Finite Group Databasebeta

← #25113376

Subgroups — #25113376

Order 11008.

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×#14583634C211yesmaximal, characteristic
2M16×Dic86C212yesmaximal, complemented
3#19135443C214yesmaximal, complemented
4#19135435C212yesmaximal
5#19134591C211yesmaximal, characteristic, complemented
6#19134586C212yesmaximal, complemented
7#19134386C212yesmaximal, complemented
8C4×#101×C43C211yesmaximal, characteristic, complemented
9#14583634—22no—
10#14583634—24no—
11#14583634—22no—
12C2×#13063322C2211yescharacteristic
13C2×#13063314C2212yes—
14C2×#13063313C412yes—
15C2×#13063310C414yes—
16C2×C4×Dic86C2211yescharacteristic
17C2×C4×D172C412yes—
18C2×#13063287C2212yes—
19#14583029—22no—
20#14583027—24no—
21C8×Dic86—22no—
22#14582923C2212yes—
23#14582922C2214yes—
24C2×#13063244C2212yes—
25C2×#13063241C2214yes—
26C2×#13063241C2214yes—
27M16×Dic43C2214yes—
28#14582668C2212yes—
29C2×C4×Q8×C43C2211yescharacteristic
30C2×#101×C43C2212yes—
31C4×M16×C43C2212yes—
32C43×#343C2211yescharacteristic
33C43×#343C2211yescharacteristic
34C22×Dic86C2×C411yescharacteristic
35C22×D172C2×C412yes—
36C22×D172C2×C412yes—
37C22×D172C2312yes—
38C22×C4×D43C2×C414yes—
39C2×#10621877C2314yes—
40#13063322—22no—
41#13063322—22no—
42#13063314—24no—
43#13063314—24no—
44#13063313—24no—
45#13063313—24no—
46#13063310—28no—
47#13063310—28no—
48C4×Dic86D412yes—
49C4×Dic86D414yes—
50C4×Dic86D412yes—
51C4×D172—24no—
52C4×D172—22no—
53C4×D172—22no—
54#13063287—24no—
55#13063287—24no—
56C2×C4⋊Dic43C2×C411yescharacteristic
57C2×Dic43⋊C4C2×C412yes—
58C2×(C2×D43)⋊C4C2×C414yes—
59C2×(C2×D43)⋊C4C2×C414yes—
60C2×(C2×D43)⋊C4C2×C414yes—
61C2×C4×Dic43C2312yes—
62C4⋊(C43⋊C8)—22no—
63C4⋊(C43⋊C8)—22no—
64Dic43⋊C8—24no—
65Dic43⋊C8—24no—
66#13063244—42no—
67#13063241—28no—
68#13063241—44no—
69#13063235—24no—
70C8×Dic43—24no—
71C22×Q8×C43C2311yescharacteristic
72C2×M16×C43C2312yes—
73C2×M16×C43C2312yes—
74C4×Q8×C43—22no—
75C4×Q8×C43—22no—
76C2×C4⋊C4×C43C2×C412yes—
77C2×C4⋊C4×C43C2311yescharacteristic
78C2×C42×C43C2×C412yes—
79C2×C42×C43C2311yescharacteristic
80#101×C43—24no—
81#97×C43—22no—
82C4×C8×C43—22no—
83C23×D43C22×C414yes—
84C2×Q8×C43—22no—
85C2×Q8×C43—22no—
86C2×Q8×C43—21no—
87C2×Q8×C43—21no—
88C22×C4×C43C22×C412yes—
89C22×C4×C43C22×C411yescharacteristic
90C22×C4×C43C22×C412yes—
91C22×C4×C43C22×C411yescharacteristic
92C22×C4×C43C2411yescharacteristic
93C22×Dic43C22×C412yes—
94C22×Dic43C22×C412yes—
95C2×D172—22no—
96C2×D172—22no—
97C2×D172—24no—
98C2×D172—24no—
99C2×D172—22no—
100C2×D172—22no—
101C2×D172—24no—
102C2×D172—44no—
103C2×D172—82no—
104C2×C4×D43—24no—
105C2×C4×D43—24no—
106C2×C4×D43—84no—
107C2×Dic86—42no—
108C2×Dic86C22⋊C412yes—
109C2×Dic86C22⋊C412yes—
110M16×C43—22no—
111M16×C43—22no—
112M16×C43—22no—
113M16×C43#4114yes—
114C2×C8×C43—22no—
115C2×C8×C43—22no—
116C4⋊C4×C43—24no—
117C4⋊C4×C43—22no—
118C42×C43—22no—
119C42×C43—22no—
120C42×C43C2×D412yes—
121C42×C43C2×D412yes—
122(C2×D43)⋊C4—28no—
123(C2×D43)⋊C4—24no—
124(C2×D43)⋊C4—24no—
125(C2×D43)⋊C4—28no—
126C4⋊Dic43C22⋊C412yes—
127C4⋊Dic43C22⋊C411yescharacteristic
128C4⋊Dic43C22⋊C411yescharacteristic
129Dic43⋊C4C22⋊C414yes—
130Dic43⋊C4C22⋊C414yes—
131C4×Dic43C2×D414yes—
132C4×Dic43C2×D414yes—
133#10621877—24no—
134#10621877—24no—
135C2×C43⋊C8—24no—
136C23×C43C23×C411yescharacteristic
137C22×D43—24no—
138C22×D43—24no—
139C22×D43—24no—
140C22×D43—44no—
141C22×D43—44no—
142Q8×C43—42no—
143Q8×C43—42no—
144C2×C4×C43—22no—
145C2×C4×C43—22no—
146C2×C4×C43—22no—
147C2×C4×C43—22no—
148C2×C4×C43—22no—
149C2×C4×C43—41no—
150C2×C4×C43—42no—
151C2×C4×C43C2×C22⋊C412yes—
152C2×C4×C43C2×C22⋊C411yescharacteristic
153C2×C4×C43C2×C22⋊C411yescharacteristic
154C2×C4×C43C22×D412yes—
155C2×C4×C43C2×#4112yes—
156C2×C4×C43H(5,𝔽2)11yescharacteristic
157C2×C4×C432−1+411yescharacteristic
158C2×Dic43—42no—
159C2×Dic43—42no—
160C2×Dic43C2×C22⋊C412yes—
161C2×Dic43C2×C22⋊C412yes—
162C2×Dic43C2×C22⋊C414yes—
163D172—44no—
164D172—44no—
165D172—44no—
166D172—44no—
167D172—84no—
168C4×D43—84no—
169C4×D43—84no—
170Dic86—24no—
171Dic86—42no—
172C8×C43—24no—
173C8×C43—42no—
174C43⋊C8—44no—
175#22539—431nomaximal, Sylow
176C22×C43—21no—
177C22×C43—21no—
178C22×C43C22×C22⋊C411yescharacteristic
179C22×C43C4×#4111yescharacteristic
180C22×C43#52011yescharacteristic
181C2×D43—44no—
182C2×D43—44no—
183C2×D43—44no—
184C2×D43—84no—
185C2×D43—84no—
186C4×C43—21no—
187C4×C43—22no—
188C4×C43—42no—
189C4×C43—42no—
190C4×C43#41312yes—
191C4×C43#54812yes—
192C4×C43#55012yes—
193Dic43—24no—
194Dic43—22no—
195Dic43—22no—
196C2×#559—431no—
197Q8×M16—432no—
198#1967—434no—
199#1965—432no—
200#1801—431no—
201#1800—432no—
202C4×#101D4311yescharacteristic, complemented
203C4×#100—432no—
204C2×C43—21no—
205C2×C43—21no—
206C2×C43#234111yescharacteristic
207C2×C43C2×#41311yescharacteristic
208C2×C43#288811yescharacteristic
209D43—84no—
210#559—862no—
211#559—864no—
212#559—862no—
213C2×#127—431no—
214C2×#124—432no—
215C2×#124—432no—
216C2×#123—434no—
217C2×C4×Q8—431no—
218C2×C4×Q8C2×D4311yescharacteristic
219C2×C4×D4—432no—
220C2×C4×D4—432no—
221#446—862no—
222#446—864no—
223C8×Q8—862no—
224#423—434no—
225#423—432no—
226C2×#101—432no—
227C2×#101C2×D4312yes—
228C2×#100—434no—
229C2×#100—434no—
230C4×M16—434no—
231C4×M16C2×D4312yes—
232#343—432no—
233#343C2×D4311yescharacteristic
234#343C2×D4311yescharacteristic
235C43#2253911yescharacteristic, complemented, Sylow
236C22×Q8—431no—
237C22×Q8C22×D4311yescharacteristic
238C22×D4—432no—
239C22×D4—432no—
240C22×D4—432no—
241C23×C4—434no—
242C2×M16—434no—
243C2×M16C22×D4312yes—
244C2×M16C22×D4312yes—
245#127—862no—
246#127—862no—
247#124—864no—
248#124—864no—
249#124—864no—
250#124—864no—
251#123—868no—
252#123—868no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—432no—
256C4×Q8—434no—
257C4×Q8—432no—
258C4×D4—864no—
259C4×D4—862no—
260C4×D4—862no—
261C4×D4—864no—
262C4×D4—864no—
263C2×C4⋊C4—431no—
264C2×C4⋊C4—432no—
265C2×C4⋊C4C4×D4312yes—
266C2×C4⋊C4C22×D4311yescharacteristic
267C2×C22⋊C4—434no—
268C2×C22⋊C4—434no—
269C2×C22⋊C4—434no—
270C2×C42—432no—
271C2×C42C4×D4312yes—
272C2×C42C22×D4311yescharacteristic
273C4⋊C8—864no—
274C4⋊C8—864no—
275C4⋊C8—862no—
276C4⋊C8—862no—
277#101—24no—
278#101—1722no—
279#100—868no—
280#100—1724no—
281#97—22no—
282#97—864no—
283C4×C8—22no—
284C4×C8—864no—
285C24—434no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—432no—
291C2×Q8—432no—
292C2×Q8—1722no—
293C2×D4—864no—
294C2×D4—862no—
295C2×D4—862no—
296C2×D4—864no—
297C2×D4—864no—
298C2×D4—862no—
299C2×D4—862no—
300C2×D4—3442no—
301C2×D4—1724no—
302C22×C4—432no—
303C22×C4—432no—
304C22×C4—864no—
305C22×C4—864no—
306C22×C4—3444no—
307C22×C4C2×C4×D4312yes—
308C22×C4C2×C4×D4311yescharacteristic
309C22×C4C2×C4×D4312yes—
310C22×C4C2×C4×D4311yescharacteristic
311C22×C4C23×D4311yescharacteristic
312M16—22no—
313M16—22no—
314M16—22no—
315M16—864no—
316M16—864no—
317M16#1062189814yes—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—864no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—432no—
324C4⋊C4—431no—
325C4⋊C4—431no—
326C4⋊C4—434no—
327C4⋊C4—434no—
328C22⋊C4—864no—
329C22⋊C4—864no—
330C22⋊C4—868no—
331C22⋊C4—868no—
332C42—22no—
333C42—22no—
334C42—434no—
335C42—434no—
336C42D4×D4312yes—
337C42D4×D4312yes—
338C23—864no—
339C23—864no—
340C23—864no—
341C23—1724no—
342C23—1724no—
343C23C22×C4×D4311yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—864no—
347Q8—1722no—
348D4—3444no—
349D4—1724no—
350D4—1724no—
351D4—1724no—
352D4—1724no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—434no—
361C2×C4—432no—
362C2×C4—432no—
363C2×C4—3444no—
364C2×C4—3444no—
365C2×C4—1722no—
366C2×C4—1722no—
367C2×C4C22⋊C4×D4311yescharacteristic
368C2×C4C22⋊C4×D4312yes—
369C2×C4C22⋊C4×D4311yescharacteristic
370C2×C4C2×D4×D4312yes—
371C2×C4C2×#1062189812yes—
372C2×C4#1306337111yescharacteristic
373C2×C4#1306337211yescharacteristic
374C8—24no—
375C8—42no—
376C8—1724no—
377C22—21no—
378C22—21no—
379C22—3444no—
380C22—3444no—
381C22—1724no—
382C22—1724no—
383C22—1724no—
384C22C2×C22⋊C4×D4311yescharacteristic
385C22C4×#1062189811yescharacteristic
386C22#1458337911yescharacteristic
387C4—21no—
388C4—22no—
389C4—42no—
390C4—42no—
391C4—864no—
392C4—862no—
393C4—862no—
394C4#1458288212yes—
395C4#1458355812yes—
396C4#1458357012yes—
397C2—21no—
398C2—21no—
399C2—3444no—
400C2#1913743511yescharacteristic
401C2C2×#1458288211yescharacteristic
402C2#1914307211yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D43
commutatorC23×C43C23×C4
FrattiniC22×C4C23×D43
FittingC4×#101×C43C2
socleC22×C43C22×C22⋊C4
prime radicalC2×#13063322C22
cosocleC22×C4×C43C24
hypercenterC4×#101D43
hypocenterC43#22539
LCSG ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
2. M16×Dic86C2semidirect
3. #19135443C2semidirect
5. #19134591C2semidirect
6. #19134586C2semidirect
7. #19134386C2semidirect
8. C4×#101×C43C2semidirect
202. C4×#101D43semidirect
235. C43#22539semidirect

Maximal subgroups

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

subgroupindex
C2×#145836342
M16×Dic862
#191354432
#191354352
#191345912
#191345862
#191343862
C4×#101×C432
#2253943