Finite Group Databasebeta

← #25012189

Subgroups — #25012189

Order 11008.

694 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19136280C211yesmaximal, characteristic, complemented
2#19136260C211yesmaximal, characteristic, complemented
3#19136236C212yesmaximal, complemented
4#19134699C212yesmaximal, complemented
5C43×#3455C211yesmaximal, characteristic
6#14582883C2212yes—
7#14582873C2212yes—
8#14582698—21no—
9#14582696—21no—
10#14582694—21no—
11#14582692—21no—
12#14582689—22no—
13#14582686—22no—
14#14582659—22no—
15#14582655—22no—
16C43×#542—21no—
17C43×#542—21no—
18C43×#541—21no—
19C43×#540—21no—
20C43×#539—22no—
21C43×#538—22no—
22C43×#533C411yescharacteristic, complemented
23C43×#532C2211yescharacteristic
24C43×#529—22no—
25C43×#529—22no—
26C43×#525C411yescharacteristic, complemented
27C43×#525C2211yescharacteristic
28C4×#41×C43C411yescharacteristic, complemented
29C4×#41×C43C2211yescharacteristic
30C2×#117×C43C411yescharacteristic, complemented
31C2×#10621877—22no—
32#13063284—22no—
33#13063244—22no—
34#13063242—22no—
35#13063240—22no—
36#13063240—22no—
37C2×#41×C43C2×C411yescharacteristic
38C2×#41×C43C2311yescharacteristic
39C23×C4×C43C2×C411yescharacteristic
40C4⋊Q8×C43—21no—
41C4⋊Q8×C43D411yescharacteristic
42#127×C43D411yescharacteristic
43#126×C43—22no—
44#126×C43—22no—
45#126×C43—22no—
46#126×C43—22no—
47#125×C43—22no—
48#125×C43—22no—
49#125×C43—21no—
50#124×C43—22no—
51#124×C43—22no—
52#124×C43D411yescharacteristic
53#124×C43D411yescharacteristic
54#123×C43—22no—
55#123×C43—22no—
56#123×C43—22no—
57#123×C43—22no—
58#123×C43—21no—
59#123×C43D412yes—
60C22⋊Q8×C43—22no—
61C22⋊Q8×C43—22no—
62C22⋊Q8×C43—21no—
63C22⋊Q8×C43—21no—
64C22⋊Q8×C43—22no—
65#121×C43—22no—
66#121×C43—22no—
67#121×C43—21no—
68#121×C43—21no—
69#121×C43—22no—
70#120×C43—21no—
71#120×C43D412yes—
72C4×Q8×C43—21no—
73C4×Q8×C43—21no—
74C4×Q8×C43—22no—
75C4×D4×C43—21no—
76C4×D4×C43—22no—
77C4×D4×C43—21no—
78C4×D4×C43—22no—
79C4×D4×C43—22no—
80C4×D4×C43—22no—
81C4×D4×C43—22no—
82#117×C43—21no—
83#117×C43—21no—
84#117×C43—21no—
85#117×C43—21no—
86#117×C43—21no—
87#117×C43C2×C411yescharacteristic
88#117×C43C2×C411yescharacteristic
89#117×C43D412yes—
90C2×C4⋊C4×C43—21no—
91C2×C22⋊C4×C43—21no—
92C2×C42×C43—21no—
93C2×C42×C43C2×C411yescharacteristic
94C2×C42×C43C2×C411yescharacteristic
95C2×C42×C43D412yes—
96C24×C43C22⋊C411yescharacteristic
97#41×C43—41no—
98#41×C43—41no—
99#41×C43—41no—
100#41×C43—41no—
101C2×Q8×C43—22no—
102C2×Q8×C43C22⋊C411yescharacteristic
103C2×Q8×C43C2×D411yescharacteristic
104C2×D4×C43—22no—
105C2×D4×C43—21no—
106C2×D4×C43—22no—
107C2×D4×C43—22no—
108C2×D4×C43C22⋊C411yescharacteristic
109C2×D4×C43C2×D412yes—
110C2×D4×C43C2×D411yescharacteristic
111C22×C4×C43—22no—
112C22×C4×C43—21no—
113C22×C4×C43—21no—
114C22×C4×C43—21no—
115C22×C4×C43—21no—
116C22×C4×C43—21no—
117C22×C4×C43—22no—
118C22×C4×C43—41no—
119C22×C4×C43—41no—
120C22×C4×C43C22⋊C411yescharacteristic
121C22×C4×C43C22×C411yescharacteristic
122C22×C4×C43C2×D412yes—
123C4⋊C4×C43—21no—
124C4⋊C4×C43—21no—
125C4⋊C4×C43—21no—
126C4⋊C4×C43—21no—
127C4⋊C4×C43—22no—
128C4⋊C4×C43—22no—
129C4⋊C4×C43—22no—
130C4⋊C4×C43—22no—
131C4⋊C4×C43—22no—
132C4⋊C4×C43—21no—
133C4⋊C4×C43—21no—
134C4⋊C4×C43—22no—
135C4⋊C4×C43—22no—
136C4⋊C4×C43—22no—
137C22⋊C4×C43—21no—
138C22⋊C4×C43—21no—
139C22⋊C4×C43—21no—
140C22⋊C4×C43—21no—
141C22⋊C4×C43—22no—
142C22⋊C4×C43—22no—
143C22⋊C4×C43—22no—
144C22⋊C4×C43—22no—
145C22⋊C4×C43—22no—
146C22⋊C4×C43—21no—
147C22⋊C4×C43—22no—
148C22⋊C4×C43—22no—
149C22⋊C4×C43—22no—
150C22⋊C4×C43C22⋊C412yes—
151C42×C43—22no—
152C42×C43—21no—
153C42×C43—21no—
154C42×C43—21no—
155C42×C43—22no—
156C42×C43C22⋊C411yescharacteristic
157C42×C43C22⋊C411yescharacteristic
158C42×C43C22⋊C411yescharacteristic
159C42×C43C22⋊C412yes—
160C42×C43C22⋊C411yescharacteristic
161C22⋊Dic43—42no—
162C4⋊Dic43—42no—
163C4×Dic43—42no—
164#10621877—42no—
165#10621877—42no—
166C2×C43⋊C8—42no—
167C23×C43—21no—
168C23×C43—21no—
169C23×C43—41no—
170C23×C43—81no—
171C23×C43C2×C22⋊C411yescharacteristic
172C23×C43#12012yes—
173Q8×C43—41no—
174Q8×C43—41no—
175Q8×C43—82no—
176D4×C43—42no—
177D4×C43—41no—
178D4×C43—41no—
179D4×C43—42no—
180D4×C43—82no—
181D4×C43—82no—
182D4×C43—82no—
183C2×C4×C43—22no—
184C2×C4×C43—22no—
185C2×C4×C43—21no—
186C2×C4×C43—21no—
187C2×C4×C43—21no—
188C2×C4×C43—21no—
189C2×C4×C43—21no—
190C2×C4×C43—22no—
191C2×C4×C43—21no—
192C2×C4×C43—22no—
193C2×C4×C43—21no—
194C2×C4×C43—21no—
195C2×C4×C43—42no—
196C2×C4×C43—41no—
197C2×C4×C43—41no—
198C2×C4×C43—41no—
199C2×C4×C43—41no—
200C2×C4×C43—41no—
201C2×C4×C43—81no—
202C2×C4×C43—82no—
203C2×C4×C43—82no—
204C2×C4×C43—81no—
205C2×C4×C43C2×C22⋊C411yescharacteristic
206C2×C4×C43C2×C22⋊C411yescharacteristic
207C2×C4×C43#12012yes—
208C2×Dic43—82no—
209C43⋊C8—82no—
210#12720—431nomaximal, Sylow
211C22×C43—21no—
212C22×C43—22no—
213C22×C43—41no—
214C22×C43—42no—
215C22×C43—41no—
216C22×C43—41no—
217C22×C43—42no—
218C22×C43—41no—
219C22×C43—41no—
220C22×C43—82no—
221C22×C43#37911yescharacteristic
222C4×C43—22no—
223C4×C43—21no—
224C4×C43—41no—
225C4×C43—41no—
226C4×C43—41no—
227C4×C43—42no—
228C4×C43—42no—
229C4×C43—41no—
230C4×C43—41no—
231C4×C43—41no—
232C4×C43#41011yescharacteristic
233C4×C43#41011yescharacteristic
234Dic43—162no—
235#3455D4311yescharacteristic
236#2132—431no—
237#2126—431no—
238#2120—432no—
239#1827—432no—
240C2×C43—21no—
241C2×C43—42no—
242C2×C43—41no—
243C2×C43—81no—
244C2×C43—81no—
245C2×C43#178211yescharacteristic
246#542—21no—
247#542—21no—
248#541—21no—
249#540—21no—
250#539—22no—
251#538—22no—
252#533Dic4311yescharacteristic, complemented
253#532C2×D4311yescharacteristic
254#529—22no—
255#529—22no—
256#525Dic4311yescharacteristic, complemented
257#525C2×D4311yescharacteristic
258C4×#41Dic4311yescharacteristic, complemented
259C4×#41C2×D4311yescharacteristic
260C2×#117Dic4311yescharacteristic, complemented
261#413—432no—
262#410—432no—
263#356—861no—
264#355—861no—
265#354—861no—
266#353—861no—
267#352—862no—
268AΣL(1,𝔽16)—862no—
269#339—862no—
270H(3,ℤ/4)—862no—
271C43#1272011yescharacteristic, complemented, Sylow
272C2×#41C2×Dic4311yescharacteristic
273C2×#41C22×D4311yescharacteristic
274C23×C4C2×Dic4311yescharacteristic
275C2×M16—862no—
276C4⋊Q8—21no—
277C4⋊Q8C22⋊D4311yescharacteristic
278#127C22⋊D4311yescharacteristic
279#126—22no—
280#126—22no—
281#126—22no—
282#126—22no—
283#125—22no—
284#125—22no—
285#125—21no—
286#124—22no—
287#124—22no—
288#124C22⋊D4311yescharacteristic
289#124C22⋊D4311yescharacteristic
290#123—22no—
291#123—22no—
292#123—22no—
293#123—22no—
294#123—21no—
295#123C22⋊D4312yes—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—22no—
301#121—22no—
302#121—22no—
303#121—21no—
304#121—21no—
305#121—22no—
306#120—21no—
307#120C22⋊D4312yes—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—22no—
311C4×D4—21no—
312C4×D4—22no—
313C4×D4—21no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—22no—
318#117—21no—
319#117—21no—
320#117—21no—
321#117—21no—
322#117—21no—
323#117—862no—
324#117C2×Dic4311yescharacteristic
325#117C2×Dic4311yescharacteristic
326#117C22⋊D4312yes—
327C2×C4⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C42—21no—
330C2×C42C2×Dic4311yescharacteristic
331C2×C42C2×Dic4311yescharacteristic
332C2×C42C22⋊D4312yes—
333#101—862no—
334#100—862no—
335#99—862no—
336#99—862no—
337C24C22⋊Dic4311yescharacteristic
338#41—41no—
339#41—41no—
340#41—41no—
341#41—41no—
342C2×Q8—22no—
343C2×Q8C22⋊Dic4311yescharacteristic
344C2×Q8C2×C22⋊D4311yescharacteristic
345C2×D4—21no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4C22⋊Dic4311yescharacteristic
350C2×D4C2×C22⋊D4311yescharacteristic
351C2×D4C2×C22⋊D4312yes—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4C22⋊Dic4311yescharacteristic
362C22×C4C22×Dic4311yescharacteristic
363C22×C4C2×C22⋊D4312yes—
364M16—1722no—
365M16—1722no—
366C2×C8—1722no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—1722no—
382C22⋊C4—21no—
383C22⋊C4—21no—
384C22⋊C4—21no—
385C22⋊C4—21no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—21no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—1722no—
396C22⋊C4C22⋊Dic4312yes—
397C42—22no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—22no—
402C42—1722no—
403C42C22⋊Dic4311yescharacteristic
404C42C22⋊Dic4311yescharacteristic
405C42C22⋊Dic4311yescharacteristic
406C42C22⋊Dic4312yes—
407C42C22⋊Dic4311yescharacteristic
408C23—21no—
409C23—21no—
410C23—41no—
411C23—81no—
412C23C2×C22⋊Dic4311yescharacteristic
413C23#1306329412yes—
414Q8—41no—
415Q8—41no—
416Q8—82no—
417D4—42no—
418D4—41no—
419D4—41no—
420D4—42no—
421D4—82no—
422D4—82no—
423D4—82no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—3442no—
447C2×C4C2×C22⋊Dic4311yescharacteristic
448C2×C4C2×C22⋊Dic4311yescharacteristic
449C2×C4#1306329412yes—
450C8—3442no—
451C22—21no—
452C22—22no—
453C22—41no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—82no—
461C22#1458275511yescharacteristic
462C4—22no—
463C4—21no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—42no—
468C4—42no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—6882no—
473C4#1458287311yescharacteristic
474C4#1458287311yescharacteristic
475C2—21no—
476C2—42no—
477C2—41no—
478C2—81no—
479C2—81no—
480C2#1913451311yescharacteristic
subgroupG/H
centerC4#14582873
commutatorC22×C4×C43C22×C4
FrattiniC2×#41C22×D43
Fitting, prime radicalC43×#3455C2
socleC2×C43#1782
cosocleC2×#41×C43C23
hypercenter#3455D43
hypocenterC43#12720
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C4 ◁ C42 ◁ C4×#41 ◁ #3455

Complements

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

NH ≅ G/Ntype
1. #19136280C2semidirect
2. #19136260C2semidirect
3. #19136236C2semidirect
4. #19134699C2semidirect
22. C43×#533C4semidirect
26. C43×#525C4semidirect
28. C4×#41×C43C4semidirect
30. C2×#117×C43C4semidirect
252. #533Dic43semidirect
256. #525Dic43semidirect
258. C4×#41Dic43semidirect
260. C2×#117Dic43semidirect
271. C43#12720semidirect

Maximal subgroups

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

subgroupindex
#191362802
#191362602
#191362362
#191346992
C43×#34552
#1272043