Finite Group Databasebeta

← #36364315

Subgroups — #36364315

Order 22144.

468 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×D173C211yesmaximal, characteristic, complemented
2#121×D173C211yesmaximal, characteristic
3#26011823C211yesmaximal, characteristic, complemented
4#26011820C211yesmaximal, characteristic, complemented
5#26011811C211yesmaximal, characteristic, complemented
6#26011808C211yesmaximal, characteristic, complemented
7#26011722C211yesmaximal, characteristic, complemented
8#26011720C211yesmaximal, characteristic, complemented
9#26011718C211yesmaximal, characteristic, complemented
10#26011643C211yesmaximal, characteristic, complemented
11#26011640C211yesmaximal, characteristic, complemented
12#26011631C211yesmaximal, characteristic, complemented
13#26011628C211yesmaximal, characteristic, complemented
14C2×#19149591C211yesmaximal, characteristic, complemented
15#465×C173C211yesmaximal, characteristic, complemented
16C2×Q8×D173C2211yescharacteristic, complemented
17C2×D4×D173—21no—
18C2×D4×D173C2211yescharacteristic
19C22×C4×D173C2211yescharacteristic
20C2×#14583815C2211yescharacteristic, complemented
21C2×#14583814C2211yescharacteristic, complemented
22C2×#14583813C2211yescharacteristic, complemented
23C22×#13063436C2211yescharacteristic, complemented
24#19149659C2211yescharacteristic
25#19149658C2211yescharacteristic, complemented
26#19149657—21no—
27#19149656—21no—
28#19149655C2211yescharacteristic
29#19149654C2211yescharacteristic
30#19149653C2211yescharacteristic
31#19149652—21no—
32#19149651—21no—
33#19149650C2211yescharacteristic
34C4⋊C4×D173—21no—
35C4⋊C4×D173C2211yescharacteristic
36C4⋊C4×D173C2211yescharacteristic, complemented
37C22⋊C4×D173—21no—
38C22⋊C4×D173—21no—
39#19149611C2211yescharacteristic, complemented
40#19149610C2211yescharacteristic, complemented
41#19149610C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149603C2211yescharacteristic, complemented
44#19149602C2211yescharacteristic, complemented
45#19149601C2211yescharacteristic, complemented
46#19149600C2211yescharacteristic, complemented
47#19149599C2211yescharacteristic, complemented
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic, complemented
52C2×SD16×C173C2211yescharacteristic, complemented
53C22×C8×C173C2211yescharacteristic, complemented
54C22⋊Q8×C173C2211yescharacteristic, complemented
55#121×C173C2211yescharacteristic
56#106×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60C23×D173D411yescharacteristic
61C2×Dic346—21no—
62C2×Dic346C2311yescharacteristic
63Q8×D173—41no—
64Q8×D173—41no—
65C2×D692—21no—
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70D4×D173—41no—
71D4×D173—41no—
72D4×D173—44no—
73C22×Dic173D411yescharacteristic
74C2×C4×D173—21no—
75C2×C4×D173—21no—
76C2×C4×D173—21no—
77C2×C4×D173—41no—
78C2×C4×D173—41no—
79C2×C4×D173D411yescharacteristic
80C2×C4×D173D411yescharacteristic
81C2×C4×D173C2311yescharacteristic
82C2×C4×D173C2311yescharacteristic
83#14583815—41no—
84#14583814—41no—
85#14583813—41no—
86C2×#13063436—41no—
87C2×#13063436C2311yescharacteristic
88C2×#13063436C2311yescharacteristic
89C4⋊Dic173—21no—
90C4⋊Dic173C2311yescharacteristic
91C4⋊Dic173C2311yescharacteristic
92#14583805—21no—
93#14583805—21no—
94#14583805—21no—
95#14583805—21no—
96C22⋊Dic173—21no—
97C22⋊Dic173—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803—21no—
101#14583803—21no—
102C2×Q8×C173C2311yescharacteristic
103C2×D4×C173—21no—
104C2×D4×C173C2311yescharacteristic
105C22×C4×C173C2311yescharacteristic
106SD16×C173—41no—
107C2×C8×C173—41no—
108C2×C8×C173C2311yescharacteristic
109C2×C8×C173C2311yescharacteristic
110C4⋊C4×C173—21no—
111C4⋊C4×C173C2311yescharacteristic
112C4⋊C4×C173C2311yescharacteristic
113C22⋊C4×C173—21no—
114C22⋊C4×C173—21no—
115C22×D173—21no—
116C22×D173—41no—
117C22×D173—41no—
118C22×D173—42no—
119C22×D173—42no—
120C22×D173—42no—
121C22×D173SD1614yescomplemented
122C22×D173C2×D411yescharacteristic
123C22×D173C2×D411yescharacteristic
124Dic346—21no—
125Dic346—21no—
126Dic346—42no—
127D692—21no—
128D692—21no—
129D692—42no—
130C22⋊D173—42no—
131C22⋊D173—42no—
132C22⋊D173—42no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—41no—
137C2×Dic173C2×D411yescharacteristic
138C2×Dic173C2×D411yescharacteristic
139C4×D173—21no—
140C4×D173—21no—
141C4×D173—21no—
142C4×D173—21no—
143C4×D173—41no—
144C4×D173—41no—
145C4×D173—42no—
146C4×D173—42no—
147C4×D173—42no—
148#13063436—21no—
149#13063436—21no—
150C23×C173—21no—
151C23×C173C2×D411yescharacteristic
152Q8×C173—21no—
153Q8×C173—21no—
154D4×C173—21no—
155D4×C173—21no—
156D4×C173—42no—
157C2×C4×C173—21no—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173C2×D411yescharacteristic
163C2×C4×C173C2411yescharacteristic
164C8×C173—21no—
165C8×C173—21no—
166C22×C173—21no—
167C22×C173—21no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173—41no—
171C22×C173C2×SD1612yes—
172C22×C173C22×D411yescharacteristic
173C2×D173—21no—
174C2×D173—21no—
175C2×D173—21no—
176C2×D173—21no—
177C2×D173—44no—
178C2×D173—41no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—84no—
182C2×D173C2×SD1612yes—
183C2×D173C2×SD1612yes—
184C4×C173—21no—
185C4×C173—41no—
186C4×C173—41no—
187C4×C173—41no—
188C4×C173H(5,𝔽2)11yescharacteristic
189C4×C173H(5,𝔽2)11yescharacteristic
190Dic173—21no—
191Dic173—21no—
192Dic173—41no—
193Dic173—41no—
194Dic173—41no—
195C2×C173—22no—
196C2×C173—81no—
197C2×C173#53411yescharacteristic
198C2×C173C22×SD1611yescharacteristic
199C2×C173#57511yescharacteristic
200D173—22no—
201D173—22no—
202D173—81no—
203C173#318711yescharacteristic, complemented, Sylow
204#3187—1731nomaximal, Sylow
205C2×C22⋊Q8—1731no—
206C2×#121—1731no—
207#483—1731no—
208#483—1731no—
209#481—1731no—
210#481—1731no—
211#465—1731no—
212#465—1731no—
213#465—1731no—
214#465D17311yescharacteristic, complemented
215#450—1731no—
216#450—1731no—
217#448—1731no—
218#448—1731no—
219C2×C22⋊C8—1731no—
220C22×Q8—1731no—
221C22×D4—3461no—
222C22×D4—1731no—
223C23×C4—1731no—
224C2×SD16—1731no—
225C2×SD16—1731no—
226C2×SD16—1731no—
227C2×SD16C2×D17311yescharacteristic, complemented
228C22×C8—1731no—
229C22×C8C2×D17311yescharacteristic, complemented
230C22⋊Q8—3461no—
231C22⋊Q8—3461no—
232C22⋊Q8—1731no—
233C22⋊Q8—1731no—
234C22⋊Q8—1731no—
235C22⋊Q8C2×D17311yescharacteristic, complemented
236#121—3461no—
237#121—3461no—
238#121—1731no—
239#121—1731no—
240#121—1731no—
241#121C2×D17311yescharacteristic
242C2×C4⋊C4—3461no—
243C2×C4⋊C4—1731no—
244C2×C4⋊C4—1731no—
245C2×C22⋊C4—3461no—
246C2×C22⋊C4—3461no—
247#106—1731no—
248#106—1731no—
249#106—1731no—
250#106C2×D17311yescharacteristic, complemented
251#103—1731no—
252#103—1731no—
253#103—1731no—
254#103C2×D17311yescharacteristic, complemented
255#102—1731no—
256#102—1731no—
257#102—1731no—
258#102C2×D17311yescharacteristic, complemented
259C22⋊C8—1731no—
260C22⋊C8—1731no—
261C22⋊C8—1731no—
262C22⋊C8—1731no—
263C24—3461no—
264C24—1731no—
265C2×Q8—3461no—
266C2×Q8—1731no—
267C2×Q8—6921no—
268C2×Q8—6921no—
269C2×Q8C22×D17311yescharacteristic
270C2×D4—21no—
271C2×D4—3461no—
272C2×D4—3461no—
273C2×D4—3461no—
274C2×D4—3461no—
275C2×D4—1731no—
276C2×D4—6921no—
277C2×D4—6921no—
278C2×D4—6924no—
279C2×D4C22×D17311yescharacteristic
280C22×C4—3461no—
281C22×C4—3461no—
282C22×C4—3461no—
283C22×C4—1731no—
284C22×C4—1731no—
285C22×C4—1731no—
286C22×C4—1731no—
287C22×C4—1731no—
288C22×C4—6921no—
289C22×C4—6921no—
290C22×C4C22×D17311yescharacteristic
291SD16—41no—
292SD16—6921no—
293SD16—6921no—
294SD16—6921no—
295C2×C8—41no—
296C2×C8—1731no—
297C2×C8—1731no—
298C2×C8—6921no—
299C2×C8C22×D17311yescharacteristic
300C2×C8C22×D17311yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—3461no—
303C4⋊C4—3461no—
304C4⋊C4—3461no—
305C4⋊C4—3461no—
306C4⋊C4—3461no—
307C4⋊C4—1731no—
308C4⋊C4—1731no—
309C4⋊C4C22×D17311yescharacteristic
310C4⋊C4C22×D17311yescharacteristic
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C23—21no—
320C23—3461no—
321C23—1731no—
322C23—1731no—
323C23—1734no—
324C23—6921no—
325C23—6921no—
326C23—6922no—
327C23—6922no—
328C23—6922no—
329C23D4×D17311yescharacteristic
330Q8—21no—
331Q8—21no—
332Q8—3461no—
333Q8—3461no—
334Q8—6922no—
335D4—21no—
336D4—21no—
337D4—42no—
338D4—3461no—
339D4—3461no—
340D4—6922no—
341D4—6922no—
342D4—6922no—
343D4—6922no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—3461no—
350C2×C4—3461no—
351C2×C4—3461no—
352C2×C4—3461no—
353C2×C4—3461no—
354C2×C4—3461no—
355C2×C4—3461no—
356C2×C4—1731no—
357C2×C4—1731no—
358C2×C4—6921no—
359C2×C4—6921no—
360C2×C4—6922no—
361C2×C4—6922no—
362C2×C4—6922no—
363C2×C4—6921no—
364C2×C4D4×D17311yescharacteristic
365C2×C4C23×D17311yescharacteristic
366C8—21no—
367C8—21no—
368C8—3461no—
369C8—3461no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—3461no—
376C22—3461no—
377C22—3461no—
378C22—3461no—
379C22—13844no—
380C22—1732no—
381C22—1732no—
382C22—6924no—
383C22—6921no—
384C22—6921no—
385C22—6921no—
386C22SD16×D17312yes—
387C22C2×D4×D17311yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—3461no—
393C4—3461no—
394C4—6921no—
395C4—6921no—
396C4—6921no—
397C4#1914972411yescharacteristic
398C4#1914972511yescharacteristic
399C2—22no—
400C2—81no—
401C2—3462no—
402C2—3462no—
403C2—13841no—
404C2#2601204911yescharacteristic
405C2C2×SD16×D17311yescharacteristic
406C2#2601228811yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#465×C173C2
socleC22×C173C22×D4
prime radical#121×D173C2
hypercenter#465D173
hypocenterC173#3187
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D173C2semidirect
3. #26011823C2semidirect
4. #26011820C2semidirect
5. #26011811C2semidirect
6. #26011808C2semidirect
7. #26011722C2semidirect
8. #26011720C2semidirect
9. #26011718C2semidirect
10. #26011643C2semidirect
11. #26011640C2semidirect
12. #26011631C2semidirect
13. #26011628C2semidirect
14. C2×#19149591C2semidirect
15. #465×C173C2semidirect
16. C2×Q8×D173C22semidirect
20. C2×#14583815C22semidirect
21. C2×#14583814C22semidirect
22. C2×#14583813C22semidirect
23. C22×#13063436C22semidirect
25. #19149658C22semidirect
36. C4⋊C4×D173C22semidirect
39. #19149611C22semidirect
40. #19149610C22semidirect
41. #19149610C22semidirect
42. #19149604C22semidirect
43. #19149603C22semidirect
44. #19149602C22semidirect
45. #19149601C22semidirect
46. #19149600C22semidirect
47. #19149599C22semidirect
50. #19149591C22semidirect
51. #19149591C22semidirect
52. C2×SD16×C173C22semidirect
53. C22×C8×C173C22semidirect
54. C22⋊Q8×C173C22semidirect
56. #106×C173C22semidirect
57. #103×C173C22semidirect
58. #102×C173C22semidirect
121. C22×D173SD16semidirect
203. C173#3187semidirect
214. #465D173semidirect
227. C2×SD16C2×D173semidirect
229. C22×C8C2×D173semidirect
235. C22⋊Q8C2×D173semidirect
250. #106C2×D173semidirect
254. #103C2×D173semidirect
258. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1732
#121×D1732
#260118232
#260118202
#260118112
#260118082
#260117222
#260117202
#260117182
#260116432
#260116402
#260116312
#260116282
C2×#191495912
#465×C1732
#3187173