Finite Group Databasebeta

← #29598370

Subgroups — #29598370

Order 15616.

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

normal subgroups only

This group has 432 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#21447514C211yesmaximal, characteristic
3#21447513C211yesmaximal, characteristic
4#21447472C211yesmaximal, characteristic
5#21445867C211yesmaximal, characteristic
6#21445621C211yesmaximal, characteristic
7C61×#2123C211yesmaximal, characteristic
8C22×#11758202C411yescharacteristic, complemented
9C2×#14519784C2211yescharacteristic
10#16840333C2211yescharacteristic
11#16840018—21no—
12#16840018—21no—
13#16840001—21no—
14#16840001—21no—
15C2×#14519712—21no—
16#16839798C411yescharacteristic, complemented
17#16839771C411yescharacteristic, complemented
18#16839770C411yescharacteristic, complemented
19#16839746C2211yescharacteristic
20#16839746C2211yescharacteristic
21#16839718—21no—
22#16839561—22no—
23#16839560—22no—
24#16839551—22no—
25#16839532—21no—
26#16839520—21no—
27C61×#536C2211yescharacteristic
28C61×#413C2211yescharacteristic
29C61×#410C2211yescharacteristic
30C61×#352—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×#11758194—21no—
40C2×#11758194C2×C411yescharacteristic
41C2×#11758194D411yescharacteristic
42C2×#11758194D411yescharacteristic
43C22×#92418C2×C411yescharacteristic
44C22×#92418D411yescharacteristic
45C22×#92418D411yescharacteristic
46#14519775—21no—
47#14519764—21no—
48#14519758—21no—
49#14519758—21no—
50#14519756—21no—
51#14519756—21no—
52#14519736C2×C411yescharacteristic
53#14519712—41no—
54#14519712—41no—
55#14519710—21no—
56#14519708—21no—
57#14519701—21no—
58#14519701—21no—
59#14519700—21no—
60#14519700—21no—
61#14519696—21no—
62#14519696D412yes—
63#14519694—21no—
64#14519694—21no—
65#14519688—41no—
66C4×#92418—41no—
67C2×#41×C61C2311yescharacteristic
68C22×Q8×C61C2×C411yescharacteristic
69C2×M16×C61—21no—
70C4⋊Q8×C61—21no—
71#124×C61—21no—
72#123×C61—21no—
73#123×C61—21no—
74C22⋊Q8×C61—21no—
75C22⋊Q8×C61—21no—
76#117×C61—21no—
77#117×C61C2×C411yescharacteristic
78#101×C61—21no—
79#100×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×C61C2×D411yescharacteristic
89C2×D4×C61C2×D411yescharacteristic
90C2×D4×C61#4111yescharacteristic
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—
101M16×C61—41no—
102M16×C61—41no—
103C2×C8×C61—41no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C22⋊C4×C61—21no—
109C22⋊C4×C61—22no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—41no—
112C42×C61—21no—
113C42×C61—41no—
114C22⋊Dic61—21no—
115C22⋊Dic61—41no—
116C22⋊Dic61—41no—
117#11758202—42no—
118#11758202—41no—
119#11758202—41no—
120C4⋊Dic61—21no—
121#11758196—41no—
122#11758196—41no—
123C4×Dic61—21no—
124#11758194—22no—
125#11758194—41no—
126#11758194—41no—
127#11758194—41no—
128#11758194—41no—
129#11758194—41no—
130#11758194—41no—
131C2×#92418—21no—
132C2×#92418—21no—
133C2×#92418—21no—
134C2×#92418—21no—
135C2×#92418—21no—
136C2×#92418—21no—
137C2×#92418—41no—
138C2×#92418—41no—
139C2×#92418C22⋊C411yescharacteristic
140C2×#92418C22⋊C411yescharacteristic
141C2×#92418C22⋊C411yescharacteristic
142C2×#92418C22⋊C411yescharacteristic
143C23×C61C4×D411yescharacteristic
144C23×C61#12111yescharacteristic
145C23×C61#12411yescharacteristic
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×C61C2×C22⋊C411yescharacteristic
164C2×C4×C61C4×D411yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12711yescharacteristic
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—81no—
170Dic122—41no—
171Dic122—41no—
172Dic122—81no—
173C8×C61—81no—
174#92418—21no—
175#92418—21no—
176#92418—21no—
177#92418—42no—
178#92418—81no—
179#92418#9911yescharacteristic
180#92418#9911yescharacteristic
181#13233—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×C61C2×#9911yescharacteristic
196C4×C61#41011yescharacteristic
197Dic61—81no—
198Dic61—81no—
199#3059—611no—
200#2135—611no—
201#2135—611no—
202#2123—611no—
203#2123D6111yescharacteristic
204#1813—611no—
205#1757—611no—
206C2×C61—21no—
207C2×C61—41no—
208C2×C61—41no—
209C2×C61—41no—
210C2×C61#196411yescharacteristic
211C22×Q16—611no—
212C2×#131—611no—
213#536—611no—
214#536C2×D6111yescharacteristic
215#470—1221no—
216#470—1221no—
217#467—1221no—
218#467—1221no—
219C2×#108—1221no—
220#427—611no—
221#419—611no—
222#419—611no—
223#413—611no—
224#413—611no—
225#413C2×D6111yescharacteristic
226#410C2×D6111yescharacteristic
227#405—1221no—
228#356—1222no—
229#356—1222no—
230#352—22no—
231#352—1222no—
232#344—1221no—
233#338—1221no—
234C61#1323311yescharacteristic, complemented, Sylow
235C2×#41C22×D6111yescharacteristic
236C22×Q8—611no—
237C22×Q8C4×D6111yescharacteristic
238C2×Q16—1221no—
239C2×Q16—1221no—
240C2×Q16—2441no—
241C2×Q16—2441no—
242#131—2441no—
243#131—2441no—
244C2×M16—21no—
245C2×M16—611no—
246C2×M16—611no—
247C2×M16—611no—
248C2×M16—1221no—
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—611no—
267#117C4×D6111yescharacteristic
268#108—2441no—
269#108—2441no—
270#107—1221no—
271#106—1221no—
272#103—1221no—
273#103—1221no—
274#103—1221no—
275#103—1221no—
276#101—21no—
277#101—612no—
278#101—1221no—
279#100—21no—
280#100—1221no—
281#100—1221no—
282#99—21no—
283#99C22⋊D6112yes—
284#97—2441no—
285C4×C8—2441no—
286#41—41no—
287#41—41no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—1221no—
293C2×Q8—1221no—
294C2×Q8—2441no—
295C2×Q8—2441no—
296C2×Q8D4×D6111yescharacteristic
297C2×D4#1175822211yescharacteristic
298C2×D4#1175822411yescharacteristic
299C2×D4C2×C22⋊D6111yescharacteristic
300C22×C4—21no—
301C22×C4—1221no—
302C22×C4C2×C4×D6111yescharacteristic
303C22×C4D4×D6111yescharacteristic
304C22×C4D4×D6111yescharacteristic
305Q16—2441no—
306Q16—2441no—
307Q16—2442no—
308M16—41no—
309M16—41no—
310M16—1222no—
311M16—2441no—
312M16—2441no—
313M16—2441no—
314M16—2441no—
315M16—2441no—
316M16—2441no—
317C2×C8—41no—
318C2×C8—611no—
319C2×C8—611no—
320C2×C8—611no—
321C2×C8—611no—
322C2×C8—1221no—
323C2×C8—1221no—
324C2×C8—1221no—
325C2×C8—1221no—
326C2×C8—1221no—
327C2×C8—1221no—
328C2×C8—2441no—
329C2×C8—2441no—
330C4⋊C4—21no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—1221no—
335C4⋊C4—2441no—
336C4⋊C4—2441no—
337C22⋊C4—21no—
338C22⋊C4—22no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—1221no—
342C22⋊C4—2441no—
343C22⋊C4—2441no—
344C42—21no—
345C42—41no—
346C42—1221no—
347C23C4×C22⋊D6111yescharacteristic
348C23#1451974911yescharacteristic
349C23#1451976311yescharacteristic
350Q8—41no—
351Q8—41no—
352Q8—41no—
353Q8—81no—
354Q8—4881no—
355Q8—2441no—
356Q8—2441no—
357D4—41no—
358D4—41no—
359D4—41no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—81no—
370C2×C4—4881no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4C22⋊C4×D6111yescharacteristic
374C2×C4#1451973711yescharacteristic
375C2×C4#1451975011yescharacteristic
376C2×C4#1451977311yescharacteristic
377C8—81no—
378C8—611no—
379C8—611no—
380C8—1221no—
381C8—1221no—
382C8—1221no—
383C8—4881no—
384C8—2442no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22#1683966211yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—81no—
396C4—82no—
397C4—81no—
398C4—4881no—
399C4—4881no—
400C4#99×D6111yescharacteristic
401C4#1683973411yescharacteristic
402C2—21no—
403C2—41no—
404C2—41no—
405C2—41no—
406C2#2144665311yescharacteristic
subgroupG/H
centerC2#21446653
commutatorC22×C4×C61C22×C4
FrattiniC2×#41C22×D61
FittingC61×#2123C2
socleC2×C61#1964
prime radical, cosocleC2×#41×C61C23
hypercenter#2123D61
hypocenterC61#13233
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #2123

Complements

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

NH ≅ G/Ntype
8. C22×#11758202C4semidirect
16. #16839798C4semidirect
17. #16839771C4semidirect
18. #16839770C4semidirect
234. C61#13233semidirect

Maximal subgroups

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

subgroupindex
#214554712
#214475142
#214475132
#214474722
#214458672
#214456212
C61×#21232
#1323361