Finite Group Databasebeta

← #25006919

Subgroups — #25006919

Order 11008.

648 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#19137810C211yesmaximal, characteristic
2#19132916C211yesmaximal, characteristic, complemented
3#19132615C212yesmaximal, complemented
4#19132591C212yesmaximal, complemented
5C4×#99×C43C211yesmaximal, characteristic, complemented
6C22×#10621898C411yescharacteristic, complemented
7C2×C4×C22⋊D43—21no—
8#14582952—21no—
9#14582819C411yescharacteristic, complemented
10#14582816C411yescharacteristic, complemented
11#14582814—21no—
12#14582811C2211yescharacteristic
13#14582808C2211yescharacteristic
14#14582807—21no—
15C4×(C2×D43)⋊C4—21no—
16C4×C22⋊Dic43C411yescharacteristic, complemented
17#14582623—21no—
18#14582623C2212yes—
19C2×C4×D4×C43C2211yescharacteristic
20C2×#99×C43—21no—
21C4×C22⋊C4×C43—21no—
22C43×#342C2212yes—
23C2×#10621898—44no—
24C2×#10621898—41no—
25C2×#10621898—41no—
26C22×Dic86C2×C411yescharacteristic
27C22×D172C2×C411yescharacteristic
28C22×C22⋊D43—21no—
29C22×C22⋊D43—21no—
30C22×C4×D43—21no—
31C4×C22⋊D43—44no—
32C2×Dic43⋊C4—21no—
33C2×C22⋊Dic43—21no—
34C2×C22⋊Dic43—21no—
35C2×(C2×D43)⋊C4—21no—
36C2×(C2×D43)⋊C4—21no—
37C2×(C2×D43)⋊C4—21no—
38C2×C4×Dic43—21no—
39C4⋊(C43⋊C8)—41no—
40C22⋊(C43⋊C8)—22no—
41C4×C43⋊C8—41no—
42C43⋊#95—21no—
43C43⋊#95—21no—
44C43⋊#95C2×C411yescharacteristic
45C43⋊#95C2×C411yescharacteristic
46C22×D4×C43D411yescharacteristic
47C23×C4×C43C2×C411yescharacteristic
48C23×C4×C43D411yescharacteristic
49C4×D4×C43—41no—
50C4×D4×C43—41no—
51C2×C4⋊C4×C43C2×C411yescharacteristic
52C2×C22⋊C4×C43—22no—
53C2×C22⋊C4×C43D411yescharacteristic
54C2×C22⋊C4×C43C2311yescharacteristic
55C2×C42×C43—41no—
56C2×C42×C43D411yescharacteristic
57#99×C43—24no—
58#95×C43—41no—
59C24×C43C22⋊C411yescharacteristic
60C24×C43C2×D411yescharacteristic
61C23×D43—21no—
62C2×D4×C43—22no—
63C2×D4×C43—22no—
64C2×D4×C43—41no—
65C22×C4×C43—21no—
66C22×C4×C43—21no—
67C22×C4×C43—21no—
68C22×C4×C43—21no—
69C22×C4×C43—42no—
70C22×C4×C43—42no—
71C22×C4×C43—42no—
72C22×C4×C43C22⋊C411yescharacteristic
73C22×C4×C43C22⋊C411yescharacteristic
74C22×C4×C43C22⋊C411yescharacteristic
75C22×C4×C43C22×C411yescharacteristic
76C22×C4×C43C2×D411yescharacteristic
77C2×C22⋊D43—42no—
78C2×C22⋊D43—42no—
79C2×C22⋊D43—42no—
80C2×C22⋊D43—42no—
81C2×C22⋊D43—42no—
82C2×C22⋊D43—42no—
83C22×Dic43—21no—
84C22×Dic43—21no—
85C22×Dic43—21no—
86#10621898—44no—
87#10621898—44no—
88#10621898—44no—
89#10621898—44no—
90C2×D172—42no—
91C2×D172—41no—
92C2×C4×D43—24no—
93C2×C4×D43—42no—
94C2×C4×D43—41no—
95C2×C4×D43—41no—
96C2×Dic86—41no—
97C2×Dic86—42no—
98C4⋊C4×C43—41no—
99C22⋊C4×C43—24no—
100C22⋊C4×C43—41no—
101C22⋊C4×C43—41no—
102C22⋊C4×C43—42no—
103C42×C43—21no—
104C42×C43—21no—
105C42×C43—82no—
106C22⋊Dic43—42no—
107C22⋊Dic43—42no—
108(C2×D43)⋊C4—42no—
109(C2×D43)⋊C4—42no—
110(C2×D43)⋊C4—42no—
111Dic43⋊C4—42no—
112C4×Dic43—42no—
113C2×C43⋊C8—42no—
114C23×C43—21no—
115C23×C43—21no—
116C23×C43—21no—
117C23×C43—21no—
118C23×C43—21no—
119C23×C43—41no—
120C23×C43—41no—
121C23×C43—41no—
122C23×C43—41no—
123C23×C43C2×C22⋊C411yescharacteristic
124C23×C43#13512yes—
125C22×D43—22no—
126C22×D43—41no—
127C22×D43—41no—
128C22×D43—41no—
129C22×D43—41no—
130C22×D43—41no—
131C22×D43—41no—
132D4×C43—44no—
133C2×C4×C43—21no—
134C2×C4×C43—21no—
135C2×C4×C43—22no—
136C2×C4×C43—21no—
137C2×C4×C43—21no—
138C2×C4×C43—21no—
139C2×C4×C43—42no—
140C2×C4×C43—42no—
141C2×C4×C43—42no—
142C2×C4×C43—41no—
143C2×C4×C43—41no—
144C2×C4×C43—41no—
145C2×C4×C43—44no—
146C2×C4×C43—81no—
147C2×C4×C43—82no—
148C2×C4×C43—82no—
149C2×C4×C43#9911yescharacteristic
150C2×C4×C43#9911yescharacteristic
151C2×C4×C43#10412yes—
152C22⋊D43—48no—
153C22⋊D43—48no—
154C2×Dic43—22no—
155C2×Dic43—22no—
156C2×Dic43—41no—
157C2×Dic43—41no—
158C2×Dic43—41no—
159C2×Dic43—41no—
160C2×Dic43—41no—
161C2×Dic43—41no—
162C2×Dic43—41no—
163D172—42no—
164D172—42no—
165C4×D43—44no—
166C4×D43—44no—
167Dic86—42no—
168Dic86—42no—
169C43⋊C8—82no—
170#11725—431nomaximal, Sylow
171C22×C43—21no—
172C22×C43—21no—
173C22×C43—22no—
174C22×C43—42no—
175C22×C43—41no—
176C22×C43—41no—
177C22×C43—41no—
178C22×C43—41no—
179C22×C43—41no—
180C22×C43—81no—
181C22×C43—81no—
182C22×C43—81no—
183C22×C43C2×#9911yescharacteristic
184C22×C43#41611yescharacteristic
185C22×C43C2×#10411yescharacteristic
186C2×D43—42no—
187C2×D43—42no—
188C2×D43—42no—
189C2×D43—81no—
190C2×D43—81no—
191C2×D43—81no—
192C2×D43—81no—
193C4×C43—21no—
194C4×C43—22no—
195C4×C43—41no—
196C4×C43—41no—
197C4×C43—41no—
198C4×C43—84no—
199C4×C43AΣL(1,𝔽16)12yes—
200Dic43—42no—
201Dic43—42no—
202Dic43—81no—
203#2388—431no—
204C4×#99D4311yescharacteristic, complemented
205#1460—431no—
206#1351—432no—
207#1343—432no—
208C2×C43—21no—
209C2×C43—21no—
210C2×C43—42no—
211C2×C43—81no—
212C2×C43#150111yescharacteristic
213C2×C43C2×AΣL(1,𝔽16)11yescharacteristic
214C2×C43#212611yescharacteristic
215D43—82no—
216C22×#41—431no—
217C2×C4×D4—861no—
218C2×C4×D4C2×D4311yescharacteristic
219#431—861no—
220C2×#99—21no—
221#395—431no—
222#394—431no—
223#394—861no—
224#393—431no—
225#392—431no—
226#392—861no—
227C4×C22⋊C4—21no—
228C4×C22⋊C4—431no—
229C4×C22⋊C4—861no—
230#342C2×D4312yes—
231#323—432no—
232#323—861no—
233C43#1172511yescharacteristic, complemented, Sylow
234C2×#41—1721no—
235C2×#41—1721no—
236C2×#41—1724no—
237C22×Q8—431no—
238C22×D4—431no—
239C22×D4—861no—
240C22×D4—861no—
241C22×D4C22⋊D4311yescharacteristic
242C23×C4—861no—
243C23×C4C4×D4311yescharacteristic
244C23×C4C22⋊D4311yescharacteristic
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—1724no—
248C2×C4⋊C4—861no—
249C2×C4⋊C4C4×D4311yescharacteristic
250C2×C22⋊C4—22no—
251C2×C22⋊C4—861no—
252C2×C22⋊C4—861no—
253C2×C22⋊C4—861no—
254C2×C22⋊C4—861no—
255C2×C22⋊C4—861no—
256C2×C22⋊C4D17211yescharacteristic
257C2×C22⋊C4C22×D4311yescharacteristic
258C2×C42—41no—
259C2×C42—861no—
260C2×C42D17211yescharacteristic
261C4⋊C8—1721no—
262#99—24no—
263C22⋊C8—862no—
264C4×C8—1721no—
265#95—41no—
266#95—431no—
267#95—431no—
268#95—861no—
269#95—861no—
270C24—861no—
271C24(C2×D43)⋊C411yescharacteristic
272C24C2×C22⋊D4311yescharacteristic
273#41—1724no—
274#41—1724no—
275#41—1724no—
276#41—1724no—
277C2×Q8—1721no—
278C2×Q8—1722no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—41no—
282C2×D4—1721no—
283C2×D4—1722no—
284C2×D4—1722no—
285C2×D4—1722no—
286C2×D4—1722no—
287C2×D4—1722no—
288C2×D4—1722no—
289C2×D4—1722no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—864no—
298C22×C4—861no—
299C22×C4—861no—
300C22×C4—861no—
301C22×C4—1722no—
302C22×C4—1721no—
303C22×C4—1721no—
304C22×C4(C2×D43)⋊C411yescharacteristic
305C22×C4(C2×D43)⋊C411yescharacteristic
306C22×C4(C2×D43)⋊C411yescharacteristic
307C22×C4C2×C4×D4311yescharacteristic
308C22×C4C2×D17211yescharacteristic
309C2×C8—1722no—
310C4⋊C4—41no—
311C4⋊C4—1722no—
312C22⋊C4—24no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—1722no—
317C22⋊C4—1722no—
318C22⋊C4—1722no—
319C22⋊C4—1722no—
320C22⋊C4—1722no—
321C42—21no—
322C42—21no—
323C42—82no—
324C42—1722no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—862no—
335C23—1721no—
336C23—1721no—
337C23—1721no—
338C23—1721no—
339C23—1721no—
340C23—1721no—
341C23C2×(C2×D43)⋊C411yescharacteristic
342C23#1306334512yes—
343Q8—1722no—
344Q8—1722no—
345D4—44no—
346D4—1722no—
347D4—1722no—
348D4—1728no—
349D4—1728no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—862no—
367C2×C4—862no—
368C2×C4—1724no—
369C2×C4—1721no—
370C2×C4—1721no—
371C2×C4—1721no—
372C2×C4—1721no—
373C2×C4—1721no—
374C2×C4—1721no—
375C2×C4—1721no—
376C2×C4—1724no—
377C2×C4#1306323911yescharacteristic
378C2×C4#1306323911yescharacteristic
379C2×C4#1306325312yes—
380C8—3442no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—3441no—
394C22—3441no—
395C22—3441no—
396C22—3441no—
397C22—1722no—
398C22—1722no—
399C22—1722no—
400C22C2×#1306323911yescharacteristic
401C22#1458289311yescharacteristic
402C22C2×#1306325311yescharacteristic
403C4—22no—
404C4—21no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—84no—
409C4—3441no—
410C4—1722no—
411C4—1722no—
412C4#1458268512yes—
413C2—21no—
414C2—21no—
415C2—42no—
416C2—81no—
417C2—3442no—
418C2#1913308411yescharacteristic
419C2C2×#1458268511yescharacteristic
420C2#1913625811yescharacteristic
subgroupG/H
centerC2×C4#13063239
commutatorC22×C4×C43C22×C4
FrattiniC2×C22⋊C4C22×D43
FittingC4×#99×C43C2
socleC22×C43C2×#99
prime radical#19137810C2
cosocleC2×C22⋊C4×C43C23
hypercenterC4×#99D43
hypocenterC43#11725
LCSG ▷ C22×C4×C43 ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #19132916C2semidirect
3. #19132615C2semidirect
4. #19132591C2semidirect
5. C4×#99×C43C2semidirect
6. C22×#10621898C4semidirect
9. #14582819C4semidirect
10. #14582816C4semidirect
16. C4×C22⋊Dic43C4semidirect
204. C4×#99D43semidirect
233. C43#11725semidirect

Maximal subgroups

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

subgroupindex
#191378102
#191329162
#191326152
#191325912
C4×#99×C432
#1172543