Finite Group Databasebeta

← #29577971

Subgroups — #29577971

Order 15616.

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

normal subgroups only

This group has 544 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#21454846C211yesmaximal, characteristic
2#21450128C211yesmaximal, characteristic
3#21445897C211yesmaximal, characteristic, complemented
4#21445897C211yesmaximal, characteristic, complemented
5#21444155C211yesmaximal, characteristic
6#21443725C211yesmaximal, characteristic, complemented
7C61×#1305C211yesmaximal, characteristic, complemented
8C2×#14519761C412yes—
9C2×C4×C22⋊D61C2211yescharacteristic
10C22×C4⋊Dic61C2211yescharacteristic
11#16839836—22no—
12#16839831—22no—
13#16839816—21no—
14#16839726C2211yescharacteristic
15#16839726C2211yescharacteristic
16C22⋊C8×D61—21no—
17#16839694—21no—
18#16839689—21no—
19#16839661—21no—
20#16839623C412yes—
21#16839619—21no—
22#16839530—22no—
23#16839530—22no—
24#16839487—21no—
25#16839485—21no—
26C61×#407C2211yescharacteristic
27C61×#407C2211yescharacteristic
28C61×#380C2211yescharacteristic
29C22×C22⋊D61C2×C411yescharacteristic
30C23×Dic61C2×C412yes—
31C22×C4×D61D411yescharacteristic
32C2×#11758194—21no—
33C2×#11758194D412yes—
34C2×C8×D61—22no—
35#14519761—44no—
36C4×C22⋊D61—21no—
37C4×C22⋊D61—21no—
38C4×C22⋊D61—21no—
39C4×C22⋊D61—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61—22no—
47C2×#11758196D411yescharacteristic
48C2×C22⋊Dic61C2×C411yescharacteristic
49C2×#11758198—22no—
50C2×#11758198D411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52#14519706—22no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57#14519689—22no—
58#14519687—22no—
59#14519684—22no—
60#14519684—22no—
61C23×C4×C61C2311yescharacteristic
62C2×M16×C61—21no—
63C2×M16×C61D412yes—
64C2×C4⋊C4×C61—21no—
65C2×C22⋊C4×C61C2×C412yes—
66C22⋊C8×C61—21no—
67C22⋊C8×C61—21no—
68C22⋊C8×C61—21no—
69C22⋊C8×C61—21no—
70#95×C61—21no—
71C24×C61C22×C411yescharacteristic
72C23×D61C22⋊C411yescharacteristic
73C22×C4×C61—22no—
74C22×C4×C61—42no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61#4111yescharacteristic
80C22×C4×C61#4111yescharacteristic
81C2×C22⋊D61—21no—
82C2×C22⋊D61—21no—
83C2×C22⋊D61—22no—
84C2×C22⋊D61—41no—
85C22×Dic61—22no—
86C22×Dic61—22no—
87C22×Dic61—22no—
88C22×Dic61—42no—
89C22×Dic61—42no—
90C22×Dic61C22⋊C411yescharacteristic
91C22×Dic61C22⋊C411yescharacteristic
92C22×Dic61C22⋊C411yescharacteristic
93C2×C4×D61—21no—
94C2×C4×D61—21no—
95C2×C4×D61—42no—
96M16×C61—42no—
97M16×C61—42no—
98C2×C8×C61—22no—
99C2×C8×C61—41no—
100C4⋊C4×C61—42no—
101C22⋊C4×C61—42no—
102C22⋊Dic61—21no—
103C22⋊Dic61—21no—
104#11758198—21no—
105#11758198—21no—
106#11758198—44no—
107C4⋊Dic61—44no—
108C4⋊Dic61—41no—
109C4⋊Dic61—41no—
110C4⋊Dic61—41no—
111C4⋊Dic61—41no—
112#11758196—21no—
113#11758196—21no—
114C4×Dic61—21no—
115C4×Dic61—21no—
116#11758194—42no—
117#11758194—42no—
118C2×#92418—22no—
119C2×#92418—41no—
120C8×D61—82no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61C2×C22⋊C411yescharacteristic
126C23×C61C4×D411yescharacteristic
127C23×C61C4×D411yescharacteristic
128C22×D61—21no—
129C22×D61—21no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61#10012yes—
133C2×C4×C61—22no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61#12011yescharacteristic
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12311yescharacteristic
146C22⋊D61—42no—
147C22⋊D61—42no—
148C2×Dic61—21no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—42no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61—82no—
159C2×Dic61—82no—
160C4×D61—42no—
161C4×D61—81no—
162C8×C61—42no—
163C8×C61—81no—
164#92418—42no—
165#92418—81no—
166#9507—611nomaximal, Sylow
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—42no—
176C22×C61—42no—
177C22×C61#38611yescharacteristic
178C22×C61C2×#10011yescharacteristic
179C22×C61#41311yescharacteristic
180C2×D61—22no—
181C2×D61—41no—
182C2×D61—41no—
183C2×D61—42no—
184C2×D61—81no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—82no—
189C4×C61#45812yes—
190Dic61—41no—
191Dic61—41no—
192Dic61—41no—
193Dic61—82no—
194#2973—611no—
195#2486—611no—
196#1821—611no—
197#1821—611no—
198#1465—611no—
199#1305—611no—
200#1305D6111yescharacteristic, complemented
201C2×C61—21no—
202C2×C61—21no—
203C2×C61—41no—
204C2×C61—41no—
205C2×C61#189011yescharacteristic
206C2×C61#190011yescharacteristic
207C2×C61#191511yescharacteristic
208D61—42no—
209C2×#123—612no—
210C2×C4×D4—611no—
211C22×C4⋊C4—611no—
212#435—1222no—
213#435—1222no—
214#432—1221no—
215#407—611no—
216#407—611no—
217#407C2×D6111yescharacteristic
218#407C2×D6111yescharacteristic
219C2×C22⋊C8—1221no—
220#398—1221no—
221#397—1221no—
222#389—1221no—
223#381—612no—
224#380—1221no—
225#380C2×D6111yescharacteristic
226#343—1222no—
227#343—1222no—
228#324—1221no—
229#323—1221no—
230C61#950711yescharacteristic, complemented, Sylow
231C22×D4—611no—
232C23×C4—611no—
233C23×C4—612no—
234C23×C4C22×D6111yescharacteristic
235C2×M16—21no—
236C2×M16—612no—
237C2×M16—1221no—
238C2×M16D24412yes—
239C22×C8—1222no—
240#123—2444no—
241C4×D4—1221no—
242C4×D4—1221no—
243C4×D4—1221no—
244C4×D4—1221no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—611no—
247C2×C4⋊C4—1222no—
248C2×C4⋊C4—1221no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4—1221no—
253C2×C4⋊C4—1221no—
254C2×C22⋊C4—611no—
255C2×C22⋊C4—611no—
256C2×C22⋊C4—1222no—
257C2×C22⋊C4C4×D6112yes—
258C2×C42—611no—
259C4⋊C8—1222no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—1222no—
265C22⋊C8—1221no—
266C22⋊C8—1221no—
267C22⋊C8—1221no—
268C22⋊C8—1221no—
269#97—1222no—
270#95—21no—
271#95—1222no—
272#95—1222no—
273C24—611no—
274C24C2×C4×D6111yescharacteristic
275C2×D4—1222no—
276C2×D4—1221no—
277C2×D4—1221no—
278C2×D4—2441no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—611no—
282C22×C4—611no—
283C22×C4—611no—
284C22×C4—1221no—
285C22×C4—1221no—
286C22×C4—1222no—
287C22×C4—1222no—
288C22×C4—1222no—
289C22×C4—2442no—
290C22×C4—2442no—
291C22×C4—2442no—
292C22×C4C2×D24411yescharacteristic
293C22×C4#1175822211yescharacteristic
294C22×C4D4×D6111yescharacteristic
295C22×C4D4×D6111yescharacteristic
296C22×C4D4×D6111yescharacteristic
297C22×C4#1175822611yescharacteristic
298M16—42no—
299M16—42no—
300M16—2442no—
301M16—2442no—
302C2×C8—22no—
303C2×C8—41no—
304C2×C8—1222no—
305C2×C8—4882no—
306C2×C8—2441no—
307C4⋊C4—42no—
308C4⋊C4—1221no—
309C4⋊C4—1221no—
310C4⋊C4—2444no—
311C4⋊C4—2441no—
312C4⋊C4—2441no—
313C4⋊C4—2441no—
314C4⋊C4—2441no—
315C22⋊C4—42no—
316C22⋊C4—1221no—
317C22⋊C4—1221no—
318C22⋊C4—1221no—
319C22⋊C4—1221no—
320C22⋊C4—2444no—
321C42—1221no—
322C42—1221no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—41no—
327C23—612no—
328C23—1221no—
329C23—1221no—
330C23—2441no—
331C23—2441no—
332C23C22⋊C4×D6111yescharacteristic
333C23#1451973911yescharacteristic
334C23C4×D24411yescharacteristic
335D4—2442no—
336D4—2442no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—1221no—
347C2×C4—1221no—
348C2×C4—1221no—
349C2×C4—1221no—
350C2×C4—1221no—
351C2×C4—4881no—
352C2×C4—4882no—
353C2×C4—4882no—
354C2×C4—2442no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2442no—
358C2×C4—2441no—
359C2×C4—2441no—
360C2×C4#1451974711yescharacteristic
361C2×C4#1451975011yescharacteristic
362C2×C4#1451975211yescharacteristic
363C2×C4#1451976211yescharacteristic
364C8—42no—
365C8—81no—
366C8—4881no—
367C8—2442no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—42no—
377C22—42no—
378C22—1222no—
379C22—4881no—
380C22—2442no—
381C22—2441no—
382C22—2441no—
383C22#1683964011yescharacteristic
384C22#100×D6111yescharacteristic
385C22#1683974511yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—82no—
390C4—4882no—
391C4—2441no—
392C4—2441no—
393C4—2441no—
394C4#1683995312yes—
395C2—21no—
396C2—21no—
397C2—41no—
398C2—41no—
399C2—2442no—
400C2#2144623111yescharacteristic
401C2#2144628111yescharacteristic
402C2#2144637011yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1305C2
socleC22×C61#386
prime radicalC22×C22⋊D61C2×C4
cosocleC23×C4×C61C23
hypercenter#1305D61
hypocenterC61#9507
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
3. #21445897C2semidirect
4. #21445897C2semidirect
6. #21443725C2semidirect
7. C61×#1305C2semidirect
200. #1305D61semidirect
230. C61#9507semidirect

Maximal subgroups

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

subgroupindex
#214548462
#214501282
#214458972
#214441552
#214437252
C61×#13052
#950761