Finite Group Databasebeta

← #29598369

Subgroups — #29598369

Order 15616.

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

normal subgroups only

This group has 448 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#21455460C211yesmaximal, characteristic
2#21447510C211yesmaximal, characteristic
3#21447473C211yesmaximal, characteristic, complemented
4#21447470C211yesmaximal, characteristic
5#21445867C211yesmaximal, characteristic, complemented
6#21445622C211yesmaximal, characteristic, complemented
7C61×#2135C211yesmaximal, characteristic, complemented
8C22×#11758201C411yescharacteristic, complemented
9C2×#14519784C2211yescharacteristic
10#16840326C2211yescharacteristic
11#16840010—21no—
12#16840010—21no—
13#16839993—21no—
14#16839993—21no—
15C2×#14519712—21no—
16#16839798C411yescharacteristic, complemented
17#16839771C411yescharacteristic, complemented
18#16839766C411yescharacteristic, complemented
19#16839746C2211yescharacteristic
20#16839736C2211yescharacteristic
21#16839720—21no—
22#16839558—22no—
23#16839552—22no—
24#16839550—22no—
25#16839534—21no—
26#16839520—21no—
27C61×#536C2211yescharacteristic
28C61×#413C2211yescharacteristic
29C61×#413C2211yescharacteristic
30C61×#356—22no—
31C22×D244C2×C411yescharacteristic
32C2×#11758201—21no—
33C2×#11758201—21no—
34C2×#11758201—41no—
35C2×#11758201—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#14519773—21no—
46#14519766—21no—
47#14519751—21no—
48#14519751—21no—
49#14519745—21no—
50#14519745—21no—
51#14519736—21no—
52#14519736C2×C411yescharacteristic
53#14519712—41no—
54#14519712—41no—
55#14519710—21no—
56#14519708—21no—
57#14519701—21no—
58#14519701—21no—
59#14519697—21no—
60#14519697—21no—
61#14519696—21no—
62#14519694—21no—
63#14519692—21no—
64#14519692D412yes—
65#14519687—41no—
66C8×Dic61—41no—
67C2×#41×C61C2311yescharacteristic
68C22×Q8×C61C2×C411yescharacteristic
69C2×M16×C61—21no—
70C2×M16×C61—21no—
71C4⋊Q8×C61—21no—
72#124×C61—21no—
73#123×C61—21no—
74#123×C61—21no—
75C22⋊Q8×C61—21no—
76C22⋊Q8×C61—21no—
77#117×C61C2×C411yescharacteristic
78#101×C61—21no—
79#101×C61D412yes—
80#100×C61—21no—
81#100×C61—21no—
82C23×D61—21no—
83#41×C61—41no—
84#41×C61—41no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—41no—
88C2×Q8×C61—41no—
89C2×Q8×C61C2×D411yescharacteristic
90C2×D4×C61C2×D411yescharacteristic
91C2×D4×C61#4111yescharacteristic
92C2×D4×C61#4111yescharacteristic
93C22×C4×C61—21no—
94C22×C4×C61C22×C411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C2×C22⋊D61—41no—
98C2×C22⋊D61—41no—
99C2×D244—21no—
100C2×D244—21no—
101C2×D244—41no—
102C2×D244—41no—
103M16×C61—22no—
104M16×C61—41no—
105M16×C61—41no—
106M16×C61—41no—
107C2×C8×C61—41no—
108C2×C8×C61—41no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C22⋊C4×C61—21no—
113C22⋊C4×C61—41no—
114C22⋊C4×C61—41no—
115C42×C61—21no—
116C22⋊Dic61—21no—
117C22⋊Dic61—22no—
118#11758201—42no—
119#11758201—41no—
120#11758201—41no—
121#11758198—41no—
122#11758198—41no—
123C4⋊Dic61—21no—
124C4⋊Dic61—41no—
125C4×Dic61—21no—
126C4×Dic61—41no—
127#11758194—41no—
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#12411yescharacteristic
146C22×D61—41no—
147C22×D61—81no—
148C22×D61—161no—
149Q8×C61—41no—
150Q8×C61—41no—
151Q8×C61—41no—
152Q8×C61—81no—
153D4×C61—41no—
154D4×C61—41no—
155D4×C61—41no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—81no—
165C2×C4×C61C2×C22⋊C411yescharacteristic
166C2×C4×C61C4×D411yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12711yescharacteristic
169C22⋊D61—161no—
170C22⋊D61—161no—
171C2×Dic61—42no—
172C2×Dic61—41no—
173D244—41no—
174D244—41no—
175D244—81no—
176C8×C61—42no—
177C8×C61—81no—
178#92418—21no—
179#92418—21no—
180#92418—21no—
181#92418—81no—
182#92418#9911yescharacteristic
183#92418#9911yescharacteristic
184#13232—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61#39011yescharacteristic
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—81no—
195C2×D61—162no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—81no—
200C4×C61—81no—
201C4×C61C2×#9911yescharacteristic
202C4×C61#41011yescharacteristic
203Dic61—81no—
204Dic61—82no—
205#3057—611no—
206#2135D6111yescharacteristic, complemented
207#2134—611no—
208#2123—611no—
209#2122—611no—
210#1813—611no—
211#1757—611no—
212C2×C61—21no—
213C2×C61—41no—
214C2×C61—41no—
215C2×C61—41no—
216C2×C61#196411yescharacteristic
217D61—161no—
218C22×SD16—611no—
219C2×#131—611no—
220#536C2×D6111yescharacteristic
221#535—611no—
222#468—1221no—
223#468—1221no—
224#465—1221no—
225#465—1221no—
226C2×#108—1221no—
227#427—611no—
228#419—611no—
229#418—611no—
230#413—611no—
231#413C2×D6111yescharacteristic
232#413C2×D6111yescharacteristic
233#410—611no—
234#405—1221no—
235#356—22no—
236#355—1222no—
237#352—1222no—
238AΣL(1,𝔽16)—1222no—
239#344—1221no—
240#338—1221no—
241C61#1323211yescharacteristic, complemented, Sylow
242C2×#41C22×D6111yescharacteristic
243C22×Q8C4×D6111yescharacteristic
244C22×D4—611no—
245C2×SD16—1221no—
246C2×SD16—1221no—
247C2×SD16—2441no—
248C2×SD16—2441no—
249#131—2441no—
250#131—2441no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—611no—
254C2×M16—611no—
255C2×M16—611no—
256C2×M16—1221no—
257C22×C8—611no—
258C22×C8—611no—
259C22×C8—611no—
260C4⋊Q8—21no—
261#127—1221no—
262#124—21no—
263#124—1221no—
264#123—21no—
265#123—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268#121—1221no—
269#121—1221no—
270#120—1221no—
271#120—1221no—
272#117—611no—
273#117—1221no—
274#117C4×D6111yescharacteristic
275#108—2441no—
276#108—2441no—
277#107—1221no—
278#106—1221no—
279#103—1221no—
280#103—1221no—
281#102—1221no—
282#102—1221no—
283#101—21no—
284#101—1221no—
285#101C22⋊D6112yes—
286#100—21no—
287#100—21no—
288#100—1221no—
289#99—612no—
290#99—1221no—
291#97—2441no—
292C4×C8—2441no—
293C24—1221no—
294#41—41no—
295#41—41no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8C2×C22⋊D6111yescharacteristic
301C2×D4—1221no—
302C2×D4—1221no—
303C2×D4—2441no—
304C2×D4—2441no—
305C2×D4—2441no—
306C2×D4—2441no—
307C2×D4#1175822211yescharacteristic
308C2×D4D4×D6111yescharacteristic
309C2×D4#1175822411yescharacteristic
310C22×C4—21no—
311C22×C4C2×C4×D6111yescharacteristic
312C22×C4D4×D6111yescharacteristic
313C22×C4D4×D6111yescharacteristic
314SD16—2441no—
315SD16—2441no—
316SD16—2442no—
317M16—22no—
318M16—41no—
319M16—41no—
320M16—41no—
321M16—2441no—
322M16—2441no—
323M16—2441no—
324M16—2441no—
325M16—2441no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—611no—
329C2×C8—611no—
330C2×C8—611no—
331C2×C8—611no—
332C2×C8—1221no—
333C2×C8—1221no—
334C2×C8—1221no—
335C2×C8—1221no—
336C2×C8—1221no—
337C2×C8—1221no—
338C2×C8—2441no—
339C4⋊C4—21no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—1221no—
343C4⋊C4—2441no—
344C22⋊C4—21no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—1221no—
348C22⋊C4—1222no—
349C22⋊C4—2441no—
350C22⋊C4—2441no—
351C42—21no—
352C42—1221no—
353C42—2441no—
354C23—9761no—
355C23—4881no—
356C23—2441no—
357C23#1451973711yescharacteristic
358C23#1451975011yescharacteristic
359C23#1451976311yescharacteristic
360Q8—41no—
361Q8—41no—
362Q8—41no—
363Q8—81no—
364D4—41no—
365D4—41no—
366D4—41no—
367D4—9761no—
368D4—9761no—
369D4—4881no—
370D4—2441no—
371D4—2441no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4—2442no—
382C2×C4—2441no—
383C2×C4C22⋊C4×D6111yescharacteristic
384C2×C4C4×C22⋊D6111yescharacteristic
385C2×C4#1451974911yescharacteristic
386C2×C4#1451977311yescharacteristic
387C8—42no—
388C8—81no—
389C8—611no—
390C8—611no—
391C8—1221no—
392C8—1221no—
393C8—1221no—
394C8—4881no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—9762no—
402C22—4881no—
403C22—4881no—
404C22—4881no—
405C22#1683966211yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—81no—
410C4—81no—
411C4—4881no—
412C4—4882no—
413C4#99×D6111yescharacteristic
414C4#1683973411yescharacteristic
415C2—21no—
416C2—41no—
417C2—41no—
418C2—41no—
419C2—9761no—
420C2#2144665311yescharacteristic
subgroupG/H
centerC2#21446653
commutatorC22×C4×C61C22×C4
FrattiniC2×#41C22×D61
FittingC61×#2135C2
socleC2×C61#1964
prime radical#16840326C22
cosocleC2×#41×C61C23
hypercenter#2135D61
hypocenterC61#13232
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #2135

Complements

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

NH ≅ G/Ntype
3. #21447473C2semidirect
5. #21445867C2semidirect
6. #21445622C2semidirect
7. C61×#2135C2semidirect
8. C22×#11758201C4semidirect
16. #16839798C4semidirect
17. #16839771C4semidirect
18. #16839766C4semidirect
206. #2135D61semidirect
241. C61#13232semidirect

Maximal subgroups

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

subgroupindex
#214554602
#214475102
#214474732
#214474702
#214458672
#214456222
C61×#21352
#1323261