Finite Group Databasebeta

← #29696792

Subgroups — #29696792

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840497C211yesmaximal, characteristic
2M16×Dic122C212yesmaximal, complemented
3#21446667C214yesmaximal, complemented
4#21446659C212yesmaximal
5#21445815C211yesmaximal, characteristic, complemented
6#21445810C212yesmaximal, complemented
7#21445610C212yesmaximal, complemented
8C4×#101×C61C211yesmaximal, characteristic, complemented
9#16840497—22no—
10#16840497—24no—
11#16840497—22no—
12C2×#14519774C2211yescharacteristic
13C2×#14519766C2212yes—
14C2×#14519765C412yes—
15C2×#14519762C414yes—
16C2×C4×Dic122C2211yescharacteristic
17C2×C4×D244C412yes—
18C2×#14519739C2212yes—
19#16839892—22no—
20#16839890—24no—
21C8×Dic122—22no—
22#16839786C2212yes—
23#16839785C2214yes—
24C2×#14519696C2212yes—
25C2×#14519693C2214yes—
26C2×#14519693C2214yes—
27M16×Dic61C2214yes—
28#16839531C2212yes—
29C2×C4×Q8×C61C2211yescharacteristic
30C2×#101×C61C2212yes—
31C4×M16×C61C2212yes—
32C61×#343C2211yescharacteristic
33C61×#343C2211yescharacteristic
34C22×Dic122C2×C411yescharacteristic
35C22×D244C2×C412yes—
36C22×D244C2×C412yes—
37C22×D244C2312yes—
38C22×C4×D61C2×C414yes—
39C2×#11758194C2314yes—
40#14519774—22no—
41#14519774—22no—
42#14519766—24no—
43#14519766—24no—
44#14519765—24no—
45#14519765—24no—
46#14519762—28no—
47#14519762—28no—
48C4×Dic122D412yes—
49C4×Dic122D414yes—
50C4×Dic122D412yes—
51C4×D244—24no—
52C4×D244—22no—
53C4×D244—22no—
54#14519739—24no—
55#14519739—24no—
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×#11758196C2×C412yes—
58C2×#11758198C2×C414yes—
59C2×#11758198C2×C414yes—
60C2×#11758198C2×C414yes—
61C2×C4×Dic61C2312yes—
62C4⋊#92418—22no—
63C4⋊#92418—22no—
64#14519706—24no—
65#14519706—24no—
66#14519696—42no—
67#14519693—28no—
68#14519693—44no—
69#14519687—24no—
70C8×Dic61—24no—
71C22×Q8×C61C2311yescharacteristic
72C2×M16×C61C2312yes—
73C2×M16×C61C2312yes—
74C4×Q8×C61—22no—
75C4×Q8×C61—22no—
76C2×C4⋊C4×C61C2×C412yes—
77C2×C4⋊C4×C61C2311yescharacteristic
78C2×C42×C61C2×C412yes—
79C2×C42×C61C2311yescharacteristic
80#101×C61—24no—
81#97×C61—22no—
82C4×C8×C61—22no—
83C23×D61C22×C414yes—
84C2×Q8×C61—22no—
85C2×Q8×C61—22no—
86C2×Q8×C61—21no—
87C2×Q8×C61—21no—
88C22×C4×C61C22×C412yes—
89C22×C4×C61C22×C411yescharacteristic
90C22×C4×C61C22×C412yes—
91C22×C4×C61C22×C411yescharacteristic
92C22×C4×C61C2411yescharacteristic
93C22×Dic61C22×C412yes—
94C22×Dic61C22×C412yes—
95C2×D244—22no—
96C2×D244—22no—
97C2×D244—24no—
98C2×D244—24no—
99C2×D244—22no—
100C2×D244—22no—
101C2×D244—24no—
102C2×D244—44no—
103C2×D244—82no—
104C2×C4×D61—24no—
105C2×C4×D61—24no—
106C2×C4×D61—84no—
107C2×Dic122—42no—
108C2×Dic122C22⋊C412yes—
109C2×Dic122C22⋊C412yes—
110M16×C61—22no—
111M16×C61—22no—
112M16×C61—22no—
113M16×C61#4114yes—
114C2×C8×C61—22no—
115C2×C8×C61—22no—
116C4⋊C4×C61—24no—
117C4⋊C4×C61—22no—
118C42×C61—22no—
119C42×C61—22no—
120C42×C61C2×D412yes—
121C42×C61C2×D412yes—
122#11758198—28no—
123#11758198—24no—
124#11758198—24no—
125#11758198—28no—
126C4⋊Dic61C22⋊C412yes—
127C4⋊Dic61C22⋊C411yescharacteristic
128C4⋊Dic61C22⋊C411yescharacteristic
129#11758196C22⋊C414yes—
130#11758196C22⋊C414yes—
131C4×Dic61C2×D414yes—
132C4×Dic61C2×D414yes—
133#11758194—24no—
134#11758194—24no—
135C2×#92418—24no—
136C23×C61C23×C411yescharacteristic
137C22×D61—24no—
138C22×D61—24no—
139C22×D61—24no—
140C22×D61—44no—
141C22×D61—44no—
142Q8×C61—42no—
143Q8×C61—42no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61C2×C22⋊C412yes—
152C2×C4×C61C2×C22⋊C411yescharacteristic
153C2×C4×C61C2×C22⋊C411yescharacteristic
154C2×C4×C61C22×D412yes—
155C2×C4×C61C2×#4112yes—
156C2×C4×C61H(5,𝔽2)11yescharacteristic
157C2×C4×C612−1+411yescharacteristic
158C2×Dic61—42no—
159C2×Dic61—42no—
160C2×Dic61C2×C22⋊C412yes—
161C2×Dic61C2×C22⋊C412yes—
162C2×Dic61C2×C22⋊C414yes—
163D244—44no—
164D244—44no—
165D244—44no—
166D244—44no—
167D244—84no—
168C4×D61—84no—
169C4×D61—84no—
170Dic122—24no—
171Dic122—42no—
172C8×C61—24no—
173C8×C61—42no—
174#92418—44no—
175#22539—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61C22×C22⋊C411yescharacteristic
179C22×C61C4×#4111yescharacteristic
180C22×C61#52011yescharacteristic
181C2×D61—44no—
182C2×D61—44no—
183C2×D61—44no—
184C2×D61—84no—
185C2×D61—84no—
186C4×C61—21no—
187C4×C61—22no—
188C4×C61—42no—
189C4×C61—42no—
190C4×C61#41312yes—
191C4×C61#54812yes—
192C4×C61#55012yes—
193Dic61—24no—
194Dic61—22no—
195Dic61—22no—
196C2×#559—611no—
197Q8×M16—612no—
198#1967—614no—
199#1965—612no—
200#1801—611no—
201#1800—612no—
202C4×#101D6111yescharacteristic, complemented
203C4×#100—612no—
204C2×C61—21no—
205C2×C61—21no—
206C2×C61#234111yescharacteristic
207C2×C61C2×#41311yescharacteristic
208C2×C61#288811yescharacteristic
209D61—84no—
210#559—1222no—
211#559—1224no—
212#559—1222no—
213C2×#127—611no—
214C2×#124—612no—
215C2×#124—612no—
216C2×#123—614no—
217C2×C4×Q8—611no—
218C2×C4×Q8C2×D6111yescharacteristic
219C2×C4×D4—612no—
220C2×C4×D4—612no—
221#446—1222no—
222#446—1224no—
223C8×Q8—1222no—
224#423—614no—
225#423—612no—
226C2×#101—612no—
227C2×#101C2×D6112yes—
228C2×#100—614no—
229C2×#100—614no—
230C4×M16—614no—
231C4×M16C2×D6112yes—
232#343—612no—
233#343C2×D6111yescharacteristic
234#343C2×D6111yescharacteristic
235C61#2253911yescharacteristic, complemented, Sylow
236C22×Q8—611no—
237C22×Q8C22×D6111yescharacteristic
238C22×D4—612no—
239C22×D4—612no—
240C22×D4—612no—
241C23×C4—614no—
242C2×M16—614no—
243C2×M16C22×D6112yes—
244C2×M16C22×D6112yes—
245#127—1222no—
246#127—1222no—
247#124—1224no—
248#124—1224no—
249#124—1224no—
250#124—1224no—
251#123—1228no—
252#123—1228no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—612no—
256C4×Q8—614no—
257C4×Q8—612no—
258C4×D4—1224no—
259C4×D4—1222no—
260C4×D4—1222no—
261C4×D4—1224no—
262C4×D4—1224no—
263C2×C4⋊C4—611no—
264C2×C4⋊C4—612no—
265C2×C4⋊C4C4×D6112yes—
266C2×C4⋊C4C22×D6111yescharacteristic
267C2×C22⋊C4—614no—
268C2×C22⋊C4—614no—
269C2×C22⋊C4—614no—
270C2×C42—612no—
271C2×C42C4×D6112yes—
272C2×C42C22×D6111yescharacteristic
273C4⋊C8—1224no—
274C4⋊C8—1224no—
275C4⋊C8—1222no—
276C4⋊C8—1222no—
277#101—24no—
278#101—2442no—
279#100—1228no—
280#100—2444no—
281#97—22no—
282#97—1224no—
283C4×C8—22no—
284C4×C8—1224no—
285C24—614no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—612no—
291C2×Q8—612no—
292C2×Q8—2442no—
293C2×D4—1224no—
294C2×D4—1224no—
295C2×D4—1222no—
296C2×D4—1222no—
297C2×D4—1224no—
298C2×D4—1222no—
299C2×D4—1222no—
300C2×D4—4882no—
301C2×D4—2444no—
302C22×C4—612no—
303C22×C4—612no—
304C22×C4—1224no—
305C22×C4—1224no—
306C22×C4—4884no—
307C22×C4C2×C4×D6112yes—
308C22×C4C2×C4×D6111yescharacteristic
309C22×C4C2×C4×D6112yes—
310C22×C4C2×C4×D6111yescharacteristic
311C22×C4C23×D6111yescharacteristic
312M16—22no—
313M16—22no—
314M16—22no—
315M16—1224no—
316M16—1224no—
317M16#1175822214yes—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—1224no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—614no—
324C4⋊C4—614no—
325C4⋊C4—612no—
326C4⋊C4—611no—
327C4⋊C4—611no—
328C22⋊C4—1224no—
329C22⋊C4—1224no—
330C22⋊C4—1228no—
331C22⋊C4—1228no—
332C42—22no—
333C42—22no—
334C42—614no—
335C42—614no—
336C42D4×D6112yes—
337C42D4×D6112yes—
338C23—1224no—
339C23—1224no—
340C23—1224no—
341C23—2444no—
342C23—2444no—
343C23C22×C4×D6111yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—1224no—
347Q8—2442no—
348D4—4884no—
349D4—2444no—
350D4—2444no—
351D4—2444no—
352D4—2444no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—614no—
361C2×C4—612no—
362C2×C4—612no—
363C2×C4—4884no—
364C2×C4—4884no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4C22⋊C4×D6111yescharacteristic
368C2×C4C22⋊C4×D6112yes—
369C2×C4C22⋊C4×D6111yescharacteristic
370C2×C4C2×D4×D6112yes—
371C2×C4C2×#1175822212yes—
372C2×C4#1451982311yescharacteristic
373C2×C4#1451982411yescharacteristic
374C8—24no—
375C8—42no—
376C8—2444no—
377C22—21no—
378C22—21no—
379C22—4884no—
380C22—4884no—
381C22—2444no—
382C22—2444no—
383C22—2444no—
384C22C2×C22⋊C4×D6111yescharacteristic
385C22C4×#1175822211yescharacteristic
386C22#1684024211yescharacteristic
387C4—21no—
388C4—22no—
389C4—42no—
390C4—42no—
391C4—1224no—
392C4—1222no—
393C4—1222no—
394C4#1683974512yes—
395C4#1684042112yes—
396C4#1684043312yes—
397C2—21no—
398C2—21no—
399C2—4884no—
400C2#2144865911yescharacteristic
401C2C2×#1683974511yescharacteristic
402C2#2145429611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#101×C61C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519774C22
cosocleC22×C4×C61C24
hypercenterC4×#101D61
hypocenterC61#22539
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. M16×Dic122C2semidirect
3. #21446667C2semidirect
5. #21445815C2semidirect
6. #21445810C2semidirect
7. #21445610C2semidirect
8. C4×#101×C61C2semidirect
202. C4×#101D61semidirect
235. C61#22539semidirect

Maximal subgroups

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

subgroupindex
C2×#168404972
M16×Dic1222
#214466672
#214466592
#214458152
#214458102
#214456102
C4×#101×C612
#2253961