Finite Group Databasebeta

← #29597371

Subgroups — #29597371

Order 15616.

668 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 103 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#21448909C211yesmaximal, characteristic, complemented
2C2×#16839561C211yesmaximal, characteristic, complemented
3#21445778C212yesmaximal
4#21443945C212yesmaximal, complemented
5C2×C61×#456C211yesmaximal, characteristic
6C2×#14519704—22no—
7#16839763—22no—
8C2×#14519696C2212yes—
9#16839669—21no—
10#16839669C2212yes—
11#16839661—22no—
12C4⋊C4×Dic61—22no—
13C4×C4⋊Dic61—21no—
14#16839561—22no—
15C2×2−1+4×C61C2211yescharacteristic
16C2×#137×C61—21no—
17C2×C4⋊Q8×C61C2211yescharacteristic
18C61×#456—22no—
19C61×#456C414yescomplemented
20C2×#104×C61—21no—
21C2×#101×C61C2211yescharacteristic
22C2×#11758194—22no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61D412yes—
27C2×C4×Dic61—22no—
28C2×C4×Dic61—21no—
29C2×C4×Dic61D412yes—
30#14519704—44no—
31#14519701—42no—
32#14519698—42no—
33#14519696—22no—
34#14519696—22no—
35#14519684—22no—
36#14519684—22no—
37#14519684—22no—
382−1+4×C61—21no—
392−1+4×C61—22no—
402−1+4×C61—42no—
412−1+4×C61C2×C412yes—
42C2×#41×C61—21no—
43C2×#41×C61—22no—
44C2×#41×C61—41no—
45C22×Q8×C61—41no—
46C22×Q8×C61D411yescharacteristic
47C22×Q8×C61C2311yescharacteristic
48#137×C61—24no—
49#137×C61—42no—
50C2×Q16×C61—41no—
51C2×SD16×C61—41no—
52C2×M16×C61—21no—
53C4⋊Q8×C61—22no—
54C4⋊Q8×C61C2×C412yes—
55C4⋊Q8×C61D412yes—
56C2×C4⋊C4×C61—22no—
57C2×C42×C61D411yescharacteristic
58#104×C61—24no—
59#101×C61—21no—
60#101×C61C2×C412yes—
61#41×C61—22no—
62#41×C61—24no—
63#41×C61—22no—
64#41×C61—24no—
65#41×C61—42no—
66#41×C61—41no—
67#41×C61—42no—
68#41×C61—42no—
69#41×C61—44no—
70#41×C61—42no—
71#41×C61—41no—
72C2×Q8×C61—21no—
73C2×Q8×C61—21no—
74C2×Q8×C61—22no—
75C2×Q8×C61—41no—
76C2×Q8×C61—41no—
77C2×Q8×C61—41no—
78C2×Q8×C61—41no—
79C2×Q8×C61—42no—
80C2×Q8×C61—42no—
81C2×Q8×C61—44no—
82C2×Q8×C61C22⋊C412yes—
83C2×Q8×C61C22×C411yescharacteristic
84C2×Q8×C61C2×D411yescharacteristic
85C2×Q8×C61#4112yes—
86C2×D4×C61—21no—
87C2×D4×C61—22no—
88C2×D4×C61—41no—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—42no—
92C22×C4×C61—41no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D412yes—
95C22×Dic61—22no—
96C22×Dic61—22no—
97Q16×C61—44no—
98SD16×C61—44no—
99M16×C61—22no—
100M16×C61—41no—
101C2×C8×C61—41no—
102C4⋊C4×C61—24no—
103C4⋊C4×C61—24no—
104C42×C61—21no—
105C42×C61C22⋊C412yes—
106C4⋊Dic61—24no—
107C4⋊Dic61—42no—
108C4⋊Dic61—42no—
109C4⋊Dic61—81no—
110C4×Dic61—44no—
111C4×Dic61—42no—
112C4×Dic61—81no—
113#11758194—42no—
114#11758194—42no—
115C2×#92418—42no—
116C23×C61—41no—
117C23×C61#12011yescharacteristic
118Q8×C61—22no—
119Q8×C61—24no—
120Q8×C61—41no—
121Q8×C61—41no—
122Q8×C61—42no—
123Q8×C61—42no—
124Q8×C61—41no—
125Q8×C61—42no—
126Q8×C61—81no—
127D4×C61—22no—
128D4×C61—24no—
129D4×C61—42no—
130D4×C61—41no—
131D4×C61—42no—
132D4×C61—42no—
133C2×C4×C61—22no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—22no—
141C2×C4×C61—44no—
142C2×C4×C61—42no—
143C2×C4×C61—44no—
144C2×C4×C61—42no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61C2×C22⋊C411yescharacteristic
149C2×C4×C61C4×D412yes—
150C2×C4×C61#12112yes—
151C2×C4×C61#12311yescharacteristic
152C2×Dic61—24no—
153C2×Dic61—42no—
154C2×Dic61—42no—
155C2×Dic61—42no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C8×C61—42no—
159#92418—82no—
160#13058—611nomaximal, Sylow
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—42no—
164C22×C61—81no—
165C22×C61—81no—
166C22×C61#38611yescharacteristic
167C22×C61B(4,𝔽2)11yescharacteristic
168C22×C61B(4,𝔽2)11yescharacteristic
169C4×C61—24no—
170C4×C61—22no—
171C4×C61—21no—
172C4×C61—22no—
173C4×C61—21no—
174C4×C61—42no—
175C4×C61—42no—
176Dic61—44no—
177Dic61—82no—
178C2×#456D6111yescharacteristic
179#2369—611no—
180C2×#356—611no—
181#1793—612no—
182#1394—612no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61—82no—
186C2×C61#188911yescharacteristic
187C2×C61#220411yescharacteristic
188C2×C61#220511yescharacteristic
189C2×2−1+4C2×D6111yescharacteristic
190C2×#137—21no—
191C2×C4⋊Q8C2×D6111yescharacteristic
192#456—22no—
193#456Dic6114yescomplemented
194C2×#104—21no—
195C2×#104—1222no—
196#417—1222no—
197C2×#101—612no—
198C2×#101C2×D6111yescharacteristic
199#391—612no—
200#391—1221no—
201#389—1222no—
202C4×C4⋊C4—1222no—
203C4×C4⋊C4—1221no—
204#356—1222no—
205C61#1305811yescharacteristic, complemented, Sylow
2062−1+4—22no—
2072−1+4—21no—
2082−1+4—42no—
2092−1+4C2×Dic6112yes—
210C2×#41—22no—
211C2×#41—21no—
212C2×#41—41no—
213C22×Q8—41no—
214C22×Q8C22⋊D6111yescharacteristic
215C22×Q8C22×D6111yescharacteristic
216#137—24no—
217#137—42no—
218C2×Q16—41no—
219C2×SD16—41no—
220C2×M16—21no—
221C2×M16—1222no—
222C4⋊Q8—22no—
223C4⋊Q8C2×Dic6112yes—
224C4⋊Q8C22⋊D6112yes—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—612no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4—1221no—
229C2×C4⋊C4—1221no—
230C2×C42—612no—
231C2×C42—1221no—
232C2×C42—1222no—
233C2×C42C22⋊D6111yescharacteristic
234#104—24no—
235#104—2444no—
236#103—2442no—
237#102—2442no—
238#101—21no—
239#101—1222no—
240#101—1222no—
241#101C2×Dic6112yes—
242#95—1222no—
243#95—1222no—
244#95—1222no—
245#41—24no—
246#41—22no—
247#41—22no—
248#41—24no—
249#41—42no—
250#41—41no—
251#41—42no—
252#41—42no—
253#41—44no—
254#41—42no—
255#41—41no—
256C2×Q8—21no—
257C2×Q8—22no—
258C2×Q8—21no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—44no—
266C2×Q8C22⋊Dic6112yes—
267C2×Q8#1175822412yes—
268C2×Q8C22×Dic6111yescharacteristic
269C2×Q8C2×C22⋊D6111yescharacteristic
270C2×D4—22no—
271C2×D4—21no—
272C2×D4—41no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—1222no—
278C22×C4—1222no—
279C22×C4D4×D6112yes—
280C22×C4C2×C22⋊D6111yescharacteristic
281Q16—44no—
282SD16—44no—
283M16—22no—
284M16—41no—
285M16—2442no—
286M16—2442no—
287C2×C8—41no—
288C2×C8—2442no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—1224no—
292C4⋊C4—4881no—
293C4⋊C4—2442no—
294C4⋊C4—2442no—
295C42—21no—
296C42—4881no—
297C42—2442no—
298C42—2444no—
299C42C22⋊Dic6112yes—
300C23—41no—
301C23#1451974511yescharacteristic
302Q8—22no—
303Q8—24no—
304Q8—42no—
305Q8—41no—
306Q8—41no—
307Q8—42no—
308Q8—42no—
309Q8—41no—
310Q8—81no—
311D4—22no—
312D4—24no—
313D4—42no—
314D4—41no—
315D4—42no—
316D4—42no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—1224no—
333C2×C4—2442no—
334C2×C4—2442no—
335C2×C4—2442no—
336C2×C4—2442no—
337C2×C4—2442no—
338C2×C4C2×C22⋊Dic6111yescharacteristic
339C2×C4D4×Dic6112yes—
340C2×C4#1451974912yes—
341C2×C4#1451975811yescharacteristic
342C8—42no—
343C8—4882no—
344C22—21no—
345C22—21no—
346C22—42no—
347C22—81no—
348C22—81no—
349C22#1683964211yescharacteristic
350C22#1683994811yescharacteristic
351C22#1683994811yescharacteristic
352C4—24no—
353C4—21no—
354C4—22no—
355C4—22no—
356C4—21no—
357C4—42no—
358C4—42no—
359C4—4882no—
360C4—2444no—
361C2—21no—
362C2—21no—
363C2—82no—
364C2#2144622811yescharacteristic
365C2#2144784411yescharacteristic
366C2#2144784811yescharacteristic
subgroupG/H
centerC22#16839948
commutatorC2×Q8×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC2×C61×#456C2
socleC22×C61B(4,𝔽2)
prime radicalC2×2−1+4×C61C22
cosocleC22×Q8×C61C23
hypercenterC2×#456D61
hypocenterC61#13058
LCSG ▷ C2×Q8×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#456

Complements

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

NH ≅ G/Ntype
1. #21448909C2semidirect
2. C2×#16839561C2semidirect
4. #21443945C2semidirect
19. C61×#456C4semidirect
193. #456Dic61semidirect
205. C61#13058semidirect

Maximal subgroups

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

subgroupindex
#214489092
C2×#168395612
#214457782
#214439452
C2×C61×#4562
#1305861