Finite Group Databasebeta

← C2×#19132540 #25005750

Subgroups — C2×#19132540 #25005750

Order 11008.

598 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#14582768C211yesmaximal, characteristic
2C2×#14582625C211yesmaximal, characteristic
3#19132540C214yesmaximal, direct product factor
4C2×C43×#324C211yesmaximal, characteristic
5C22×C4⋊Dic43C411yescharacteristic
6C22×Dic43⋊C4—21no—
7C22×Dic43⋊C4—21no—
8C22×C4×Dic43C411yescharacteristic
9C2×C22⋊(C43⋊C8)—21no—
10#14582768—21no—
11#14582768—21no—
12#14582768C412yes—
13#14582768C2212yes—
14#14582625—21no—
15#14582625C2212yes—
16C22×C4⋊C4×C43C2211yescharacteristic
17C2×C22⋊C8×C43—21no—
18C43×#324—21no—
19C43×#324C2212yes—
20C23×Dic43—21no—
21C23×Dic43—21no—
22C22×C43⋊C8—41no—
23C2×C4⋊Dic43—21no—
24C2×C4⋊Dic43—41no—
25C2×C4⋊Dic43—41no—
26C2×C4⋊Dic43C2×C412yes—
27C2×Dic43⋊C4—22no—
28C2×Dic43⋊C4—22no—
29C2×Dic43⋊C4—22no—
30C2×Dic43⋊C4—22no—
31C2×Dic43⋊C4—44no—
32C2×Dic43⋊C4—44no—
33C2×C4×Dic43—21no—
34C2×C4×Dic43—22no—
35C2×C4×Dic43C814yescomplemented
36C2×C4×Dic43C2×C412yes—
37C22⋊(C43⋊C8)—22no—
38C22⋊(C43⋊C8)—41no—
39C23×C4×C43C2×C411yescharacteristic
40C23×C4×C43D411yescharacteristic
41C23×C4×C43D411yescharacteristic
42C22×C8×C43—41no—
43C2×C4⋊C4×C43—41no—
44C2×C4⋊C4×C43—41no—
45C2×C4⋊C4×C43C2×C411yescharacteristic
46C2×C4⋊C4×C43D411yescharacteristic
47C2×C4⋊C4×C43D411yescharacteristic
48C2×C4⋊C4×C43C2311yescharacteristic
49C22⋊C8×C43—22no—
50C22⋊C8×C43—41no—
51C24×C43C22⋊C411yescharacteristic
52C22×C4×C43—21no—
53C22×C4×C43—21no—
54C22×C4×C43—21no—
55C22×C4×C43—21no—
56C22×C4×C43—21no—
57C22×C4×C43—21no—
58C22×C4×C43—22no—
59C22×C4×C43—42no—
60C22×C4×C43—42no—
61C22×C4×C43C22⋊C411yescharacteristic
62C22×C4×C43C22⋊C411yescharacteristic
63C22×C4×C43C22⋊C411yescharacteristic
64C22×C4×C43C2×C812yes—
65C22×C4×C43M1612yes—
66C22×C4×C43C22×C411yescharacteristic
67C22×C4×C43C2×D411yescharacteristic
68C22×C4×C43C2×D411yescharacteristic
69C22×Dic43—24no—
70C22×Dic43—24no—
71C22×Dic43—24no—
72C22×Dic43—22no—
73C22×Dic43—22no—
74C22×Dic43—42no—
75C22×Dic43—42no—
76C22×Dic43—41no—
77C22×Dic43—41no—
78C2×C8×C43—42no—
79C2×C8×C43—82no—
80C4⋊C4×C43—41no—
81C4⋊C4×C43—41no—
82C4⋊C4×C43—41no—
83C4⋊C4×C43—41no—
84C4⋊Dic43—42no—
85C4⋊Dic43—42no—
86Dic43⋊C4—44no—
87Dic43⋊C4—44no—
88Dic43⋊C4—44no—
89Dic43⋊C4—44no—
90C4×Dic43—24no—
91C4×Dic43—22no—
92C4×Dic43C2×C814yescomplemented
93C2×C43⋊C8—42no—
94C2×C43⋊C8—82no—
95C23×C43—22no—
96C23×C43—22no—
97C23×C43C22⋊C812yes—
98C23×C43#9911yescharacteristic
99C23×C43#9911yescharacteristic
100C23×C43#10112yes—
101C23×C43C2×C22⋊C411yescharacteristic
102C2×C4×C43—22no—
103C2×C4×C43—22no—
104C2×C4×C43—22no—
105C2×C4×C43—22no—
106C2×C4×C43—21no—
107C2×C4×C43—21no—
108C2×C4×C43—21no—
109C2×C4×C43—21no—
110C2×C4×C43—21no—
111C2×C4×C43—21no—
112C2×C4×C43—21no—
113C2×C4×C43—21no—
114C2×C4×C43—42no—
115C2×C4×C43—42no—
116C2×C4×C43—41no—
117C2×C4×C43—41no—
118C2×C4×C43—41no—
119C2×C4×C43—41no—
120C2×C4×C43—81no—
121C2×C4×C43—81no—
122C2×C4×C43C22⋊C812yes—
123C2×C4×C43C22×C811yescharacteristic
124C2×C4×C43C2×M1611yescharacteristic
125C2×Dic43—24no—
126C2×Dic43—216no—
127C2×Dic43—24no—
128C2×Dic43—24no—
129C2×Dic43—42no—
130C2×Dic43—42no—
131C2×Dic43—42no—
132C2×Dic43—82no—
133C2×Dic43—82no—
134C8×C43—82no—
135C43⋊C8—82no—
136C2×#1325—431nomaximal, Sylow
137C22×C43—24no—
138C22×C43—24no—
139C22×C43—21no—
140C22×C43—21no—
141C22×C43—21no—
142C22×C43—21no—
143C22×C43#32412yes—
144C22×C43#32412yes—
145C22×C43#35312yes—
146C22×C43#35612yes—
147C22×C43C2×C22⋊C811yescharacteristic
148C22×C43C2×#9911yescharacteristic
149C22×C43C2×#10111yescharacteristic
150C4×C43—22no—
151C4×C43—21no—
152C4×C43—21no—
153C4×C43—41no—
154C4×C43—41no—
155C4×C43—41no—
156C4×C43—41no—
157Dic43—28no—
158Dic43—82no—
159C2×#384—431no—
160C2×#324—431no—
161C2×#324D4311yescharacteristic
162#1325—434no—
163C2×C43—21no—
164C2×C43—22no—
165C2×C43—21no—
166C2×C43#132514yescomplemented
167C2×C43C2×#32411yescharacteristic
168C2×C43C2×#35311yescharacteristic
169C2×C43C2×#35611yescharacteristic
170C22×C4⋊C4—431no—
171C22×C4⋊C4—861no—
172C22×C4⋊C4—861no—
173C22×C4⋊C4C2×D4311yescharacteristic
174C22×C42—431no—
175C2×C22⋊C8—21no—
176C2×C22⋊C8—861no—
177#384—432no—
178#384—432no—
179#384—861no—
180#384—861no—
181#324—21no—
182#324—432no—
183#324—861no—
184#324C2×D4312yes—
185C43C2×#132511yescharacteristic, complemented, Sylow
186C23×C4—861no—
187C23×C4—861no—
188C23×C4C4×D4311yescharacteristic
189C23×C4D17211yescharacteristic
190C23×C4C22⋊D4311yescharacteristic
191C22×C8—41no—
192C22×C8—1721no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—432no—
196C2×C4⋊C4—861no—
197C2×C4⋊C4—862no—
198C2×C4⋊C4—862no—
199C2×C4⋊C4—862no—
200C2×C4⋊C4—862no—
201C2×C4⋊C4—1721no—
202C2×C4⋊C4—1721no—
203C2×C4⋊C4—1724no—
204C2×C4⋊C4—1724no—
205C2×C4⋊C4C4×D4311yescharacteristic
206C2×C4⋊C4D17211yescharacteristic
207C2×C4⋊C4C22⋊D4311yescharacteristic
208C2×C4⋊C4C22×D4311yescharacteristic
209C2×C42—434no—
210C2×C42—432no—
211C2×C42—862no—
212C2×C42—861no—
213C22⋊C8—22no—
214C22⋊C8—41no—
215C22⋊C8—862no—
216C22⋊C8—1721no—
217C24(C2×D43)⋊C411yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—864no—
228C22×C4—864no—
229C22×C4—864no—
230C22×C4—862no—
231C22×C4—862no—
232C22×C4—1722no—
233C22×C4—1722no—
234C22×C4—1721no—
235C22×C4—1721no—
236C22×C4C8×D4312yes—
237C22×C4#1062187212yes—
238C22×C4(C2×D43)⋊C411yescharacteristic
239C22×C4(C2×D43)⋊C411yescharacteristic
240C22×C4(C2×D43)⋊C411yescharacteristic
241C22×C4C2×C4×D4311yescharacteristic
242C22×C4C2×D17211yescharacteristic
243C22×C4C2×C22⋊D4311yescharacteristic
244C2×C8—42no—
245C2×C8—82no—
246C2×C8—3442no—
247C2×C8—1722no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—1722no—
253C4⋊C4—1722no—
254C4⋊C4—1724no—
255C4⋊C4—1724no—
256C4⋊C4—1724no—
257C4⋊C4—1724no—
258C42—434no—
259C42—864no—
260C42—862no—
261C23—22no—
262C23—22no—
263C23(C2×D43)⋊C812yes—
264C23#1306323911yescharacteristic
265C23#1306323911yescharacteristic
266C23#1306324312yes—
267C23C2×(C2×D43)⋊C411yescharacteristic
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—864no—
289C2×C4—864no—
290C2×C4—8616no—
291C2×C4—864no—
292C2×C4—3442no—
293C2×C4—3442no—
294C2×C4—1722no—
295C2×C4—1722no—
296C2×C4—1722no—
297C2×C4(C2×D43)⋊C812yes—
298C2×C4C2×C8×D4311yescharacteristic
299C2×C4C2×#1062187211yescharacteristic
300C8—82no—
301C8—3442no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22#1458262412yes—
309C22#1458262412yes—
310C22#1458269112yes—
311C22#1458269712yes—
312C22C2×(C2×D43)⋊C811yescharacteristic
313C22C2×#1306323911yescharacteristic
314C22C2×#1306324311yescharacteristic
315C4—22no—
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—868no—
323C4—3442no—
324C2—21no—
325C2—21no—
326C2—22no—
327C2#1913254014yesdirect product factor
328C2C2×#1458262411yescharacteristic
329C2C2×#1458269111yescharacteristic
330C2C2×#1458269711yescharacteristic
subgroupG/H
centerC23#13063239
commutatorC2×C4×C43C22×C8
FrattiniC2×C4⋊C4C22×D43
FittingC2×C43×#324C2
socleC23×C43#99
prime radicalC24×C43C22⋊C4
cosocleC2×C4⋊C4×C43C23
hypercenterC2×#324D43
hypocenterC43C2×#1325
LCSG ▷ C2×C4×C43 ▷ C22×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. #19132540C2direct (N × H)
35. C2×C4×Dic43C8semidirect
92. C4×Dic43C2×C8semidirect
166. C2×C43#1325semidirect
185. C43C2×#1325semidirect
327. C2#19132540direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#145827682
C2×#145826252
#191325402
C2×C43×#3242
C2×#132543