Finite Group Databasebeta

← #36365078

Subgroups — #36365078

Order 22144.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149712C211yesmaximal, characteristic, complemented
2C22×#14583815C211yesmaximal, characteristic, complemented
3D4×Dic346C211yesmaximal, characteristic, complemented
4#26012137C211yesmaximal, characteristic
5#26011885C211yesmaximal, characteristic, complemented
6#26011737C211yesmaximal, characteristic, complemented
7#26011732C211yesmaximal, characteristic, complemented
8#26011714C211yesmaximal, characteristic, complemented
9#26011697C211yesmaximal, characteristic, complemented
10#26011655C211yesmaximal, characteristic, complemented
11#26011649C211yesmaximal, characteristic, complemented
12#26011644C211yesmaximal, characteristic, complemented
13#26011632C211yesmaximal, characteristic, complemented
14#26011593C211yesmaximal, characteristic, complemented
15#435×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C22×Dic346C2211yescharacteristic, complemented
19C22×D692C2211yescharacteristic
20#19149712—24no—
21C2×Dic692C2211yescharacteristic, complemented
22C2×Dic692C2211yescharacteristic, complemented
23C2×Dic692C2211yescharacteristic, complemented
24C2×#14583815—24no—
25C2×#14583815C2211yescharacteristic
26C2×#14583815C2211yescharacteristic
27C2×#14583815C2211yescharacteristic
28C2×#14583815C2211yescharacteristic
29C2×#14583815C2211yescharacteristic
30C2×#14583815C2211yescharacteristic
31#19149679C2211yescharacteristic, complemented
32#19149678—21no—
33#19149667C2211yescharacteristic
34#19149664—21no—
35#19149658C2211yescharacteristic
36#19149658C2211yescharacteristic
37#19149658C2211yescharacteristic, complemented
38#19149656—21no—
39#19149656—21no—
40#19149653C2211yescharacteristic
41#19149652—21no—
42#19149647—21no—
43C4×Dic346C2211yescharacteristic, complemented
44C4×D692C2211yescharacteristic
45D4×Dic173—21no—
46C22⋊C4×D173—21no—
47#19149613C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149604C2211yescharacteristic, complemented
50#19149604C2211yescharacteristic, complemented
51#19149604C2211yescharacteristic, complemented
52#19149601C2211yescharacteristic
53#19149601C2211yescharacteristic
54C2×M16×C173C2211yescharacteristic, complemented
55C22×C8×C173C2211yescharacteristic
56C4×D4×C173C2211yescharacteristic
57C4⋊C8×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic
59C22⋊C8×C173C2211yescharacteristic, complemented
60#97×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583826—22no—
63#14583826—41no—
64C2×Dic346—22no—
65C2×Dic346—22no—
66C2×Dic346—21no—
67C2×Dic346—21no—
68C2×Dic346—41no—
69C2×Dic346—41no—
70C2×Dic346C2311yescharacteristic
71C2×Dic346C2311yescharacteristic
72C2×Dic346C2311yescharacteristic
73C2×Dic346C2311yescharacteristic
74C2×D692—22no—
75C2×D692—21no—
76C2×D692—41no—
77C2×D692C2311yescharacteristic
78C2×D692C2311yescharacteristic
79C2×C22⋊D173—21no—
80C2×C22⋊D173—21no—
81C22×Dic173—21no—
82C22×Dic173—21no—
83C2×C4×D173—21no—
84C2×C4×D173—21no—
85Dic692—22no—
86Dic692—22no—
87Dic692—41no—
88#14583815—22no—
89#14583815—22no—
90#14583815—22no—
91#14583815—22no—
92#14583815—41no—
93#14583815D414yescomplemented
94C4⋊Dic173—21no—
95C4⋊Dic173—21no—
96C4⋊Dic173C2311yescharacteristic
97C4⋊Dic173C2311yescharacteristic
98#14583805—21no—
99#14583805—21no—
100#14583805—21no—
101#14583805—21no—
102C22⋊Dic173—21no—
103C22⋊Dic173—21no—
104C22⋊Dic173—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803—21no—
108#14583803—21no—
109C4×Dic173—21no—
110C2×D4×C173D411yescharacteristic
111C22×C4×C173C2311yescharacteristic
112C22×C4×C173C2311yescharacteristic
113M16×C173—22no—
114C2×C8×C173—22no—
115C2×C8×C173C2311yescharacteristic
116C2×C8×C173C2311yescharacteristic
117C2×C8×C173C2311yescharacteristic
118C2×C8×C173C2311yescharacteristic
119C4⋊C4×C173D411yescharacteristic
120C22⋊C4×C173D411yescharacteristic
121C22⋊C4×C173D411yescharacteristic
122C42×C173C2311yescharacteristic
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—42no—
126C22×D173—42no—
127C22×D173—81no—
128Dic346—21no—
129Dic346—21no—
130Dic346—21no—
131Dic346—21no—
132Dic346—21no—
133Dic346—21no—
134Dic346—21no—
135Dic346—42no—
136Dic346—42no—
137Dic346C2×D412yes—
138D692—21no—
139D692—21no—
140D692—21no—
141D692—42no—
142D692C2×D412yes—
143C22⋊D173—42no—
144C22⋊D173—81no—
145C2×Dic173—21no—
146C2×Dic173—21no—
147C2×Dic173—21no—
148C2×Dic173—21no—
149C2×Dic173—21no—
150C2×Dic173—21no—
151C2×Dic173—42no—
152C2×Dic173—42no—
153C4×D173—42no—
154C4×D173—81no—
155C23×C173C2×D411yescharacteristic
156C23×C173C2×D411yescharacteristic
157D4×C173—22no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173C2×D411yescharacteristic
163C2×C4×C173C2×D411yescharacteristic
164C2×C4×C173C2×D411yescharacteristic
165C2×C4×C173C2×D411yescharacteristic
166C2×C4×C173C2411yescharacteristic
167C8×C173—21no—
168C8×C173—21no—
169C8×C173—21no—
170C8×C173C2×D412yes—
171C22×C173—21no—
172C22×C173—21no—
173C22×C173—21no—
174C22×C173—21no—
175C22×C173—21no—
176C22×C173#13712yescomplemented
177C22×C173C22×D411yescharacteristic
178C2×D173—22no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—41no—
183C2×D173—41no—
184C2×D173—84no—
185C4×C173—21no—
186C4×C173—21no—
187C4×C173—21no—
188C4×C173—21no—
189C4×C173C22×D411yescharacteristic
190C4×C173H(5,𝔽2)11yescharacteristic
191Dic173—22no—
192Dic173—41no—
193Dic173—41no—
194Dic173—41no—
195Dic173—41no—
196Dic173—41no—
197C2×C173—22no—
198C2×C173—41no—
199C2×C173D4211yescharacteristic
200C2×C173C2×#13711yescharacteristic
201C2×C173#57711yescharacteristic
202D173—42no—
203D173—81no—
204C173#326611yescharacteristic, complemented, Sylow
205#3266—1731nomaximal, Sylow
206C2×#137—1731no—
207C22×SD16—1731no—
208D4×Q8—1731no—
209#546—1731no—
210#497—1731no—
211#468—1731no—
212#467—1731no—
213#464—1731no—
214#461—1731no—
215#452—1731no—
216#451—1731no—
217#450—1731no—
218#448—1731no—
219#440—1731no—
220#435D17311yescharacteristic, complemented
221C2×#41—1731no—
222C22×Q8—1731no—
223C22×Q8—1731no—
224C22×D4—1731no—
225#137—3464no—
226C2×Q16—1731no—
227C2×Q16—1731no—
228C2×Q16—1731no—
229C2×SD16—3464no—
230C2×SD16—1731no—
231C2×SD16—1731no—
232C2×SD16—1731no—
233C2×SD16—1731no—
234C2×SD16—1731no—
235C2×SD16—1731no—
236C2×M16C2×D17311yescharacteristic, complemented
237C22×C8C2×D17311yescharacteristic
238C4⋊Q8—3461no—
239C4⋊Q8—1731no—
240#124—1731no—
241#123—3461no—
242C22⋊Q8—3461no—
243C22⋊Q8—3461no—
244C22⋊Q8—1731no—
245C22⋊Q8—1731no—
246C22⋊Q8—1731no—
247#121—3461no—
248#121—1731no—
249#120—3461no—
250C4×Q8—1731no—
251C4×D4—3461no—
252C4×D4—1731no—
253C4×D4C2×D17311yescharacteristic
254C2×C22⋊C4—3461no—
255#107—1731no—
256C4⋊C8C2×D17311yescharacteristic, complemented
257#103—1731no—
258#103—1731no—
259#103—1731no—
260#103—1731no—
261#102—1731no—
262#102—1731no—
263C22⋊C8C2×D17311yescharacteristic
264C22⋊C8C2×D17311yescharacteristic, complemented
265#97C2×D17311yescharacteristic, complemented
266C24—3461no—
267#41—3462no—
268#41—6921no—
269C2×Q8—3462no—
270C2×Q8—3462no—
271C2×Q8—3461no—
272C2×Q8—3461no—
273C2×Q8—1731no—
274C2×Q8—1731no—
275C2×Q8—1731no—
276C2×Q8—1731no—
277C2×Q8—6921no—
278C2×Q8—6921no—
279C2×D4—3462no—
280C2×D4—3461no—
281C2×D4—3461no—
282C2×D4—3461no—
283C2×D4—1731no—
284C2×D4—1731no—
285C2×D4—6921no—
286C2×D4D69211yescharacteristic
287C22×C4—3461no—
288C22×C4—3461no—
289C22×C4—3461no—
290C22×C4—3461no—
291C22×C4C22×D17311yescharacteristic
292C22×C4C22×D17311yescharacteristic
293Q16—3462no—
294Q16—3462no—
295Q16—6921no—
296SD16—3462no—
297SD16—3462no—
298SD16—3462no—
299SD16—3462no—
300SD16—1734no—
301SD16—6921no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D17311yescharacteristic
305C2×C8C22×D17311yescharacteristic
306C2×C8C22×D17311yescharacteristic
307C2×C8C22×D17311yescharacteristic
308C4⋊C4—3461no—
309C4⋊C4—3461no—
310C4⋊C4—3461no—
311C4⋊C4—3461no—
312C4⋊C4—3461no—
313C4⋊C4—3461no—
314C4⋊C4—1731no—
315C4⋊C4—1731no—
316C4⋊C4D69211yescharacteristic
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—3461no—
320C22⋊C4—3461no—
321C22⋊C4—3461no—
322C22⋊C4—3461no—
323C22⋊C4—3461no—
324C22⋊C4D69211yescharacteristic
325C22⋊C4D69211yescharacteristic
326C42—3461no—
327C42C22×D17311yescharacteristic
328C23—3461no—
329C23—3461no—
330C23—13841no—
331C23—6922no—
332C23—6922no—
333C23C2×D69211yescharacteristic
334C23C2×D69211yescharacteristic
335Q8—3461no—
336Q8—3461no—
337Q8—3461no—
338Q8—3461no—
339Q8—3461no—
340Q8—3461no—
341Q8—3461no—
342Q8—1732no—
343Q8—6922no—
344Q8—6922no—
345D4—22no—
346D4—3461no—
347D4—3461no—
348D4—3461no—
349D4—13841no—
350D4—1732no—
351D4—6922no—
352D4—6922no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—3461no—
358C2×C4—3461no—
359C2×C4—3461no—
360C2×C4—3461no—
361C2×C4—3461no—
362C2×C4—3461no—
363C2×C4—13841no—
364C2×C4—6922no—
365C2×C4—6922no—
366C2×C4—6922no—
367C2×C4C2×D69211yescharacteristic
368C2×C4C2×D69211yescharacteristic
369C2×C4C2×D69211yescharacteristic
370C2×C4C2×D69211yescharacteristic
371C2×C4C23×D17311yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D17312yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—3462no—
382C22—13844no—
383C22—6921no—
384C22—6921no—
385C22—6921no—
386C22—6921no—
387C22—6921no—
388C22#1914971212yescomplemented
389C22C22×D69211yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—3462no—
395C4—6921no—
396C4—6921no—
397C4—6921no—
398C4—6921no—
399C4—6921no—
400C4C2×D4×D17311yescharacteristic
401C4#1914972511yescharacteristic
402C2—22no—
403C2—41no—
404C2—13841no—
405C2—6922no—
406C2D4×D69211yescharacteristic
407C2C2×#1914971211yescharacteristic
408C2#2601230311yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#435×C173C2
socleC22×C173C22×D4
prime radical#26012137C2
hypercenter#435D173
hypocenterC173#3266
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#19149712C2semidirect
2. C22×#14583815C2semidirect
3. D4×Dic346C2semidirect
5. #26011885C2semidirect
6. #26011737C2semidirect
7. #26011732C2semidirect
8. #26011714C2semidirect
9. #26011697C2semidirect
10. #26011655C2semidirect
11. #26011649C2semidirect
12. #26011644C2semidirect
13. #26011632C2semidirect
14. #26011593C2semidirect
15. #435×C173C2semidirect
18. C22×Dic346C22semidirect
21. C2×Dic692C22semidirect
22. C2×Dic692C22semidirect
23. C2×Dic692C22semidirect
31. #19149679C22semidirect
37. #19149658C22semidirect
43. C4×Dic346C22semidirect
47. #19149613C22semidirect
48. #19149604C22semidirect
49. #19149604C22semidirect
50. #19149604C22semidirect
51. #19149604C22semidirect
54. C2×M16×C173C22semidirect
57. C4⋊C8×C173C22semidirect
59. C22⋊C8×C173C22semidirect
60. #97×C173C22semidirect
93. #14583815D4semidirect
176. C22×C173#137semidirect
204. C173#3266semidirect
220. #435D173semidirect
236. C2×M16C2×D173semidirect
256. C4⋊C8C2×D173semidirect
264. C22⋊C8C2×D173semidirect
265. #97C2×D173semidirect
388. C22#19149712semidirect

Maximal subgroups

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

subgroupindex
C2×#191497122
C22×#145838152
D4×Dic3462
#260121372
#260118852
#260117372
#260117322
#260117142
#260116972
#260116552
#260116492
#260116442
#260116322
#260115932
#435×C1732
#3266173