Finite Group Databasebeta

← #29580712

Subgroups — #29580712

Order 15616.

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

normal subgroups only

This group has 562 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#21456616C211yesmaximal, characteristic
2C2×#16840080C211yesmaximal, characteristic, complemented
3#21445899C211yesmaximal, characteristic, complemented
4#21444179C211yesmaximal, characteristic, complemented
5#21444178C211yesmaximal, characteristic, complemented
6#21443707C211yesmaximal, characteristic, complemented
7C2×C61×#480C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9#16840080—22no—
10#16840080—22no—
11#16840080—21no—
12#16840078—21no—
13#16840013—22no—
14#16839896—21no—
15#16839895—21no—
16C2×#14519708—21no—
17C2×#14519697—21no—
18#16839726C2211yescharacteristic
19#16839726C2211yescharacteristic
20C2×C22⋊#92418C2211yescharacteristic
21#16839531—22no—
22#16839486—22no—
23#16839485—22no—
24C2×#121×C61C2211yescharacteristic
25C22×C4⋊C4×C61C2211yescharacteristic
26C61×#480—22no—
27C61×#480—22no—
28C2×#107×C61—21no—
29C2×#102×C61—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C22×D244D411yescharacteristic
32C22×C22⋊D61—21no—
33C2×#11758199—22no—
34C2×#11758194—21no—
35C2×#11758194D412yes—
36C22×#92418—21no—
37#14519751—21no—
38#14519751—21no—
39#14519751—21no—
40#14519751—21no—
41C2×C4⋊Dic61D411yescharacteristic
42C2×#11758198—21no—
43#14519708—22no—
44#14519708—24no—
45#14519698—22no—
46#14519697—24no—
47#14519697—22no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53C22⋊#92418D412yes—
54C22×D4×C61—21no—
55C22×D4×C61D411yescharacteristic
56C23×C4×C61—21no—
57C23×C4×C61C2311yescharacteristic
58C22×C8×C61—21no—
59#121×C61—21no—
60#121×C61—21no—
61#121×C61—21no—
62#121×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61D411yescharacteristic
69C2×C4⋊C4×C61D412yes—
70C2×C22⋊C4×C61—21no—
71#107×C61—24no—
72#102×C61—24no—
73C22⋊C8×C61—21no—
74C22⋊C8×C61D412yes—
75C24×C61—21no—
76C24×C61#4111yescharacteristic
77C23×D61—21no—
78C2×D4×C61—21no—
79C2×D4×C61—21no—
80C2×D4×C61—42no—
81C2×D4×C61—41no—
82C2×D4×C61—41no—
83C2×D4×C61—82no—
84C22×C4×C61—21no—
85C22×C4×C61—22no—
86C22×C4×C61—42no—
87C22×C4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61—82no—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C2×C22⋊D61—41no—
97C2×C22⋊D61—41no—
98C2×C22⋊D61—82no—
99C22×Dic61—21no—
100C2×D244—21no—
101C2×D244—21no—
102C2×D244—42no—
103C2×C8×C61—22no—
104C2×C8×C61—42no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113#11758199—82no—
114#11758198—41no—
115#11758198—41no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118#11758194—42no—
119#11758194—42no—
120C2×#92418—22no—
121C2×#92418—22no—
122C2×#92418—42no—
123C2×#92418—41no—
124C23×C61—22no—
125C23×C61—22no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61—42no—
129C23×C61—81no—
130C23×C61—81no—
131C23×C61#12111yescharacteristic
132C23×C61#12111yescharacteristic
133C23×C61#12111yescharacteristic
134C22×D61—41no—
135C22×D61—42no—
136C22×D61—81no—
137C22×D61—81no—
138D4×C61—42no—
139D4×C61—42no—
140D4×C61—82no—
141D4×C61—82no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61—82no—
151C2×C4×C61—82no—
152C2×C4×C61—82no—
153C2×C4×C61—81no—
154C2×C4×C61#12011yescharacteristic
155C2×C4×C61#12011yescharacteristic
156C2×C4×C61#12011yescharacteristic
157C2×C4×C61#12711yescharacteristic
158C22⋊D61—82no—
159C22⋊D61—82no—
160C2×Dic61—41no—
161C2×Dic61—81no—
162D244—42no—
163D244—42no—
164C8×C61—42no—
165#92418—42no—
166#92418—42no—
167#92418—81no—
168#9992—611nomaximal, Sylow
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—44no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61—81no—
180C22×C61—162no—
181C22×C61#39211yescharacteristic
182C22×C61#47112yes—
183C22×C61#47212yes—
184C2×D61—81no—
185C2×D61—81no—
186C2×D61—81no—
187C2×D61—162no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—82no—
192C4×C61—81no—
193C4×C61#45812yes—
194Dic61—81no—
195#3192—611no—
196C2×#481—611no—
197C2×#480D6111yescharacteristic, complemented
198#1821—611no—
199#1471—611no—
200#1471—611no—
201#1298—611no—
202C2×C61—22no—
203C2×C61—22no—
204C2×C61—41no—
205C2×C61—161no—
206C2×C61#201811yescharacteristic
207C2×C61#201911yescharacteristic
208C2×C61#202211yescharacteristic
209D61—161no—
210C2×#121—611no—
211C2×#121C2×D6111yescharacteristic
212C22×C4⋊C4C2×D6111yescharacteristic
213#481—1222no—
214#481—1221no—
215#481—1222no—
216#481—1221no—
217#480—22no—
218#480—22no—
219#469—1222no—
220#447—1221no—
221#447—1221no—
222C2×#107—21no—
223C2×#106—1221no—
224C2×#102—21no—
225C2×#102—1221no—
226#407—611no—
227#407—611no—
228C2×C22⋊C8—611no—
229C2×C22⋊C8C2×D6111yescharacteristic
230#343—1222no—
231#323—1222no—
232#323—1222no—
233C61#999211yescharacteristic, complemented, Sylow
234C22×D4—21no—
235C22×D4—611no—
236C22×D4—1221no—
237C22×D4C22⋊D6111yescharacteristic
238C23×C4—21no—
239C23×C4C22×D6111yescharacteristic
240C2×D8—1222no—
241C2×M16—612no—
242C2×M16—1221no—
243C22×C8—21no—
244C22×C8—1221no—
245#121—21no—
246#121—21no—
247#121—21no—
248#121—21no—
249#121—1221no—
250#121—1221no—
251#121—1221no—
252#121—1221no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—611no—
259C2×C4⋊C4C22⋊D6112yes—
260C2×C4⋊C4C22⋊D6111yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—1221no—
263#107—24no—
264#106—1224no—
265#106—1222no—
266#102—24no—
267#102—1224no—
268#102—1222no—
269#102—1222no—
270C22⋊C8—21no—
271C22⋊C8—612no—
272C22⋊C8—1221no—
273C22⋊C8—1221no—
274C22⋊C8—1221no—
275C22⋊C8—1221no—
276C22⋊C8—1221no—
277C22⋊C8C22⋊D6112yes—
278C24—21no—
279C24—1221no—
280C24#1175822411yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—42no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—82no—
287C2×D4—1221no—
288C2×D4—1221no—
289C2×D4—4882no—
290C2×D4—2442no—
291C2×D4—2441no—
292C2×D4—2441no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—82no—
299C22×C4—1221no—
300C22×C4D4×D6111yescharacteristic
301C22×C4D4×D6111yescharacteristic
302C22×C4D4×D6111yescharacteristic
303C22×C4C2×C22⋊D6111yescharacteristic
304C22×C4C2×C22⋊D6111yescharacteristic
305C22×C4C2×C22⋊D6111yescharacteristic
306D8—4882no—
307M16—2442no—
308M16—2442no—
309C2×C8—22no—
310C2×C8—42no—
311C2×C8—1222no—
312C2×C8—1222no—
313C2×C8—2441no—
314C2×C8—2442no—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—21no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—1221no—
322C4⋊C4—1221no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—2441no—
326C22⋊C4—2441no—
327C23—22no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—42no—
332C23—81no—
333C23—81no—
334C23—4881no—
335C23—4881no—
336C23—2442no—
337C23—2441no—
338C23#1451974811yescharacteristic
339C23#1451974911yescharacteristic
340C23#1451974911yescharacteristic
341D4—42no—
342D4—42no—
343D4—82no—
344D4—82no—
345D4—4882no—
346D4—4882no—
347D4—2442no—
348D4—2442no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—4881no—
362C2×C4—2441no—
363C2×C4#1451974511yescharacteristic
364C2×C4#1451974511yescharacteristic
365C2×C4#1451974611yescharacteristic
366C2×C4#1451977311yescharacteristic
367C8—42no—
368C8—4881no—
369C8—2442no—
370C8—2442no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—44no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—162no—
383C22—9762no—
384C22—4881no—
385C22—4881no—
386C22—4881no—
387C22#1683967011yescharacteristic
388C22#1684002712yes—
389C22#1684002912yes—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—82no—
394C4—81no—
395C4—4881no—
396C4#1683995112yes—
397C2—22no—
398C2—22no—
399C2—41no—
400C2—161no—
401C2—9761no—
402C2#2144692611yescharacteristic
403C2#2144693011yescharacteristic
404C2#2144694711yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#480C2
socleC22×C61#392
prime radical#21456616C2
hypercenterC2×#480D61
hypocenterC61#9992
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
2. C2×#16840080C2semidirect
3. #21445899C2semidirect
4. #21444179C2semidirect
5. #21444178C2semidirect
6. #21443707C2semidirect
7. C2×C61×#480C2semidirect
197. C2×#480D61semidirect
233. C61#9992semidirect

Maximal subgroups

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

subgroupindex
#214566162
C2×#168400802
#214458992
#214441792
#214441782
#214437072
C2×C61×#4802
#999261