Finite Group Databasebeta

← #25113178

Subgroups — #25113178

Order 11008.

938 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#10621877C212yesmaximal
2#19135589C214yesmaximal
3#19135585C212yesmaximal
4#19134591C211yesmaximal, characteristic
5#19134591C212yesmaximal
6C4×#13063244C211yesmaximal, characteristic
7C4×#13063244C212yesmaximal
8C2×C43×#557C211yesmaximal, characteristic
9#14583030—24no—
10#14583030—22no—
11Q8×C43⋊C8—22no—
12#14582923C2212yes—
13#14582923C2214yes—
14C2×#13063244C2211yescharacteristic
15C2×#13063244C2212yes—
16C2×#13063244C2211yescharacteristic
17C4×#10621877C2212yes—
18C4×#10621877C2214yes—
19#14582668C2214yes—
20#14582668C2214yes—
21#14582668C2212yes—
22#14582668C2212yes—
23C43×#557—22no—
24C43×#557—24no—
25C43×#557—22no—
26C2×C4⋊Q8×C43C412yes—
27C2×C4⋊Q8×C43C2211yescharacteristic
28C2×#125×C43C414yes—
29C2×#125×C43C2212yes—
30C2×C4×Q8×C43C412yes—
31C2×C4×Q8×C43C2212yes—
32C2×C4×Q8×C43C2211yescharacteristic
33C2×C4×Q8×C43C2211yescharacteristic
34C2×#10621877C2312yes—
35C2×#10621877C2314yes—
36C2×#10621877C2312yes—
37C4⋊(C43⋊C8)—24no—
38C4⋊(C43⋊C8)—22no—
39C4⋊(C43⋊C8)—24no—
40C4⋊(C43⋊C8)—22no—
41#13063244—22no—
42#13063244—24no—
43#13063244Q814yescomplemented
44#13063236—24no—
45#13063236—22no—
46C4×C43⋊C8—24no—
47C4×C43⋊C8—22no—
48C22×Q8×C43C2×C412yes—
49C22×Q8×C43C2×C411yescharacteristic
50C22×Q8×C43C2311yescharacteristic
51C4⋊Q8×C43—24no—
52C4⋊Q8×C43—24no—
53C4⋊Q8×C43—22no—
54C4⋊Q8×C43—22no—
55#125×C43—28no—
56#125×C43—28no—
57#125×C43—24no—
58#125×C43—24no—
59C4×Q8×C43—24no—
60C4×Q8×C43—22no—
61C4×Q8×C43—22no—
62C4×Q8×C43—24no—
63C4×Q8×C43—24no—
64C4×Q8×C43—22no—
65C4×Q8×C43—22no—
66C4×Q8×C43D412yes—
67C4×Q8×C43D414yes—
68C4×Q8×C43D412yes—
69C2×C4⋊C4×C43C2×C414yes—
70C2×C4⋊C4×C43C2×C412yes—
71C2×C4⋊C4×C43C2×C414yes—
72C2×C4⋊C4×C43C2×C412yes—
73C2×C4⋊C4×C43C2×C414yes—
74C2×C4⋊C4×C43C2×C411yescharacteristic
75C2×C4⋊C4×C43C2×C412yes—
76C2×C4⋊C4×C43C2311yescharacteristic
77C2×C4⋊C4×C43C2312yes—
78C2×C42×C43C2×C414yes—
79C2×C42×C43C2×C412yes—
80C2×C42×C43C2312yes—
81C2×C42×C43C2311yescharacteristic
82C2×Q8×C43—22no—
83C2×Q8×C43—22no—
84C2×Q8×C43—22no—
85C2×Q8×C43—22no—
86C2×Q8×C43—42no—
87C2×Q8×C43—44no—
88C2×Q8×C43C22⋊C412yes—
89C2×Q8×C43C22⋊C412yes—
90C2×Q8×C43C2×Q812yes—
91C2×Q8×C43#4112yes—
92C22×C4×C43C22×C411yescharacteristic
93C22×C4×C43C22×C412yes—
94C22×C4×C43C22×C412yes—
95C22×C4×C43C22×C414yes—
96C22×C4×C43C22×C412yes—
97C22×C4×C43C22×C411yescharacteristic
98C22×C4×C43C22×C412yes—
99C22×C4×C43C2411yescharacteristic
100C4⋊C4×C43—24no—
101C4⋊C4×C43—24no—
102C4⋊C4×C43—24no—
103C4⋊C4×C43—28no—
104C4⋊C4×C43—28no—
105C4⋊C4×C43—24no—
106C4⋊C4×C43—22no—
107C4⋊C4×C43—24no—
108C4⋊C4×C43C22⋊C412yes—
109C4⋊C4×C43C22⋊C411yescharacteristic
110C4⋊C4×C43C22⋊C411yescharacteristic
111C4⋊C4×C43C22⋊C414yes—
112C4⋊C4×C43C22⋊C414yes—
113C42×C43—24no—
114C42×C43—24no—
115C42×C43—22no—
116C42×C43—22no—
117C42×C43C2×D414yes—
118C42×C43C2×D414yes—
119C42×C43C2×D412yes—
120C42×C43C2×D412yes—
121#10621877—22no—
122#10621877—22no—
123#10621877—24no—
124#10621877—22no—
125#10621877—24no—
126#10621877C2×Q814yes—
127C2×C43⋊C8—22no—
128C2×C43⋊C8—24no—
129C2×C43⋊C8—22no—
130C23×C43C23×C411yescharacteristic
131Q8×C43—24no—
132Q8×C43—24no—
133Q8×C43—24no—
134Q8×C43—42no—
135Q8×C43—44no—
136Q8×C43—44no—
137C2×C4×C43—22no—
138C2×C4×C43—22no—
139C2×C4×C43—22no—
140C2×C4×C43—22no—
141C2×C4×C43—24no—
142C2×C4×C43—42no—
143C2×C4×C43—41no—
144C2×C4×C43—42no—
145C2×C4×C43—42no—
146C2×C4×C43—44no—
147C2×C4×C43C2×C22⋊C412yes—
148C2×C4×C43C2×C22⋊C412yes—
149C2×C4×C43C2×C22⋊C411yescharacteristic
150C2×C4×C43C2×C22⋊C412yes—
151C2×C4×C43C2×C22⋊C414yes—
152C2×C4×C43C2×C22⋊C411yescharacteristic
153C2×C4×C43C4×Q814yes—
154C2×C4×C43C22×D412yes—
155C2×C4×C43C22×Q811yescharacteristic
156C2×C4×C43C2×#4111yescharacteristic
157C2×C4×C432−1+412yes—
158C43⋊C8—24no—
159C43⋊C8—44no—
160C43⋊C8—42no—
161#22517—431nomaximal, Sylow
162C22×C43—21no—
163C22×C43—21no—
164C22×C43C22×C22⋊C411yescharacteristic
165C22×C43C2×C4×Q811yescharacteristic
166C22×C43#51911yescharacteristic
167C4×C43—24no—
168C4×C43—21no—
169C4×C43—24no—
170C4×C43—22no—
171C4×C43—22no—
172C4×C43—22no—
173C4×C43—44no—
174C4×C43—42no—
175C4×C43#41312yes—
176C4×C43#54812yes—
177C4×C43D4×Q812yes—
178C2×#557D4311yescharacteristic
179Q8×M16—432no—
180#1998—434no—
181#1997—432no—
182#1801—432no—
183#1801—431no—
184C4×#101—432no—
185C4×#101—431no—
186C2×C43—21no—
187C2×C43—21no—
188C2×C43Q8×C22⋊C411yescharacteristic
189C2×C43C2×#41311yescharacteristic
190C2×C43#288911yescharacteristic
191#557—22no—
192#557—24no—
193#557—22no—
194C2×C4⋊Q8Dic4312yes—
195C2×C4⋊Q8C2×D4311yescharacteristic
196C2×#125Dic4314yes—
197C2×#125C2×D4312yes—
198C2×C4×Q8Dic4312yes—
199C2×C4×Q8C2×D4312yes—
200C2×C4×Q8C2×D4311yescharacteristic
201C2×C4×Q8C2×D4311yescharacteristic
202#446—864no—
203#446—862no—
204C8×Q8—862no—
205#423—432no—
206#423—434no—
207C2×#101—431no—
208C2×#101—431no—
209C2×#101—432no—
210C4×M16—432no—
211C4×M16—434no—
212#343—434no—
213#343—432no—
214#343—434no—
215#343—432no—
216C43#2251711yescharacteristic, complemented, Sylow
217C22×Q8C2×Dic4312yes—
218C22×Q8C2×Dic4311yescharacteristic
219C22×Q8C22×D4311yescharacteristic
220C2×M16—432no—
221C2×M16—432no—
222C2×M16—434no—
223C4⋊Q8—24no—
224C4⋊Q8—24no—
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227#125—28no—
228#125—28no—
229#125—24no—
230#125—24no—
231C4×Q8—24no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8C22⋊D4312yes—
239C4×Q8C22⋊D4314yes—
240C4×Q8C22⋊D4312yes—
241C2×C4⋊C4C2×Dic4314yes—
242C2×C4⋊C4C2×Dic4312yes—
243C2×C4⋊C4C2×Dic4314yes—
244C2×C4⋊C4C2×Dic4312yes—
245C2×C4⋊C4C2×Dic4314yes—
246C2×C4⋊C4C2×Dic4311yescharacteristic
247C2×C4⋊C4C2×Dic4312yes—
248C2×C4⋊C4C22×D4311yescharacteristic
249C2×C4⋊C4C22×D4312yes—
250C2×C42C2×Dic4314yes—
251C2×C42C2×Dic4312yes—
252C2×C42C22×D4312yes—
253C2×C42C22×D4311yescharacteristic
254C4⋊C8—864no—
255C4⋊C8—862no—
256C4⋊C8—864no—
257C4⋊C8—862no—
258#101—434no—
259#101—862no—
260#101—864no—
261#97—864no—
262#97—862no—
263C4×C8—864no—
264C4×C8—862no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—42no—
270C2×Q8—44no—
271C2×Q8C22⋊Dic4312yes—
272C2×Q8C22⋊Dic4312yes—
273C2×Q8Q8×D4312yes—
274C2×Q8#1062190212yes—
275C22×C4C22×Dic4311yescharacteristic
276C22×C4C22×Dic4312yes—
277C22×C4C22×Dic4312yes—
278C22×C4C22×Dic4314yes—
279C22×C4C22×Dic4312yes—
280C22×C4C22×Dic4311yescharacteristic
281C22×C4C22×Dic4312yes—
282C22×C4C23×D4311yescharacteristic
283M16—434no—
284M16—862no—
285M16—862no—
286M16—864no—
287M16—862no—
288M16—864no—
289C2×C8—862no—
290C2×C8—864no—
291C2×C8—862no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4C22⋊Dic4312yes—
301C4⋊C4C22⋊Dic4311yescharacteristic
302C4⋊C4C22⋊Dic4311yescharacteristic
303C4⋊C4C22⋊Dic4314yes—
304C4⋊C4C22⋊Dic4314yes—
305C42—24no—
306C42—24no—
307C42—22no—
308C42—22no—
309C42C2×C22⋊D4314yes—
310C42C2×C22⋊D4312yes—
311C42C2×C22⋊D4314yes—
312C42C2×C22⋊D4312yes—
313C23C23×Dic4311yescharacteristic
314Q8—24no—
315Q8—24no—
316Q8—24no—
317Q8—42no—
318Q8—44no—
319Q8—44no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4C2×C22⋊Dic4312yes—
331C2×C4C2×C22⋊Dic4312yes—
332C2×C4C2×C22⋊Dic4311yescharacteristic
333C2×C4C2×C22⋊Dic4312yes—
334C2×C4C2×C22⋊Dic4314yes—
335C2×C4C2×C22⋊Dic4311yescharacteristic
336C2×C4Q8×Dic4314yes—
337C2×C4C22×C22⋊D4312yes—
338C2×C4C2×Q8×D4311yescharacteristic
339C2×C4C2×#1062190211yescharacteristic
340C2×C4#1306337312yes—
341C8—864no—
342C8—1724no—
343C8—1722no—
344C22—21no—
345C22—21no—
346C22C22×C22⋊Dic4311yescharacteristic
347C22C2×Q8×Dic4311yescharacteristic
348C22#1458337711yescharacteristic
349C4—24no—
350C4—21no—
351C4—24no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—44no—
356C4—42no—
357C4#1458288312yes—
358C4#1458356012yes—
359C4Q8×C22⋊D4312yes—
360C2—21no—
361C2—21no—
362C2Q8×C22⋊Dic4311yescharacteristic
363C2C2×#1458288311yescharacteristic
364C2#1914307811yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic43
commutator, prime radicalC23×C43C23×C4
FrattiniC22×C4C23×D43
FittingC2×C43×#557C2
socleC22×C43C22×C22⋊C4
cosocleC22×C4×C43C24
hypercenterC2×#557D43
hypocenterC43#22517
LCSG ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
43. #13063244Q8semidirect
216. C43#22517semidirect

Maximal subgroups

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

subgroupindex
Q8×#106218772
#191355892
#191355852
#191345912
C4×#130632442
C2×C43×#5572
#2251743