Finite Group Databasebeta

← #29598355

Subgroups — #29598355

Order 15616.

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

normal subgroups only

This group has 436 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#21455471C211yesmaximal, characteristic
2#21447493C211yesmaximal, characteristic
3#21447473C211yesmaximal, characteristic
4#21447472C211yesmaximal, characteristic
5#21445863C211yesmaximal, characteristic
6#21445621C211yesmaximal, characteristic
7C61×#2129C211yesmaximal, characteristic
8C22×#11758202C411yescharacteristic, complemented
9C2×#14519784C2211yescharacteristic
10#16840333C2211yescharacteristic
11#16840018—21no—
12#16840018—21no—
13#16840001—21no—
14#16840001—21no—
15#16839798—21no—
16#16839798C411yescharacteristic, complemented
17#16839771C411yescharacteristic, complemented
18#16839770C411yescharacteristic, complemented
19#16839746C2211yescharacteristic
20#16839736C2211yescharacteristic
21#16839718—21no—
22#16839556—22no—
23#16839552—22no—
24#16839551—22no—
25#16839532—21no—
26#16839518—21no—
27C61×#536C2211yescharacteristic
28C61×#410C2211yescharacteristic
29C61×#410C2211yescharacteristic
30C61×#354—22no—
31C22×Dic122C2×C411yescharacteristic
32C2×#11758202—21no—
33C2×#11758202—21no—
34C2×#11758202—41no—
35C2×#11758202—41no—
36#14519784—41no—
37#14519784—41no—
38C2×#11758194—21no—
39C2×#11758194C2×C411yescharacteristic
40C2×#11758194D411yescharacteristic
41C2×#11758194D411yescharacteristic
42C22×#92418C2×C411yescharacteristic
43C22×#92418D411yescharacteristic
44C22×#92418D411yescharacteristic
45#14519775—21no—
46#14519764—21no—
47#14519758—21no—
48#14519758—21no—
49#14519756—21no—
50#14519756—21no—
51#14519736—21no—
52#14519736C2×C411yescharacteristic
53#14519710—21no—
54#14519710—41no—
55#14519708—21no—
56#14519708—41no—
57#14519701—21no—
58#14519701—21no—
59#14519700—21no—
60#14519700—21no—
61#14519696—21no—
62#14519694D412yes—
63#14519692—21no—
64#14519692—21no—
65#14519688—41no—
66C4×#92418—41no—
67C2×#41×C61C2311yescharacteristic
68C22×Q8×C61C2×C411yescharacteristic
69C4⋊Q8×C61—21no—
70#124×C61—21no—
71#123×C61—21no—
72#123×C61—21no—
73C22⋊Q8×C61—21no—
74C22⋊Q8×C61—21no—
75#117×C61—21no—
76#117×C61—21no—
77#117×C61C2×C411yescharacteristic
78#99×C61—21no—
79#99×C61—21no—
80#99×C61—21no—
81#99×C61D412yes—
82#41×C61—41no—
83#41×C61—41no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—41no—
87C2×Q8×C61—41no—
88C2×Q8×C61#4111yescharacteristic
89C2×D4×C61C2×D411yescharacteristic
90C2×D4×C61C2×D411yescharacteristic
91C2×D4×C61#4111yescharacteristic
92C22×C4×C61—21no—
93C22×C4×C61C22×C411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×Dic61—21no—
97C2×Dic122—21no—
98C2×Dic122—21no—
99C2×Dic122—41no—
100C2×Dic122—41no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—41no—
103C4⋊C4×C61—41no—
104C4⋊C4×C61—41no—
105C4⋊C4×C61—41no—
106C22⋊C4×C61—21no—
107C22⋊C4×C61—22no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C42×C61—21no—
112C42×C61—41no—
113C42×C61—41no—
114C22⋊Dic61—21no—
115C22⋊Dic61—41no—
116C22⋊Dic61—41no—
117C22⋊Dic61—41no—
118#11758202—42no—
119#11758202—41no—
120#11758202—41no—
121C4⋊Dic61—21no—
122C4⋊Dic61—41no—
123#11758196—41no—
124#11758196—41no—
125C4×Dic61—21no—
126C4×Dic61—41no—
127#11758194—22no—
128#11758194—41no—
129#11758194—41no—
130#11758194—41no—
131#11758194—41no—
132C2×#92418—21no—
133C2×#92418—21no—
134C2×#92418—21no—
135C2×#92418—21no—
136C2×#92418—21no—
137C2×#92418—21no—
138C2×#92418—41no—
139C2×#92418C22⋊C411yescharacteristic
140C2×#92418C22⋊C411yescharacteristic
141C2×#92418C22⋊C411yescharacteristic
142C2×#92418C22⋊C411yescharacteristic
143C23×C61C4×D411yescharacteristic
144C23×C61#12111yescharacteristic
145C23×C61#12711yescharacteristic
146Q8×C61—41no—
147Q8×C61—41no—
148Q8×C61—41no—
149Q8×C61—81no—
150D4×C61—41no—
151D4×C61—41no—
152D4×C61—41no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—42no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—81no—
163C2×C4×C61—81no—
164C2×C4×C61C2×C22⋊C411yescharacteristic
165C2×C4×C61C4×D411yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12411yescharacteristic
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—81no—
171C2×Dic61—81no—
172Dic122—41no—
173Dic122—41no—
174Dic122—81no—
175#92418—21no—
176#92418—21no—
177#92418—21no—
178#92418—42no—
179#92418#9911yescharacteristic
180#92418#9911yescharacteristic
181#13230—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—41no—
188C22×C61#39011yescharacteristic
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—81no—
193C4×C61—82no—
194C4×C61—81no—
195C4×C61—161no—
196C4×C61C2×#9911yescharacteristic
197C4×C61#41011yescharacteristic
198Dic61—81no—
199Dic61—81no—
200Dic61—161no—
201#3059—611no—
202#2129—611no—
203#2129D6111yescharacteristic
204#2123—611no—
205#2123—611no—
206#1812—611no—
207#1757—611no—
208C2×C61—21no—
209C2×C61—41no—
210C2×C61—41no—
211C2×C61—41no—
212C2×C61#196411yescharacteristic
213C22×Q16—611no—
214C2×#131—611no—
215#536—611no—
216#536C2×D6111yescharacteristic
217#470—1221no—
218#470—1221no—
219#467—1221no—
220#467—1221no—
221#427—611no—
222#427—1221no—
223#419—611no—
224#419—611no—
225#413—611no—
226#410—611no—
227#410C2×D6111yescharacteristic
228#410C2×D6111yescharacteristic
229#405—1221no—
230#354—22no—
231#354—1222no—
232#352—1222no—
233#352—1222no—
234#344—1221no—
235H(3,ℤ/4)—1221no—
236C61#1323011yescharacteristic, complemented, Sylow
237C2×#41C22×D6111yescharacteristic
238C22×Q8—611no—
239C22×Q8C4×D6111yescharacteristic
240C2×Q16—1221no—
241C2×Q16—1221no—
242C2×Q16—2441no—
243C2×Q16—2441no—
244#131—2441no—
245#131—2441no—
246C2×M16—611no—
247C2×M16—611no—
248C2×M16—611no—
249C2×M16—1221no—
250C22×C8—611no—
251C22×C8—611no—
252C22×C8—611no—
253C4⋊Q8—21no—
254C4⋊Q8—1221no—
255#124—21no—
256#124—1221no—
257#123—21no—
258#123—21no—
259#123—1221no—
260#123—1221no—
261C22⋊Q8—21no—
262C22⋊Q8—21no—
263C22⋊Q8—1221no—
264C22⋊Q8—1221no—
265#117—21no—
266#117—21no—
267#117—611no—
268#117—1221no—
269#117C4×D6111yescharacteristic
270#107—1221no—
271#107—2441no—
272#106—1221no—
273#106—2441no—
274#103—1221no—
275#103—1221no—
276#103—1221no—
277#103—1221no—
278#101—1221no—
279#100—612no—
280#99—21no—
281#99—21no—
282#99—21no—
283#99—1221no—
284#99—1221no—
285#99C22⋊D6112yes—
286#97—2441no—
287C4×C8—2441no—
288#41—41no—
289#41—41no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—1221no—
295C2×Q8—1221no—
296C2×Q8—2441no—
297C2×Q8—2441no—
298C2×Q8#1175822211yescharacteristic
299C2×D4D4×D6111yescharacteristic
300C2×D4#1175822411yescharacteristic
301C2×D4C2×C22⋊D6111yescharacteristic
302C22×C4—21no—
303C22×C4—1221no—
304C22×C4C2×C4×D6111yescharacteristic
305C22×C4D4×D6111yescharacteristic
306C22×C4D4×D6111yescharacteristic
307Q16—2441no—
308Q16—2441no—
309Q16—2442no—
310M16—1222no—
311M16—2441no—
312M16—2441no—
313M16—2441no—
314M16—2441no—
315C2×C8—611no—
316C2×C8—611no—
317C2×C8—611no—
318C2×C8—611no—
319C2×C8—1221no—
320C2×C8—1221no—
321C2×C8—1221no—
322C2×C8—1221no—
323C2×C8—1221no—
324C2×C8—1221no—
325C2×C8—2441no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—1221no—
332C4⋊C4—2441no—
333C4⋊C4—2441no—
334C4⋊C4—2441no—
335C22⋊C4—21no—
336C22⋊C4—22no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—1221no—
341C22⋊C4—2441no—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C42—21no—
345C42—41no—
346C42—41no—
347C42—1221no—
348C42—2441no—
349C23#1451973711yescharacteristic
350C23#1451974911yescharacteristic
351C23#1451977311yescharacteristic
352Q8—41no—
353Q8—41no—
354Q8—41no—
355Q8—81no—
356Q8—4881no—
357Q8—2441no—
358Q8—2441no—
359D4—41no—
360D4—41no—
361D4—41no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—4881no—
374C2×C4—4881no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4C22⋊C4×D6111yescharacteristic
378C2×C4C4×C22⋊D6111yescharacteristic
379C2×C4#1451975011yescharacteristic
380C2×C4#1451976311yescharacteristic
381C8—611no—
382C8—611no—
383C8—1221no—
384C8—1221no—
385C8—1221no—
386C8—2442no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22#1683966211yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—81no—
398C4—82no—
399C4—81no—
400C4—161no—
401C4—9761no—
402C4—4881no—
403C4—4881no—
404C4#99×D6111yescharacteristic
405C4#1683973411yescharacteristic
406C2—21no—
407C2—41no—
408C2—41no—
409C2—41no—
410C2#2144665311yescharacteristic
subgroupG/H
centerC2#21446653
commutatorC22×C4×C61C22×C4
FrattiniC2×#41C22×D61
FittingC61×#2129C2
socleC2×C61#1964
prime radical, cosocleC2×#41×C61C23
hypercenter#2129D61
hypocenterC61#13230
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #2129

Complements

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

NH ≅ G/Ntype
8. C22×#11758202C4semidirect
16. #16839798C4semidirect
17. #16839771C4semidirect
18. #16839770C4semidirect
236. C61#13230semidirect

Maximal subgroups

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

subgroupindex
#214554712
#214474932
#214474732
#214474722
#214458632
#214456212
C61×#21292
#1323061