Finite Group Databasebeta

← #29576355

Subgroups — #29576355

Order 15616.

464 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839984C211yesmaximal, characteristic
2#21444418C211yesmaximal, characteristic
3#21444266C211yesmaximal, characteristic, complemented
4#21444170C211yesmaximal, characteristic
5#21443688C211yesmaximal, characteristic, complemented
6#21443668C211yesmaximal, characteristic, complemented
7C61×#2984C211yesmaximal, characteristic, complemented
8C22×#11758202C411yescharacteristic
9C22×#11758201C411yescharacteristic
10C2×#14519765C2211yescharacteristic
11#16839984—21no—
12#16839984—21no—
13#16839984—21no—
14#16839984—21no—
15#16839819C2211yescharacteristic
16C2×C4⋊#92418C2211yescharacteristic
17C2×#14519700C411yescharacteristic
18C2×#14519697C411yescharacteristic
19C4×#11758194C2211yescharacteristic
20#16839501—22no—
21Q8⋊#92418—21no—
22Q8⋊#92418—21no—
23#16839485—21no—
24#16839485—21no—
25C2×C4×Q8×C61C2211yescharacteristic
26C61×#446—21no—
27C61×#446—22no—
28C8×Q8×C61—21no—
29C61×#432—21no—
30C61×#432C2211yescharacteristic
31C61×#431—21no—
32C61×#431C2211yescharacteristic
33C22×Dic122C2×C411yescharacteristic
34C22×D244C2×C411yescharacteristic
35C2×#11758202—21no—
36C2×#11758202—21no—
37C2×#11758202—42no—
38C2×#11758201—21no—
39C2×#11758201—21no—
40C2×#11758201—42no—
41C2×#11758194—21no—
42C22×#92418C2×C411yescharacteristic
43C22×#92418C2×C411yescharacteristic
44#14519765—21no—
45#14519765—21no—
46#14519765—21no—
47#14519765—21no—
48C2×#11758198—21no—
49C2×#11758198—21no—
50C4⋊#92418—21no—
51C4⋊#92418—21no—
52C4⋊#92418—21no—
53#14519700—21no—
54#14519700—21no—
55#14519697—21no—
56#14519697—21no—
57C22⋊#92418—21no—
58#14519688—21no—
59#14519688—21no—
60C4×#92418—21no—
61C22×Q8×C61C2×C411yescharacteristic
62C4×Q8×C61—21no—
63C4×Q8×C61—21no—
64C4×Q8×C61D411yescharacteristic
65C4×Q8×C61D411yescharacteristic
66C4×Q8×C61D411yescharacteristic
67C4×Q8×C61D411yescharacteristic
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61C2×C411yescharacteristic
70C2×C42×C61—21no—
71C2×C42×C61C2311yescharacteristic
72C4⋊C8×C61—22no—
73C4⋊C8×C61—22no—
74C4⋊C8×C61—21no—
75C4⋊C8×C61—21no—
76C22⋊C8×C61—21no—
77C22⋊C8×C61D411yescharacteristic
78C22⋊C8×C61D411yescharacteristic
79#97×C61—22no—
80#97×C61D412yes—
81C4×C8×C61—22no—
82C4×C8×C61—21no—
83C23×D61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—22no—
86C2×Q8×C61—41no—
87C2×Q8×C61C22⋊C411yescharacteristic
88C2×Q8×C61C22⋊C411yescharacteristic
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61C22×C411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C22×Dic61—21no—
95C2×D244—21no—
96C2×D244—21no—
97C2×D244—41no—
98C2×D244—41no—
99C2×Dic122—21no—
100C2×Dic122—21no—
101C2×Dic122—41no—
102C2×Dic122—41no—
103C2×C8×C61—21no—
104C2×C8×C61—21no—
105C2×C8×C61—22no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61C22⋊C411yescharacteristic
112C4⋊C4×C61C22⋊C411yescharacteristic
113C42×C61—21no—
114C42×C61—21no—
115C42×C61—21no—
116C42×C61—21no—
117C42×C61C2×D411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61#4111yescharacteristic
121#11758202—42no—
122#11758202—42no—
123#11758201—42no—
124#11758201—42no—
125#11758198—41no—
126#11758198—41no—
127#11758198—41no—
128#11758198—41no—
129#11758194—22no—
130#11758194—41no—
131C2×#92418—21no—
132C2×#92418—21no—
133C2×#92418—41no—
134C2×#92418—41no—
135C2×#92418—41no—
136C2×#92418—41no—
137C23×C61C4×D411yescharacteristic
138C22×D61—41no—
139C22×D61—41no—
140C22×D61—41no—
141C22×D61—81no—
142C22×D61—81no—
143Q8×C61—21no—
144Q8×C61—21no—
145Q8×C61—21no—
146Q8×C61—42no—
147Q8×C61#10012yes—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61C2×C22⋊C411yescharacteristic
161C2×C4×C61C4×D411yescharacteristic
162C2×C4×C61#12011yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12311yescharacteristic
166C2×Dic61—41no—
167C2×Dic61—81no—
168D244—41no—
169D244—41no—
170D244—81no—
171Dic122—41no—
172Dic122—41no—
173Dic122—81no—
174C8×C61—41no—
175C8×C61—41no—
176C8×C61—42no—
177#92418—42no—
178#92418—41no—
179#92418—41no—
180#92418—81no—
181#9244—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61#38611yescharacteristic
185C22×C61#44311yescharacteristic
186C22×C61#44411yescharacteristic
187C2×D61—81no—
188C2×D61—81no—
189C2×D61—81no—
190C2×D61—161no—
191C2×D61—161no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—22no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61—41no—
201C4×C61C2×#10011yescharacteristic
202C4×C61#41311yescharacteristic
203Dic61—81no—
204C2×#464—611no—
205#2984D6111yescharacteristic, complemented
206#1527—611no—
207#1491—611no—
208#1469—611no—
209#1290—611no—
210#1283—611no—
211C2×C61—21no—
212C2×C61—21no—
213C2×C61#187611yescharacteristic
214C2×C61#190011yescharacteristic
215C2×C61#190511yescharacteristic
216D61—161no—
217C22×Q16—611no—
218C22×SD16—611no—
219C2×#124—611no—
220C2×C4×Q8C2×D6111yescharacteristic
221#464—1221no—
222#464—1221no—
223#464—1221no—
224#464—1221no—
225#446—21no—
226#446—22no—
227C8×Q8—21no—
228#432—21no—
229#432—611no—
230#432C2×D6111yescharacteristic
231#431—21no—
232#431C2×D6111yescharacteristic
233C2×C4⋊C8—611no—
234C2×#103—611no—
235C2×#102—611no—
236C4×M16—611no—
237#329—1222no—
238Q8⋊C8—1221no—
239Q8⋊C8—1221no—
240#323—1221no—
241#323—1221no—
242C61#924411yescharacteristic, complemented, Sylow
243C22×Q8—611no—
244C22×Q8C4×D6111yescharacteristic
245C22×D4—611no—
246C2×Q16—1221no—
247C2×Q16—1221no—
248C2×Q16—2442no—
249C2×SD16—1221no—
250C2×SD16—1221no—
251C2×SD16—2442no—
252C2×M16—1221no—
253C22×C8—611no—
254C22×C8—611no—
255#124—1221no—
256#124—1221no—
257#124—1221no—
258#124—1221no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8D24411yescharacteristic
262C4×Q8D24411yescharacteristic
263C4×Q8C22⋊D6111yescharacteristic
264C4×Q8C22⋊D6111yescharacteristic
265C2×C4⋊C4—21no—
266C2×C4⋊C4C4×D6111yescharacteristic
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C42—21no—
270C2×C42C22×D6111yescharacteristic
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—1221no—
276C4⋊C8—1221no—
277C4⋊C8—1221no—
278#103—1221no—
279#103—1221no—
280#102—1221no—
281#102—1221no—
282C22⋊C8—21no—
283C22⋊C8—1221no—
284C22⋊C8C22⋊D6111yescharacteristic
285C22⋊C8C22⋊D6111yescharacteristic
286#97—22no—
287#97—1221no—
288#97—1221no—
289#97C22⋊D6112yes—
290C4×C8—22no—
291C4×C8—21no—
292C4×C8—1221no—
293C24—1221no—
294C2×Q8—21no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—1221no—
298C2×Q8—1221no—
299C2×Q8—2441no—
300C2×Q8—2441no—
301C2×Q8#1175819811yescharacteristic
302C2×Q8#1175819811yescharacteristic
303C2×D4—1221no—
304C2×D4—1221no—
305C2×D4—2441no—
306C2×D4—2441no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—1221no—
310C22×C4C2×C4×D6111yescharacteristic
311C22×C4#1175822211yescharacteristic
312C22×C4C2×C22⋊D6111yescharacteristic
313Q16—2442no—
314Q16—2442no—
315SD16—2442no—
316SD16—2442no—
317M16—1222no—
318M16—2441no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—1221no—
323C2×C8—1221no—
324C2×C8—2441no—
325C2×C8—2441no—
326C2×C8—2441no—
327C2×C8—2441no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4#1175819811yescharacteristic
334C4⋊C4#1175819811yescharacteristic
335C22⋊C4—2441no—
336C22⋊C4—2441no—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42C2×D24411yescharacteristic
344C42#1175822211yescharacteristic
345C42C2×C22⋊D6111yescharacteristic
346C42C2×C22⋊D6111yescharacteristic
347C23—4881no—
348C23—4881no—
349C23—2441no—
350C23—2441no—
351C23—2441no—
352C23C4×C22⋊D6111yescharacteristic
353Q8—21no—
354Q8—21no—
355Q8—21no—
356Q8—42no—
357Q8—4881no—
358Q8—2441no—
359Q8—2441no—
360Q8#1451969312yes—
361D4—4881no—
362D4—2441no—
363D4—2441no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—4881no—
377C2×C4—2441no—
378C2×C4C2×#1175819811yescharacteristic
379C2×C4C4×C22⋊D6111yescharacteristic
380C2×C4#1451974611yescharacteristic
381C2×C4#1451975111yescharacteristic
382C2×C4#1451975111yescharacteristic
383C2×C4#1451976011yescharacteristic
384C8—41no—
385C8—41no—
386C8—42no—
387C8—4881no—
388C8—2442no—
389C8—2441no—
390C8—2441no—
391C22—21no—
392C22—21no—
393C22—9761no—
394C22—9761no—
395C22—4881no—
396C22—4881no—
397C22—4881no—
398C22#1683964111yescharacteristic
399C22#1683987711yescharacteristic
400C22#1683988211yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—22no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—4881no—
411C4C2×#1451969311yescharacteristic
412C4#1683974411yescharacteristic
413C2—21no—
414C2—21no—
415C2—9761no—
416C2#2144614611yescharacteristic
417C2#2144628211yescharacteristic
418C2#2144631311yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2984C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#2984D61
hypocenterC61#9244
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2984

Complements

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

NH ≅ G/Ntype
3. #21444266C2semidirect
5. #21443688C2semidirect
6. #21443668C2semidirect
7. C61×#2984C2semidirect
205. #2984D61semidirect
242. C61#9244semidirect

Maximal subgroups

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

subgroupindex
C2×#168399842
#214444182
#214442662
#214441702
#214436882
#214436682
C61×#29842
#924461