Finite Group Databasebeta

← #29580716

Subgroups — #29580716

Order 15616.

562 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 87 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#21456619C211yesmaximal, characteristic
2C2×#16840080C211yesmaximal, characteristic, complemented
3C2×#16840075C211yesmaximal, characteristic, complemented
4#21444180C211yesmaximal, characteristic, complemented
5#21444180C211yesmaximal, characteristic, complemented
6#21443706C211yesmaximal, characteristic, complemented
7C61×#1821C211yesmaximal, characteristic, complemented
8C2×#14519751C2211yescharacteristic
9C2×#14519751C2211yescharacteristic
10#16840081—21no—
11#16840081—21no—
12#16840080—22no—
13#16840080—22no—
14#16840075—22no—
15#16840075—22no—
16#16840015—22no—
17#16839898—21no—
18#16839898—21no—
19C2×#14519710—21no—
20C2×#14519708—21no—
21C2×#14519697—21no—
22C2×#14519697—21no—
23C2×C22⋊#92418C2211yescharacteristic
24C2×C22⋊#92418C2211yescharacteristic
25#16839485—22no—
26#16839485—22no—
27C22×C4⋊C4×C61C2211yescharacteristic
28C61×#407C2211yescharacteristic
29C61×#407C2211yescharacteristic
30C61×#343—22no—
31C22×D244D411yescharacteristic
32C22×D244D411yescharacteristic
33C22×C22⋊D61—21no—
34C22×C22⋊D61—21no—
35C2×D488—22no—
36C22×#92418—21no—
37C22×#92418—21no—
38#14519751—21no—
39#14519751—21no—
40#14519751—21no—
41#14519751—21no—
42#14519751—21no—
43#14519751—21no—
44#14519751—21no—
45#14519751—21no—
46C2×C4⋊Dic61D411yescharacteristic
47C2×C4⋊Dic61D411yescharacteristic
48C2×#11758198—21no—
49C2×#11758198—21no—
50#14519710—24no—
51#14519709—22no—
52#14519708—24no—
53#14519699—22no—
54#14519699—22no—
55#14519697—24no—
56#14519697—24no—
57C22⋊#92418—21no—
58C22⋊#92418—21no—
59C22⋊#92418D412yes—
60C22⋊#92418D412yes—
61C23×C4×C61—21no—
62C23×C4×C61C2311yescharacteristic
63C2×M16×C61—21no—
64C2×M16×C61D412yes—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61D412yes—
71C22⋊C8×C61—21no—
72C22⋊C8×C61—21no—
73C22⋊C8×C61—21no—
74C22⋊C8×C61—21no—
75C24×C61#4111yescharacteristic
76C23×D61—21no—
77C23×D61—21no—
78C22×C4×C61—22no—
79C22×C4×C61—42no—
80C22×C4×C61—41no—
81C22×C4×C61—41no—
82C22×C4×C61—82no—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C2×C22⋊D61—41no—
90C2×C22⋊D61—41no—
91C2×C22⋊D61—41no—
92C2×C22⋊D61—41no—
93C2×C22⋊D61—82no—
94C2×C22⋊D61—82no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C2×D244—21no—
98C2×D244—21no—
99C2×D244—21no—
100C2×D244—21no—
101C2×D244—42no—
102C2×D244—42no—
103M16×C61—42no—
104M16×C61—42no—
105C2×C8×C61—22no—
106C2×C8×C61—41no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111#11758198—41no—
112#11758198—41no—
113#11758198—41no—
114#11758198—41no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C2×#92418—22no—
120C2×#92418—22no—
121C2×#92418—42no—
122C2×#92418—42no—
123D488—82no—
124C23×C61—22no—
125C23×C61—22no—
126C23×C61—41no—
127C23×C61#12111yescharacteristic
128C23×C61#12111yescharacteristic
129C23×C61#12111yescharacteristic
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—42no—
133C22×D61—42no—
134C22×D61—81no—
135C22×D61—81no—
136C22×D61—81no—
137C22×D61—81no—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—22no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—82no—
146C2×C4×C61—82no—
147C2×C4×C61—82no—
148C2×C4×C61#12011yescharacteristic
149C2×C4×C61#12011yescharacteristic
150C2×C4×C61#12011yescharacteristic
151C2×C4×C61#12711yescharacteristic
152C22⋊D61—82no—
153C22⋊D61—82no—
154C22⋊D61—82no—
155C22⋊D61—82no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—81no—
159C2×Dic61—81no—
160D244—42no—
161D244—42no—
162D244—42no—
163D244—42no—
164C8×C61—42no—
165C8×C61—81no—
166#92418—42no—
167#92418—42no—
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#39211yescharacteristic
178C22×C61#47112yes—
179C22×C61#47212yes—
180C2×D61—81no—
181C2×D61—81no—
182C2×D61—81no—
183C2×D61—81no—
184C2×D61—81no—
185C2×D61—81no—
186C2×D61—162no—
187C2×D61—162no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—82no—
192C4×C61#45812yes—
193Dic61—81no—
194Dic61—81no—
195#3192—611no—
196C2×#481—611no—
197C2×#480—611no—
198#1821D6111yescharacteristic, complemented
199#1471—611no—
200#1471—611no—
201#1298—611no—
202C2×C61—22no—
203C2×C61—22no—
204C2×C61—41no—
205C2×C61#201811yescharacteristic
206C2×C61#201911yescharacteristic
207C2×C61#202211yescharacteristic
208D61—161no—
209D61—161no—
210C2×#121—611no—
211C2×#121—611no—
212C22×C4⋊C4C2×D6111yescharacteristic
213#481—1221no—
214#481—1221no—
215#481—1222no—
216#481—1222no—
217#480—1222no—
218#480—1222no—
219#469—1222no—
220#447—1221no—
221#447—1221no—
222C2×#107—1221no—
223C2×#106—1221no—
224C2×#102—1221no—
225C2×#102—1221no—
226#407C2×D6111yescharacteristic
227#407C2×D6111yescharacteristic
228C2×C22⋊C8—611no—
229C2×C22⋊C8—611no—
230#343—22no—
231#323—1222no—
232#323—1222no—
233C61#999211yescharacteristic, complemented, Sylow
234C22×D4—611no—
235C22×D4—611no—
236C22×D4—1221no—
237C22×D4—1221no—
238C23×C4—21no—
239C23×C4C22×D6111yescharacteristic
240C2×D8—1222no—
241C2×M16—21no—
242C2×M16C22⋊D6112yes—
243C22×C8—1221no—
244C22×C8—1221no—
245#121—1221no—
246#121—1221no—
247#121—1221no—
248#121—1221no—
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⋊C4—611no—
260C2×C4⋊C4D24412yes—
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1221no—
263#107—1224no—
264#106—1224no—
265#106—1222no—
266#102—1224no—
267#102—1224no—
268#102—1222no—
269#102—1222no—
270C22⋊C8—21no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—21no—
274C22⋊C8—612no—
275C22⋊C8—612no—
276C22⋊C8—1221no—
277C22⋊C8—1221no—
278C24—1221no—
279C24—1221no—
280C24#1175822211yescharacteristic
281C2×D4—1221no—
282C2×D4—1221no—
283C2×D4—1221no—
284C2×D4—1221no—
285C2×D4—4882no—
286C2×D4—4882no—
287C2×D4—2442no—
288C2×D4—2442no—
289C2×D4—2441no—
290C2×D4—2441no—
291C2×D4—2441no—
292C2×D4—2441no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—82no—
298C22×C4—1221no—
299C22×C4—1221no—
300C22×C4C2×D24411yescharacteristic
301C22×C4D4×D6111yescharacteristic
302C22×C4D4×D6111yescharacteristic
303C22×C4D4×D6111yescharacteristic
304C22×C4D4×D6111yescharacteristic
305C22×C4C2×C22⋊D6111yescharacteristic
306D8—4882no—
307M16—42no—
308M16—42no—
309C2×C8—22no—
310C2×C8—41no—
311C2×C8—1222no—
312C2×C8—1222no—
313C2×C8—2442no—
314C2×C8—2442no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—1221no—
320C4⋊C4—1221no—
321C4⋊C4—1221no—
322C4⋊C4—1221no—
323C22⋊C4—2441no—
324C22⋊C4—2441no—
325C22⋊C4—2441no—
326C22⋊C4—2441no—
327C23—22no—
328C23—22no—
329C23—41no—
330C23—4881no—
331C23—4881no—
332C23—4881no—
333C23—4881no—
334C23—2442no—
335C23—2442no—
336C23—2441no—
337C23—2441no—
338C23#1451975011yescharacteristic
339C23#1451975011yescharacteristic
340C23#1451975111yescharacteristic
341D4—4882no—
342D4—4882no—
343D4—4882no—
344D4—4882no—
345D4—2442no—
346D4—2442no—
347D4—2442no—
348D4—2442no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—4881no—
360C2×C4—4881no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4#1451974511yescharacteristic
364C2×C4#1451974711yescharacteristic
365C2×C4#1451974711yescharacteristic
366C2×C4#1451977311yescharacteristic
367C8—42no—
368C8—81no—
369C8—2442no—
370C8—2442no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—44no—
379C22—9762no—
380C22—9762no—
381C22—4881no—
382C22—4881no—
383C22—4881no—
384C22—4881no—
385C22—4881no—
386C22—4881no—
387C22#1683967111yescharacteristic
388C22#1684002812yes—
389C22#1684003112yes—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—82no—
394C4—4881no—
395C4—4881no—
396C4#1683995112yes—
397C2—22no—
398C2—22no—
399C2—41no—
400C2—9761no—
401C2—9761no—
402C2#2144692711yescharacteristic
403C2#2144693211yescharacteristic
404C2#2144695011yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1821C2
socleC22×C61#392
prime radical#21456619C2
hypercenter#1821D61
hypocenterC61#9992
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
2. C2×#16840080C2semidirect
3. C2×#16840075C2semidirect
4. #21444180C2semidirect
5. #21444180C2semidirect
6. #21443706C2semidirect
7. C61×#1821C2semidirect
198. #1821D61semidirect
233. C61#9992semidirect

Maximal subgroups

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

subgroupindex
#214566192
C2×#168400802
C2×#168400752
#214441802
#214437062
C61×#18212
#999261