×C4—1222no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4C4⋊C4×D6111yescharacteristic
355C2×C4C4×C22⋊D6111yescharacteristic
356C2×C4#1451973911yescharacteristic
357C2×C4#1451975311yescharacteristic
358C2×C4#1451975511yescharacteristic
359C2×C4#1451976211yescharacteristic
360C8—42no—
361C8—4881no—
362C8—2442no—
363C22—21no—
364C22—21no—
365C22#1683963311yescharacteristic
366C22#1683994811yescharacteristic
367C22#1683995111yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—1224no—
379C4—4881no—
380C4—2442no—
381C4—2442no—
382C2—21no—
383C2—21no—
384C2#2144609811yescharacteristic
385C2#2144784311yescharacteristic
386C2#2144784811yescharacteristic
subgroupG/H
centerC22#16839948
commutatorC2×Q8×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#1857C2
socleC22×C61B(4,𝔽2)
prime radicalC23×C61#120
cosocleC22×Q8×C61C23
hypercenter#1857D61
hypocenterC61#12973
LCSG ▷ C2×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1857

Complements

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

NH ≅ G/Ntype
20. #16839560C4semidirect
221. C61#12973semidirect

Maximal subgroups

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

subgroupindex
#214489072
C2×#168395602
#214462122
#214460342
#214439462
#214439452
C61×#18572
#1297361