Finite Group Databasebeta

← #29577709

Subgroups — #29577709

Order 15616.

554 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 95 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#21454847C211yesmaximal, characteristic
2#21450773C211yesmaximal, characteristic
3#21444536C211yesmaximal, characteristic, complemented
4#21444159C211yesmaximal, characteristic
5#21443676C211yesmaximal, characteristic, complemented
6#21443674C211yesmaximal, characteristic, complemented
7C61×#1556C211yesmaximal, characteristic, complemented
8C2×#14519753—21no—
9C2×#14519753—21no—
10C2×C4×D244C2211yescharacteristic
11#16839835—22no—
12#16839832—22no—
13#16839819C2211yescharacteristic
14#16839819C2211yescharacteristic
15#16839724—21no—
16C22⋊C8×D61—21no—
17#16839697—21no—
18#16839697—21no—
19#16839663C412yes—
20#16839631C412yes—
21C4×C4⋊Dic61C2211yescharacteristic
22#16839508—22no—
23#16839485—21no—
24#16839485—21no—
25C2×C4⋊Q8×C61C2211yescharacteristic
26C61×#432C2211yescharacteristic
27C61×#432C2211yescharacteristic
28C61×#332—22no—
29C22×D244C2×C411yescharacteristic
30C22×C4×D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C2×#11758189—22no—
33C2×C8×D61—22no—
34#14519753—44no—
35#14519753—44no—
36C4×D244—21no—
37C4×D244—21no—
38C4×D244—21no—
39C4×D244—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×#11758196—22no—
43C2×#11758196—22no—
44C2×#11758198—22no—
45C2×#11758198—22no—
46C2×#11758198D411yescharacteristic
47C2×#11758198D411yescharacteristic
48C2×C4×Dic61C2×C412yes—
49C4⋊#92418—21no—
50C4⋊#92418D412yes—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53#14519689—22no—
54#14519689—22no—
55#14519688—21no—
56#14519688—21no—
57#14519684—21no—
58C22×Q8×C61—21no—
59C4⋊Q8×C61—21no—
60C4⋊Q8×C61—21no—
61C4⋊Q8×C61—21no—
62C4⋊Q8×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C4⋊C4×C61C2×C412yes—
65C2×C42×C61C2311yescharacteristic
66C4⋊C8×C61—21no—
67C4⋊C8×C61D412yes—
68C22⋊C8×C61—21no—
69C22⋊C8×C61—21no—
70#97×C61—21no—
71#97×C61—21no—
72C23×D61C22⋊C411yescharacteristic
73C23×D61C22⋊C411yescharacteristic
74C2×Q8×C61—42no—
75C2×Q8×C61—42no—
76C2×Q8×C61—41no—
77C2×Q8×C61—41no—
78C22×C4×C61—22no—
79C22×C4×C61C22×C411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×Dic61—22no—
83C22×Dic61C22⋊C411yescharacteristic
84C22×Dic61C22⋊C411yescharacteristic
85C2×D244—22no—
86C2×D244—21no—
87C2×D244—21no—
88C2×D244—41no—
89C2×C4×D61—21no—
90C2×C4×D61—21no—
91C2×C4×D61—21no—
92C2×C4×D61—21no—
93C2×C4×D61—42no—
94C2×C4×D61—42no—
95C2×C8×C61—22no—
96C2×C8×C61—41no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61—41no—
100C4⋊C4×C61—41no—
101C42×C61C2×D411yescharacteristic
102C42×C61C2×D411yescharacteristic
103C42×C61#4111yescharacteristic
104C42×C61#4111yescharacteristic
105#11758198—21no—
106#11758198—21no—
107#11758198—21no—
108#11758198—21no—
109#11758198—44no—
110#11758198—44no—
111C4⋊Dic61—24no—
112C4⋊Dic61—21no—
113C4⋊Dic61—21no—
114#11758196—44no—
115#11758196—44no—
116C4×Dic61—42no—
117C2×#92418—22no—
118C2×#92418—41no—
119#11758189—82no—
120C8×D61—82no—
121C23×C61C2×C22⋊C411yescharacteristic
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—21no—
125C22×D61—21no—
126C22×D61—21no—
127C22×D61—41no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61#10112yes—
132Q8×C61—82no—
133Q8×C61—82no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—22no—
137C2×C4×C61—41no—
138C2×C4×C61—42no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61C4×D411yescharacteristic
142C2×C4×C61C4×D411yescharacteristic
143C2×C4×C61#12011yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12111yescharacteristic
146C2×C4×C61#12311yescharacteristic
147C2×Dic61—24no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154D244—42no—
155D244—42no—
156C4×D61—42no—
157C4×D61—42no—
158C4×D61—81no—
159C4×D61—81no—
160C8×C61—42no—
161C8×C61—81no—
162#92418—42no—
163#92418—81no—
164#9455—611nomaximal, Sylow
165C22×C61—21no—
166C22×C61—21no—
167C22×C61#38611yescharacteristic
168C22×C61C2×#10111yescharacteristic
169C22×C61#41311yescharacteristic
170C2×D61—22no—
171C2×D61—41no—
172C2×D61—41no—
173C2×D61—42no—
174C2×D61—42no—
175C2×D61—41no—
176C2×D61—41no—
177C2×D61—41no—
178C2×D61—81no—
179C2×D61—81no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—21no—
185C4×C61—82no—
186C4×C61#45312yes—
187Dic61—44no—
188Dic61—41no—
189Dic61—41no—
190#2973—611no—
191#2542—611no—
192#1556—611no—
193#1556D6111yescharacteristic, complemented
194#1466—611no—
195#1285—611no—
196#1285—611no—
197C2×C61—21no—
198C2×C61—21no—
199C2×C61#188711yescharacteristic
200C2×C61#190111yescharacteristic
201C2×C61#190911yescharacteristic
202D61—42no—
203D61—81no—
204C2×C4⋊Q8C2×D6111yescharacteristic
205C2×C22⋊Q8—1221no—
206C2×C22⋊Q8—1221no—
207C2×C4×D4—611no—
208#435—1222no—
209#435—1222no—
210#432—611no—
211#432—611no—
212#432C2×D6111yescharacteristic
213#432C2×D6111yescharacteristic
214#407—1221no—
215C2×C22⋊C8—1221no—
216#399—1221no—
217#399—1221no—
218#390—612no—
219#384—612no—
220C4×C4⋊C4—611no—
221#332—22no—
222#332—1222no—
223#323—1221no—
224#323—1221no—
225C61#945511yescharacteristic, complemented, Sylow
226C22×Q8—21no—
227C22×D4—611no—
228C23×C4—611no—
229C23×C4—611no—
230C2×M16—1222no—
231C22×C8—1222no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C22⋊Q8—2444no—
237C22⋊Q8—2444no—
238C4×D4—1221no—
239C4×D4—1221no—
240C4×D4—1221no—
241C4×D4—1221no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—611no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4—1222no—
246C2×C4⋊C4—1222no—
247C2×C4⋊C4C4×D6112yes—
248C2×C22⋊C4—611no—
249C2×C22⋊C4—611no—
250C2×C22⋊C4—1222no—
251C2×C22⋊C4—1222no—
252C2×C42—612no—
253C2×C42C22×D6111yescharacteristic
254C4⋊C8—21no—
255C4⋊C8—612no—
256C4⋊C8—1221no—
257C4⋊C8C22⋊D6112yes—
258C22⋊C8—21no—
259C22⋊C8—21no—
260C22⋊C8—1222no—
261C22⋊C8—1222no—
262C22⋊C8—1221no—
263C22⋊C8—1221no—
264#97—21no—
265#97—21no—
266#97—1221no—
267#97—1221no—
268#95—1221no—
269C24—611no—
270C24—611no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×D4—1222no—
276C2×D4—1221no—
277C2×D4—1221no—
278C2×D4—2441no—
279C22×C4—22no—
280C22×C4—611no—
281C22×C4—611no—
282C22×C4—1222no—
283C22×C4—1221no—
284C22×C4—1221no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—2442no—
288C22×C4—2442no—
289C22×C4C2×C4×D6111yescharacteristic
290C22×C4D4×D6111yescharacteristic
291C22×C4D4×D6111yescharacteristic
292M16—4882no—
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—1222no—
296C2×C8—4882no—
297C2×C8—2441no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—1224no—
303C4⋊C4—1221no—
304C4⋊C4—1221no—
305C4⋊C4—2444no—
306C4⋊C4—2444no—
307C22⋊C4—1221no—
308C22⋊C4—1221no—
309C22⋊C4—1221no—
310C22⋊C4—1221no—
311C22⋊C4—2444no—
312C22⋊C4—2444no—
313C42—2442no—
314C42#1175822211yescharacteristic
315C42D4×D6111yescharacteristic
316C42#1175822411yescharacteristic
317C42C2×C22⋊D6111yescharacteristic
318C23—612no—
319C23—1221no—
320C23—1221no—
321C23—1221no—
322C23—1221no—
323C23—1221no—
324C23—2441no—
325C23—2441no—
326C23—2441no—
327C23—2441no—
328C23C22⋊C4×D6111yescharacteristic
329Q8—82no—
330Q8—82no—
331D4—2442no—
332D4—2442no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—1224no—
341C2×C4—1221no—
342C2×C4—1221no—
343C2×C4—4881no—
344C2×C4—4881no—
345C2×C4—2442no—
346C2×C4—2442no—
347C2×C4—2442no—
348C2×C4—2442no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4#1451973711yescharacteristic
352C2×C4C4×C22⋊D6111yescharacteristic
353C2×C4#1451974511yescharacteristic
354C2×C4#1451974811yescharacteristic
355C2×C4#1451975011yescharacteristic
356C2×C4#1451975911yescharacteristic
357C8—42no—
358C8—81no—
359C8—4881no—
360C8—2442no—
361C22—21no—
362C22—21no—
363C22—1222no—
364C22—4881no—
365C22—4881no—
366C22—2442no—
367C22—2442no—
368C22—2441no—
369C22—2441no—
370C22—2441no—
371C22—2441no—
372C22—2441no—
373C22#1683963811yescharacteristic
374C22#101×D6111yescharacteristic
375C22#1683974511yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—82no—
382C4—2444no—
383C4—2441no—
384C4—2441no—
385C4#1683992812yes—
386C2—21no—
387C2—21no—
388C2—4881no—
389C2—2442no—
390C2#2144621411yescharacteristic
391C2#2144628411yescharacteristic
392C2#2144633811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1556C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1556D61
hypocenterC61#9455
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
3. #21444536C2semidirect
5. #21443676C2semidirect
6. #21443674C2semidirect
7. C61×#1556C2semidirect
193. #1556D61semidirect
225. C61#9455semidirect

Maximal subgroups

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

subgroupindex
#214548472
#214507732
#214445362
#214441592
#214436762
#214436742
C61×#15562
#945561