Finite Group Databasebeta

← #36365197

Subgroups — #36365197

Order 22144.

498 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149710C211yesmaximal, characteristic, complemented
2C2×#19149706C211yesmaximal, characteristic, complemented
3#26012152C211yesmaximal, characteristic, complemented
4#26012138C211yesmaximal, characteristic, complemented
5#26011872C211yesmaximal, characteristic, complemented
6#26011748C211yesmaximal, characteristic, complemented
7#26011742C211yesmaximal, characteristic, complemented
8#26011711C211yesmaximal, characteristic, complemented
9#26011709C211yesmaximal, characteristic, complemented
10#26011656C211yesmaximal, characteristic, complemented
11#26011640C211yesmaximal, characteristic, complemented
12#26011638C211yesmaximal, characteristic, complemented
13#26011576C211yesmaximal, characteristic, complemented
14#26011566C211yesmaximal, characteristic, complemented
15#448×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826C2211yescharacteristic
18C2×D4×D173C2211yescharacteristic
19#19149710—24no—
20#19149706—24no—
21C2×#14583816C2211yescharacteristic, complemented
22C2×#14583816C2211yescharacteristic, complemented
23C2×#14583815C2211yescharacteristic, complemented
24C2×#14583813C2211yescharacteristic
25C2×#14583813C2211yescharacteristic
26C2×D1384C2211yescharacteristic
27C2×#14583811C2211yescharacteristic
28C2×#14583810C2211yescharacteristic, complemented
29C2×#14583809C2211yescharacteristic
30#19149668—21no—
31#19149668C2211yescharacteristic, complemented
32#19149666C2211yescharacteristic
33#19149661—21no—
34#19149659C2211yescharacteristic
35#19149655—21no—
36#19149655C2211yescharacteristic
37#19149653—21no—
38#19149653C2211yescharacteristic, complemented
39#19149650—21no—
40#19149647—21no—
41Q8×Dic173C2211yescharacteristic, complemented
42#19149641C2211yescharacteristic
43C4×C22⋊D173—21no—
44C4×C22⋊D173C2211yescharacteristic
45C22⋊C4×D173—21no—
46#19149610C2211yescharacteristic, complemented
47#19149608C2211yescharacteristic, complemented
48#19149604C2211yescharacteristic, complemented
49#19149601C2211yescharacteristic
50#19149600C2211yescharacteristic
51#19149599C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic
53C8×Dic173C2211yescharacteristic, complemented
54C22×Q8×C173C2211yescharacteristic, complemented
55C2×SD16×C173C2211yescharacteristic
56C2×SD16×C173C2211yescharacteristic
57#121×C173C2211yescharacteristic
58#103×C173C2211yescharacteristic, complemented
59#103×C173C2211yescharacteristic, complemented
60C22⋊C8×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583827—22no—
63#14583827—41no—
64#14583826—22no—
65#14583826—41no—
66C2×Dic346C2311yescharacteristic
67C2×D692—21no—
68C2×D692C2311yescharacteristic
69C2×D692C2311yescharacteristic
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173D411yescharacteristic
74D4×D173—22no—
75D4×D173—41no—
76C2×C4×D173—21no—
77C2×C4×D173—21no—
78C2×C4×D173—21no—
79C2×C4×D173C2311yescharacteristic
80#14583816—22no—
81#14583816—22no—
82#14583815—41no—
83#14583813—22no—
84#14583813D414yescomplemented
85D1384—22no—
86#14583811—22no—
87#14583810—22no—
88#14583809—22no—
89C2×#13063436C2311yescharacteristic
90C2×#13063436C2311yescharacteristic
91C4⋊Dic173—21no—
92C4⋊Dic173—21no—
93C4⋊Dic173C2311yescharacteristic
94#14583805—21no—
95#14583805D411yescharacteristic
96C22⋊Dic173—21no—
97C22⋊Dic173—21no—
98C22⋊Dic173D411yescharacteristic
99#14583803—21no—
100#14583803—21no—
101#14583803—21no—
102#14583803—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803D411yescharacteristic
106C4×Dic173—21no—
107C4×Dic173C2311yescharacteristic
108C2×Q8×C173—21no—
109C2×Q8×C173—22no—
110C2×Q8×C173—41no—
111C2×Q8×C173C2311yescharacteristic
112C2×Q8×C173C2311yescharacteristic
113C2×D4×C173—21no—
114C2×D4×C173C2311yescharacteristic
115C22×C4×C173—21no—
116C22×C4×C173C2311yescharacteristic
117SD16×C173—22no—
118SD16×C173—41no—
119C2×C8×C173C2311yescharacteristic
120C2×C8×C173C2311yescharacteristic
121C4⋊C4×C173C2311yescharacteristic
122C22⋊C4×C173—21no—
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—42no—
126C22×D173—42no—
127C22×D173—81no—
128C22×D173C2×D411yescharacteristic
129Dic346—21no—
130Dic346—21no—
131D692—21no—
132D692—21no—
133D692—21no—
134D692—42no—
135D692C2×D412yes—
136C22⋊D173—41no—
137C22⋊D173—42no—
138C22⋊D173—42no—
139C22⋊D173—42no—
140C2×Dic173—21no—
141C2×Dic173—21no—
142C2×Dic173C2×D411yescharacteristic
143C2×Dic173C2×D411yescharacteristic
144C2×Dic173C2×D411yescharacteristic
145C4×D173—21no—
146C4×D173—21no—
147C4×D173—42no—
148C4×D173—42no—
149C4×D173—81no—
150#13063436—21no—
151#13063436C2×D412yes—
152C23×C173—21no—
153C23×C173C2×D411yescharacteristic
154Q8×C173—21no—
155Q8×C173—21no—
156Q8×C173—21no—
157Q8×C173—42no—
158Q8×C173C2×D412yes—
159D4×C173—21no—
160D4×C173—21no—
161D4×C173—81no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—42no—
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2411yescharacteristic
170C8×C173—21no—
171C8×C173—21no—
172C22×C173—21no—
173C22×C173—21no—
174C22×C173—21no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173C22×D411yescharacteristic
179C2×D173—21no—
180C2×D173—22no—
181C2×D173—21no—
182C2×D173—21no—
183C2×D173—41no—
184C2×D173—41no—
185C2×D173—41no—
186C2×D173—41no—
187C2×D173—41no—
188C2×D173—84no—
189C4×C173—21no—
190C4×C173—22no—
191C4×C173—41no—
192C4×C173—41no—
193C4×C173C22×D411yescharacteristic
194C4×C173H(5,𝔽2)11yescharacteristic
195Dic173—21no—
196Dic173—21no—
197Dic173—21no—
198Dic173—41no—
199Dic173—41no—
200C2×C173—41no—
201C2×C173—81no—
202C2×C173D4211yescharacteristic
203C2×C173#57511yescharacteristic
204C2×C173#57711yescharacteristic
205D173—41no—
206D173—42no—
207D173—81no—
208C173#327511yescharacteristic, complemented, Sylow
209#3275—1731nomaximal, Sylow
210C2×#137—1731no—
211C2×AGL(1,ℤ/8)—1731no—
212#548—1731no—
213#546—1731no—
214#495—1731no—
215#470—1731no—
216#469—1731no—
217#464—1731no—
218#463—1731no—
219#452—1731no—
220#450—1731no—
221#449—1731no—
222#448D17311yescharacteristic, complemented
223C4×SD16—1731no—
224#436—1731no—
225C2×#41—1731no—
226C2×#41—1731no—
227C22×Q8C2×D17311yescharacteristic, complemented
228C22×D4—1731no—
229#137—3464no—
230AGL(1,ℤ/8)—3464no—
231C2×Q16—1731no—
232C2×Q16—1731no—
233C2×SD16—1731no—
234C2×SD16—1731no—
235C2×SD16—1731no—
236C2×SD16C2×D17311yescharacteristic
237C2×SD16C2×D17311yescharacteristic
238C2×D8—1731no—
239C2×D8—1731no—
240C2×M16—1731no—
241C2×M16—1731no—
242#124—3461no—
243#124—1731no—
244#124—1731no—
245#123—3461no—
246C22⋊Q8—3461no—
247C22⋊Q8—1731no—
248C22⋊Q8—1731no—
249#121—3461no—
250#121—3461no—
251#121—1731no—
252#121C2×D17311yescharacteristic
253#120—3461no—
254C4×Q8—1731no—
255C4×D4—3461no—
256C4×D4—1731no—
257C4×D4—1731no—
258C2×C22⋊C4—3461no—
259#106—1731no—
260C4⋊C8—1731no—
261#103—1731no—
262#103C2×D17311yescharacteristic, complemented
263#103C2×D17311yescharacteristic, complemented
264#102—1731no—
265#102—1731no—
266#102—1731no—
267C22⋊C8—1731no—
268C22⋊C8C2×D17311yescharacteristic, complemented
269C4×C8—1731no—
270C24—3461no—
271#41—3462no—
272#41—3462no—
273#41—6921no—
274#41—6921no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—1731no—
279C2×Q8C22×D17311yescharacteristic
280C2×Q8C22×D17311yescharacteristic
281C2×D4—21no—
282C2×D4—3461no—
283C2×D4—3462no—
284C2×D4—3461no—
285C2×D4—3461no—
286C2×D4—3461no—
287C2×D4—1731no—
288C2×D4—1731no—
289C2×D4—1731no—
290C2×D4—6921no—
291C2×D4C22×D17311yescharacteristic
292C22×C4—21no—
293C22×C4—3461no—
294C22×C4—3461no—
295C22×C4—3461no—
296C22×C4—1731no—
297C22×C4C22×D17311yescharacteristic
298Q16—3462no—
299Q16—3462no—
300SD16—22no—
301SD16—41no—
302SD16—3462no—
303SD16—1734no—
304SD16—6921no—
305D8—3462no—
306D8—3462no—
307M16—3462no—
308M16—3462no—
309C2×C8—1731no—
310C2×C8—1731no—
311C2×C8C22×D17311yescharacteristic
312C2×C8C22×D17311yescharacteristic
313C4⋊C4—3461no—
314C4⋊C4—3461no—
315C4⋊C4—3461no—
316C4⋊C4—1731no—
317C4⋊C4—1731no—
318C4⋊C4C22×D17311yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—3461no—
321C22⋊C4—3461no—
322C22⋊C4—3461no—
323C22⋊C4—3461no—
324C22⋊C4—3461no—
325C22⋊C4—3461no—
326C22⋊C4—3461no—
327C22⋊C4—3461no—
328C22⋊C4—1731no—
329C22⋊C4—1731no—
330C42—3461no—
331C42—1731no—
332C23—21no—
333C23—3461no—
334C23—3461no—
335C23—13841no—
336C23—1731no—
337C23—6922no—
338C23—6922no—
339C23D4×D17311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—42no—
344Q8—3461no—
345Q8—3461no—
346Q8D4×D17312yes—
347D4—21no—
348D4—21no—
349D4—81no—
350D4—3461no—
351D4—3461no—
352D4—3461no—
353D4—1732no—
354D4—6922no—
355D4—6921no—
356D4—6922no—
357D4—6922no—
358D4—6922no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—3461no—
368C2×C4—3461no—
369C2×C4—13841no—
370C2×C4—1731no—
371C2×C4—1731no—
372C2×C4—1731no—
373C2×C4—6922no—
374C2×C4—6922no—
375C2×C4D4×D17311yescharacteristic
376C2×C4C23×D17311yescharacteristic
377C8—21no—
378C8—21no—
379C8—3461no—
380C8—1732no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—3461no—
388C22—3461no—
389C22—3461no—
390C22—3462no—
391C22—13844no—
392C22—6921no—
393C22—6921no—
394C22—6921no—
395C22—6921no—
396C22—6921no—
397C22C2×D4×D17311yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—3461no—
403C4—3461no—
404C4—3461no—
405C4—6921no—
406C4—6921no—
407C4C2×D4×D17311yescharacteristic
408C4#1914972411yescharacteristic
409C2—41no—
410C2—81no—
411C2—13841no—
412C2—6921no—
413C2—6922no—
414C2#2601212311yescharacteristic
415C2#2601229011yescharacteristic
416C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#448×C173C2
socleC22×C173C22×D4
hypercenter#448D173
hypocenterC173#3275
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×#19149710C2semidirect
2. C2×#19149706C2semidirect
3. #26012152C2semidirect
4. #26012138C2semidirect
5. #26011872C2semidirect
6. #26011748C2semidirect
7. #26011742C2semidirect
8. #26011711C2semidirect
9. #26011709C2semidirect
10. #26011656C2semidirect
11. #26011640C2semidirect
12. #26011638C2semidirect
13. #26011576C2semidirect
14. #26011566C2semidirect
15. #448×C173C2semidirect
21. C2×#14583816C22semidirect
22. C2×#14583816C22semidirect
23. C2×#14583815C22semidirect
28. C2×#14583810C22semidirect
31. #19149668C22semidirect
38. #19149653C22semidirect
41. Q8×Dic173C22semidirect
46. #19149610C22semidirect
47. #19149608C22semidirect
48. #19149604C22semidirect
51. #19149599C22semidirect
53. C8×Dic173C22semidirect
54. C22×Q8×C173C22semidirect
58. #103×C173C22semidirect
59. #103×C173C22semidirect
60. C22⋊C8×C173C22semidirect
84. #14583813D4semidirect
208. C173#3275semidirect
222. #448D173semidirect
227. C22×Q8C2×D173semidirect
262. #103C2×D173semidirect
263. #103C2×D173semidirect
268. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497102
C2×#191497062
#260121522
#260121382
#260118722
#260117482
#260117422
#260117112
#260117092
#260116562
#260116402
#260116382
#260115762
#260115662
#448×C1732
#3275173