Finite Group Databasebeta

← D4×#13063271 #25253323

Subgroups — D4×#13063271 #25253323

Order 11008.

664 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13063271C212yesmaximal, complemented
2D4×Dic172C211yesmaximal, characteristic
3#19136847C211yesmaximal, characteristic, complemented
4#19136684C212yesmaximal, complemented
5#19136645C211yesmaximal, characteristic, complemented
6#19136642C211yesmaximal, characteristic, complemented
7#19136529C212yesmaximal, complemented
8#19136528C212yesmaximal, complemented
9C4×#13063271C211yesmaximal, characteristic, complemented
10D4×C43⋊C16C211yesmaximal, characteristic
11D4×Q16×C43C211yesmaximal, characteristic
12C22×Dic172C2212yes—
13D4×Dic86—21no—
14C2×#13063271—28no—
15C2×#13063271C2212yes—
16C2×#13063271C2212yes—
17C2×#13063271C2212yes—
18C2×#13063271C2212yes—
19C2×#13063271C2211yescharacteristic
20C22×C43⋊C16C2212yes—
21#14583279C2211yescharacteristic
22#14583140C2212yes—
23#14583110—21no—
24#14583057—22no—
25C4×Dic172C2211yescharacteristic
26#14582724C2211yescharacteristic
27C4⋊(C43⋊C16)C2211yescharacteristic
28#14582703C2211yescharacteristic
29#14582703C2212yes—
30#14582702C2211yescharacteristic
31#14582702C2212yes—
32C22⋊(C43⋊C16)C2212yes—
33C4×C43⋊C16C2211yescharacteristic
34C22×Q16×C43C2212yes—
35D4×Q8×C43—21no—
36C43×#494C2211yescharacteristic
37C43×#467C2212yes—
38C43×#462—21no—
39C43×#451—22no—
40C4×Q16×C43C2211yescharacteristic
41C8×D4×C43C2211yescharacteristic
42C22×Dic86—22no—
43C2×Dic172—24no—
44C2×Dic172—22no—
45C2×Dic172—21no—
46C2×Dic172—42no—
47C2×Dic172C2312yes—
48C2×Dic172C2311yescharacteristic
49#13063325—21no—
50#13063324—41no—
51#13063304—22no—
52#13063302—42no—
53C4×Dic86—21no—
54D4×Dic43—41no—
55#13063271—24no—
56#13063271—24no—
57#13063271—24no—
58#13063271—42no—
59#13063271D414yesdirect product factor
60C2×C43⋊C16—24no—
61C2×C43⋊C16C2312yes—
62C2×C43⋊C16C2311yescharacteristic
63C2×C43⋊C16C2311yescharacteristic
64#13063259C2311yescharacteristic
65#13063250—21no—
66#13063250—22no—
67C22×Q8×C43—22no—
68C2×Q16×C43—24no—
69C2×Q16×C43—22no—
70C2×Q16×C43—21no—
71C2×Q16×C43—42no—
72C2×Q16×C43C2312yes—
73C2×Q16×C43C2311yescharacteristic
74C22×C8×C43C2312yes—
75C4⋊Q8×C43—21no—
76C4⋊Q8×C43—41no—
77C22⋊Q8×C43—22no—
78C22⋊Q8×C43—42no—
79C4×Q8×C43—21no—
80C4×D4×C43—41no—
81C4×D4×C43D411yescharacteristic
82#107×C43C2311yescharacteristic
83C4⋊C8×C43D411yescharacteristic
84#103×C43—21no—
85#103×C43—22no—
86C22⋊C8×C43D412yes—
87C4×C8×C43C2311yescharacteristic
88C2×Q8×C43—21no—
89C2×Q8×C43—22no—
90C2×Q8×C43—44no—
91C2×Q8×C43—42no—
92C2×Q8×C43—82no—
93C2×D4×C43D811yescharacteristic
94C22×C4×C43—42no—
95C22×C4×C43C2×D412yes—
96C22×Dic43—42no—
97C2×Dic86—21no—
98C2×Dic86—22no—
99C2×Dic86—44no—
100C2×Dic86—42no—
101C2×Dic86—82no—
102Q16×C43—22no—
103Q16×C43—21no—
104Q16×C43—22no—
105Q16×C43—44no—
106Q16×C43—81no—
107Q16×C43C2×D412yes—
108C2×C8×C43—22no—
109C2×C8×C43—22no—
110C2×C8×C43C2×D411yescharacteristic
111C2×C8×C43C2×D412yes—
112C2×C8×C43C2411yescharacteristic
113C4⋊C4×C43—21no—
114C4⋊C4×C43—41no—
115C4⋊C4×C43—41no—
116C4⋊C4×C43—42no—
117C4⋊C4×C43—41no—
118C4⋊C4×C43D811yescharacteristic
119C22⋊C4×C43—42no—
120C22⋊C4×C43D812yes—
121C42×C43—41no—
122C42×C43C2×D411yescharacteristic
123C22⋊Dic43—42no—
124C4⋊Dic43—21no—
125C4⋊Dic43—41no—
126C4⋊Dic43—41no—
127Dic43⋊C4—41no—
128Dic43⋊C4—42no—
129C4×Dic43—41no—
130Dic172—22no—
131Dic172—21no—
132Dic172—22no—
133Dic172—44no—
134Dic172—81no—
135Dic172C2×D412yes—
136C43⋊C16—22no—
137C43⋊C16—21no—
138C43⋊C16C2×D412yes—
139C23×C43C2×D812yes—
140Q8×C43—22no—
141Q8×C43—41no—
142Q8×C43—42no—
143Q8×C43—42no—
144Q8×C43—84no—
145D4×C43Q3214yescomplemented
146C2×C4×C43—22no—
147C2×C4×C43—22no—
148C2×C4×C43—41no—
149C2×C4×C43—41no—
150C2×C4×C43—42no—
151C2×C4×C43—84no—
152C2×C4×C43C2×D811yescharacteristic
153C2×C4×C43C2×D812yes—
154C2×C4×C43C2×D811yescharacteristic
155C2×C4×C43C22×D411yescharacteristic
156C2×Dic43—41no—
157C2×Dic43—41no—
158C2×Dic43—42no—
159C2×Dic43—84no—
160Dic86—22no—
161Dic86—41no—
162Dic86—42no—
163Dic86—42no—
164Dic86—84no—
165C8×C43—22no—
166C8×C43—21no—
167C8×C43C22×D411yescharacteristic
168C8×C43H(5,𝔽2)11yescharacteristic
169D4×Q32—431nomaximal, Sylow
170C22×C43—22no—
171C22×C43—22no—
172C22×C43C2×Q3214yescomplemented
173C22×C43C22×D811yescharacteristic
174C4×C43—21no—
175C4×C43—22no—
176C4×C43—42no—
177C4×C43—81no—
178C4×C43—82no—
179C4×C43C2×Q3212yescomplemented
180C4×C43D4211yescharacteristic
181C4×C43#57611yescharacteristic
182Dic43—42no—
183Dic43—81no—
184Dic43—82no—
185C22×Q32—432no—
186D4×Q16—431no—
187D4×Q16D4311yescharacteristic
188#2247—431no—
189#2216—432no—
190#2209—431no—
191#2209—431no—
192#2188—432no—
193#2188—432no—
194C4×Q32—431no—
195D4×C16—431no—
196C2×C43—24no—
197C2×C43D4×D811yescharacteristic
198C2×C43C22×Q3211yescharacteristic
199C2×C43#341711yescharacteristic
200C22×Q16—432no—
201C22×Q16C2×D4312yes—
202D4×Q8—21no—
203D4×Q8—861no—
204C2×Q32—431no—
205C2×Q32—432no—
206C2×Q32—432no—
207C2×Q32—432no—
208C2×Q32—432no—
209C2×Q32—868no—
210C22×C16—432no—
211#494—431no—
212#494C2×D4311yescharacteristic
213#467—432no—
214#467C2×D4312yes—
215#462—21no—
216#462—861no—
217#451—22no—
218#451—862no—
219C4×Q16—431no—
220C4×Q16C2×D4311yescharacteristic
221C8×D4C2×D4311yescharacteristic
222#366—431no—
223C4⋊C16—431no—
224#358—431no—
225#358—432no—
226#358—431no—
227#358—432no—
228C22⋊C16—432no—
229C4×C16—431no—
230C43D4×Q3211yescharacteristic, complemented, Sylow
231C22×Q8—22no—
232C22×Q8—862no—
233C2×Q16—24no—
234C2×Q16—22no—
235C2×Q16—21no—
236C2×Q16—42no—
237C2×Q16—431no—
238C2×Q16—432no—
239C2×Q16—864no—
240C2×Q16—861no—
241C2×Q16—862no—
242C2×Q16—1722no—
243C2×Q16C22×D4312yes—
244C2×Q16C22×D4311yescharacteristic
245C22×C8C22×D4312yes—
246C4⋊Q8—21no—
247C4⋊Q8—41no—
248C4⋊Q8—861no—
249C4⋊Q8—1721no—
250C22⋊Q8—22no—
251C22⋊Q8—42no—
252C22⋊Q8—862no—
253C22⋊Q8—1722no—
254C4×Q8—21no—
255C4×Q8—861no—
256C4×D4—41no—
257C4×D4—1721no—
258C4×D4C22⋊D4311yescharacteristic
259Q32—434no—
260Q32—864no—
261Q32—864no—
262Q32—864no—
263Q32—1722no—
264C2×C16—431no—
265C2×C16—431no—
266C2×C16—432no—
267C2×C16—864no—
268#107—431no—
269#107C22×D4311yescharacteristic
270C4⋊C8C22⋊D4311yescharacteristic
271#103—21no—
272#103—22no—
273#103—861no—
274#103—862no—
275C22⋊C8C22⋊D4312yes—
276C4×C8C22×D4311yescharacteristic
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—44no—
280C2×Q8—42no—
281C2×Q8—82no—
282C2×Q8—861no—
283C2×Q8—862no—
284C2×Q8—3442no—
285C2×Q8—1724no—
286C2×Q8—1722no—
287C2×D4#1062188211yescharacteristic
288C22×C4—42no—
289C22×C4—1722no—
290C22×C4C2×C22⋊D4312yes—
291Q16—22no—
292Q16—21no—
293Q16—22no—
294Q16—44no—
295Q16—81no—
296Q16—432no—
297Q16—862no—
298Q16—861no—
299Q16—862no—
300Q16—3441no—
301Q16—1724no—
302Q16D4×D4312yes—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8C2×C22⋊D4311yescharacteristic
306C2×C8C2×C22⋊D4312yes—
307C2×C8C23×D4311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—861no—
314C4⋊C4—1721no—
315C4⋊C4—1721no—
316C4⋊C4—1722no—
317C4⋊C4—1721no—
318C4⋊C4#1062188211yescharacteristic
319C22⋊C4—42no—
320C22⋊C4—1722no—
321C22⋊C4#1062188212yes—
322C42—41no—
323C42—1721no—
324C42C2×C22⋊D4311yescharacteristic
325C16—432no—
326C16—862no—
327C16—861no—
328C23C2×#1062188212yes—
329Q8—22no—
330Q8—41no—
331Q8—42no—
332Q8—42no—
333Q8—84no—
334Q8—862no—
335Q8—3444no—
336Q8—1722no—
337Q8—1722no—
338Q8—1721no—
339D4#1306327114yesdirect product factor
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—84no—
346C2×C4—3444no—
347C2×C4—1721no—
348C2×C4—1722no—
349C2×C4—1721no—
350C2×C4C2×#1062188211yescharacteristic
351C2×C4C2×#1062188212yes—
352C2×C4C2×#1062188211yescharacteristic
353C2×C4C22×C22⋊D4311yescharacteristic
354C8—21no—
355C8—22no—
356C8C2×D4×D4311yescharacteristic
357C8#1306337011yescharacteristic
358C22—22no—
359C22—22no—
360C22C2×#1306327114yescomplemented
361C22C22×#1062188211yescharacteristic
362C4—22no—
363C4—21no—
364C4—42no—
365C4—82no—
366C4—81no—
367C4—3442no—
368C4—3441no—
369C4—1722no—
370C4C2×#1306327112yescomplemented
371C4D4×C22⋊D4311yescharacteristic
372C4#1458370211yescharacteristic
373C2—24no—
374C2D4×#1062188211yescharacteristic
375C2C22×#1306327111yescharacteristic
376C2#1914774611yescharacteristic
subgroupG/H
centerC22C22×#10621882
commutator, cosocleC2×C8×C43C24
FrattiniC2×C8C23×D43
FittingD4×Q16×C43C2
socleC22×C43C22×D8
prime radicalC2×D4×C43D8
hypercenterD4×Q16D43
hypocenterC43D4×Q32
LCSG ▷ C2×C8×C43 ▷ C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
1. C22×#13063271C2semidirect
3. #19136847C2semidirect
4. #19136684C2semidirect
5. #19136645C2semidirect
6. #19136642C2semidirect
7. #19136529C2semidirect
8. #19136528C2semidirect
9. C4×#13063271C2semidirect
59. #13063271D4direct (N × H)
145. D4×C43Q32semidirect
172. C22×C43C2×Q32semidirect
179. C4×C43C2×Q32semidirect
230. C43D4×Q32semidirect
339. D4#13063271direct (N × H)
360. C22C2×#13063271semidirect
370. C4C2×#13063271semidirect

Maximal subgroups

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

subgroupindex
C22×#130632712
D4×Dic1722
#191368472
#191366842
#191366452
#191366422
#191365292
#191365282
C4×#130632712
D4×C43⋊C162
D4×Q16×C432
D4×Q3243