Finite Group Databasebeta

← #15684951

Subgroups — #15684951

Order 2944.

478 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×#10625089C211yesmaximal, characteristic, complemented
2C2×#10625087C211yesmaximal, characteristic, complemented
3#13093075C211yesmaximal, characteristic, complemented
4#13093070C211yesmaximal, characteristic, complemented
5#13092660C211yesmaximal, characteristic, complemented
6#13092655C211yesmaximal, characteristic, complemented
7#13092629C211yesmaximal, characteristic, complemented
8#13092624C211yesmaximal, characteristic, complemented
9#13092578C211yesmaximal, characteristic, complemented
10#13092576C211yesmaximal, characteristic, complemented
11#13092571C211yesmaximal, characteristic, complemented
12#13092561C211yesmaximal, characteristic, complemented
13#13092519C211yesmaximal, characteristic, complemented
14#13092490C211yesmaximal, characteristic, complemented
15C23×#497C211yesmaximal, characteristic, complemented
16C2×#67292C2211yescharacteristic
17C2×#67292C2211yescharacteristic
18C2×Q8×D23C2211yescharacteristic
19C2×#67290C2211yescharacteristic
20C2×Q16×C23C2211yescharacteristic
21C2×Q16×C23C2211yescharacteristic
22C2×SD16×C23C2211yescharacteristic
23C2×SD16×C23C2211yescharacteristic
24C23×C4⋊Q8C2211yescharacteristic
25C23×#124C2211yescharacteristic
26#10625117C2211yescharacteristic
27#10625117C2211yescharacteristic
28#10625117C2211yescharacteristic
29C2×#67275C2211yescharacteristic
30C2×#67275C2211yescharacteristic
31C2×#67274C2211yescharacteristic
32#10625109C2211yescharacteristic
33C2×#67272C2211yescharacteristic
34#10625089—24no—
35#10625087—24no—
36C2×#67263C2211yescharacteristic
37C2×#67262C2211yescharacteristic
38C2×C8×D23C2211yescharacteristic
39#10625066—21no—
40#10625065—21no—
41#10625065—21no—
42C4⋊C4×D23—21no—
43#10625056—21no—
44#10625055—21no—
45(C22⋊D23)⋊C4—21no—
46#10625049—21no—
47C4×D92C2211yescharacteristic
48C4×D92C2211yescharacteristic
49C4×Dic46C2211yescharacteristic
50C23×#97C2211yescharacteristic
51#10625017C2211yescharacteristic
52#10625017C2211yescharacteristic
53#10625017C2211yescharacteristic
54#10625014C2211yescharacteristic
55#10625003C2211yescharacteristic
56(C2×D23)⋊C8C2211yescharacteristic
57(C2×D23)⋊C8C2211yescharacteristic
58#10625000C2211yescharacteristic
59#10624998C2211yescharacteristic
60C4⋊(C23⋊C8)C2211yescharacteristic
61C2×Q8×C23C2311yescharacteristic
62C2×Q8×C23C2311yescharacteristic
63C2×Q8×C23C2311yescharacteristic
64C2×D4×C23C2311yescharacteristic
65C2×C22⋊D23—21no—
66C2×C22⋊D23—21no—
67C22×Dic23—21no—
68#67292—22no—
69#67292—22no—
70#67292—41no—
71#67292—41no—
72Q8×D23—22no—
73Q8×D23—41no—
74#67290—22no—
75#67290—41no—
76C2×D92—21no—
77C2×D92—21no—
78C2×D92D411yescharacteristic
79C2×D92C2311yescharacteristic
80C2×C4×D23—21no—
81C2×C4×D23—21no—
82C2×C4×D23—21no—
83C2×C4×D23—21no—
84C2×C4×D23C2311yescharacteristic
85C2×C4×D23C2311yescharacteristic
86C2×Dic46—21no—
87C2×Dic46C2311yescharacteristic
88Q16×C23—22no—
89Q16×C23—41no—
90SD16×C23—22no—
91SD16×C23—41no—
92C2×C8×C23C2311yescharacteristic
93C2×C8×C23C2311yescharacteristic
94C4⋊C4×C23—21no—
95C4⋊C4×C23—21no—
96C22⋊C4×C23—21no—
97C22⋊C4×C23—21no—
98C42×C23C2311yescharacteristic
99C22⋊Dic23—21no—
100#67275—22no—
101#67275—22no—
102#67274—22no—
103#67272—22no—
104(C2×D23)⋊C4—21no—
105(C2×D23)⋊C4—21no—
106(C2×D23)⋊C4—21no—
107(C2×D23)⋊C4—21no—
108(C2×D23)⋊C4D411yescharacteristic
109(C2×D23)⋊C4D411yescharacteristic
110C4⋊Dic23D411yescharacteristic
111C4⋊Dic23C2311yescharacteristic
112C4⋊Dic23C2311yescharacteristic
113Dic23⋊C4—21no—
114Dic23⋊C4—21no—
115Dic23⋊C4—21no—
116Dic23⋊C4—21no—
117C4×Dic23—21no—
118C2×C23⋊C8C2311yescharacteristic
119C2×C23⋊C8C2311yescharacteristic
120#67263D414yescomplemented
121#67262—22no—
122C8×D23—22no—
123C23×C23—21no—
124C22×D23—21no—
125C22×D23C2×D411yescharacteristic
126C22×D23C2×D411yescharacteristic
127Q8×C23—21no—
128Q8×C23—21no—
129Q8×C23—21no—
130Q8×C23—21no—
131Q8×C23—21no—
132Q8×C23—21no—
133D4×C23—21no—
134D4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23C2×D411yescharacteristic
139C2×C4×C23C2×D411yescharacteristic
140C2×C4×C23C2411yescharacteristic
141C22⋊D23—42no—
142C22⋊D23—42no—
143C2×Dic23—21no—
144C2×Dic23—21no—
145C2×Dic23—21no—
146C2×Dic23—42no—
147C2×Dic23C2×D411yescharacteristic
148C2×Dic23C2×D411yescharacteristic
149D92—21no—
150D92—41no—
151D92—42no—
152D92—42no—
153D92C2×D412yes—
154C4×D23—21no—
155C4×D23—21no—
156C4×D23—21no—
157C4×D23—21no—
158C4×D23—42no—
159C4×D23—42no—
160C4×D23—42no—
161C4×D23—42no—
162Dic46—21no—
163Dic46—42no—
164Dic46C2×D412yes—
165C8×C23—21no—
166C8×C23C2×D412yes—
167C23⋊C8—21no—
168C23⋊C8—21no—
169#3269—231nomaximal, Sylow
170C22×C23—41no—
171C22×C23—41no—
172C22×C23—41no—
173C22×C23C22×D411yescharacteristic
174C2×D23—21no—
175C2×D23—21no—
176C2×D23—22no—
177C2×D23—21no—
178C2×D23—21no—
179C2×D23—21no—
180C2×D23—21no—
181C2×D23—41no—
182C2×D23—41no—
183C4×C23—21no—
184C4×C23—21no—
185C4×C23—41no—
186C4×C23—41no—
187C4×C23—41no—
188C4×C23C22×D411yescharacteristic
189C4×C23H(5,𝔽2)11yescharacteristic
190Dic23—22no—
191Dic23—21no—
192Dic23—21no—
193Dic23—41no—
194Dic23—41no—
195C2×#137—231no—
196C2×#135—231no—
197#548—231no—
198#547—231no—
199#497D2311yescharacteristic, complemented
200#468—231no—
201#467—231no—
202#463—231no—
203#462—231no—
204#452—231no—
205#452—231no—
206#451—231no—
207#449—231no—
208#440—231no—
209#435—231no—
210C2×C23—81no—
211C2×C23D4211yescharacteristic
212C2×C23#57511yescharacteristic
213C2×C23#57811yescharacteristic
214D23—41no—
215D23—41no—
216D23—42no—
217C2×#41—231no—
218C2×#41—231no—
219C2×#41—231no—
220C22×Q8—231no—
221#137—464no—
222#135—464no—
223C2×Q16—231no—
224C2×Q16—231no—
225C2×Q16C2×D2311yescharacteristic
226C2×Q16C2×D2311yescharacteristic
227C2×SD16—231no—
228C2×SD16—231no—
229C2×SD16C2×D2311yescharacteristic
230C2×SD16C2×D2311yescharacteristic
231C2×D8—231no—
232C2×M16—231no—
233C22×C8—231no—
234C4⋊Q8C2×D2311yescharacteristic
235#124—461no—
236#124C2×D2311yescharacteristic
237#123—461no—
238#123—461no—
239C22⋊Q8—231no—
240C22⋊Q8—231no—
241C22⋊Q8—231no—
242C22⋊Q8—461no—
243#121—231no—
244#121—461no—
245#121—461no—
246C4×Q8—231no—
247C4×D4—231no—
248C4×D4—231no—
249C4×D4—461no—
250C2×C4⋊C4—461no—
251#107—231no—
252C4⋊C8—231no—
253#103—231no—
254#103—231no—
255#103—231no—
256#103—231no—
257#102—231no—
258#102—231no—
259C22⋊C8—231no—
260C22⋊C8—231no—
261#97C2×D2311yescharacteristic
262C23#326911yescharacteristic, complemented, Sylow
263#41—462no—
264#41—462no—
265#41—462no—
266#41—921no—
267#41—921no—
268#41—921no—
269C2×Q8—231no—
270C2×Q8—461no—
271C2×Q8—462no—
272C2×Q8—921no—
273C2×Q8C22×D2311yescharacteristic
274C2×Q8C22×D2311yescharacteristic
275C2×Q8C22×D2311yescharacteristic
276C2×D4—231no—
277C2×D4—231no—
278C2×D4—461no—
279C2×D4—461no—
280C2×D4—461no—
281C2×D4—461no—
282C2×D4C22×D2311yescharacteristic
283C22×C4—231no—
284C22×C4—231no—
285C22×C4—461no—
286C22×C4—461no—
287C22×C4—461no—
288C22×C4—461no—
289C22×C4—461no—
290Q16—22no—
291Q16—41no—
292Q16—462no—
293Q16—462no—
294SD16—22no—
295SD16—41no—
296SD16—234no—
297SD16—462no—
298D8—462no—
299M16—462no—
300C2×C8—231no—
301C2×C8—231no—
302C2×C8—462no—
303C2×C8C22×D2311yescharacteristic
304C2×C8C22×D2311yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—231no—
308C4⋊C4—231no—
309C4⋊C4—231no—
310C4⋊C4—461no—
311C4⋊C4—461no—
312C4⋊C4—461no—
313C4⋊C4—461no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—231no—
317C22⋊C4—231no—
318C22⋊C4—461no—
319C22⋊C4—461no—
320C22⋊C4—461no—
321C22⋊C4—461no—
322C22⋊C4—461no—
323C42—461no—
324C42C22×D2311yescharacteristic
325C23—21no—
326C23—231no—
327C23—231no—
328C23—461no—
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—21no—
335Q8—232no—
336Q8—461no—
337Q8—922no—
338D4—21no—
339D4—21no—
340D4—232no—
341D4—461no—
342D4—921no—
343D4—922no—
344D4—922no—
345D4—922no—
346D4—922no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—231no—
351C2×C4—231no—
352C2×C4—461no—
353C2×C4—461no—
354C2×C4—461no—
355C2×C4—461no—
356C2×C4—461no—
357C2×C4—461no—
358C2×C4—461no—
359C2×C4—922no—
360C2×C4—922no—
361C2×C4—922no—
362C2×C4—922no—
363C2×C4—922no—
364C2×C4D4×D2311yescharacteristic
365C2×C4D4×D2311yescharacteristic
366C2×C4C23×D2311yescharacteristic
367C8—21no—
368C8—461no—
369C8—461no—
370C8D4×D2312yes—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—461no—
375C22—461no—
376C22—461no—
377C22—461no—
378C22—461no—
379C22—461no—
380C22—462no—
381C22—921no—
382C22—921no—
383C22C2×D4×D2311yescharacteristic
384C4—21no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—461no—
390C4—461no—
391C4—462no—
392C4—921no—
393C4—921no—
394C4C2×D4×D2311yescharacteristic
395C4#1062515511yescharacteristic
396C2—81no—
397C2—921no—
398C2—921no—
399C2—922no—
400C2#1309305211yescharacteristic
401C2#1309321611yescharacteristic
402C2#1309323311yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#497C2
socleC22×C23C22×D4
hypercenter#497D23
hypocenterC23#3269
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#10625089C2semidirect
2. C2×#10625087C2semidirect
3. #13093075C2semidirect
4. #13093070C2semidirect
5. #13092660C2semidirect
6. #13092655C2semidirect
7. #13092629C2semidirect
8. #13092624C2semidirect
9. #13092578C2semidirect
10. #13092576C2semidirect
11. #13092571C2semidirect
12. #13092561C2semidirect
13. #13092519C2semidirect
14. #13092490C2semidirect
15. C23×#497C2semidirect
120. #67263D4semidirect
199. #497D23semidirect
262. C23#3269semidirect

Maximal subgroups

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

subgroupindex
C2×#106250892
C2×#106250872
#130930752
#130930702
#130926602
#130926552
#130926292
#130926242
#130925782
#130925762
#130925712
#130925612
#130925192
#130924902
C23×#4972
#326923