Finite Group Databasebeta

← #36365188

Subgroups — #36365188

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×#19149706C211yesmaximal, characteristic, complemented
2#26012152C211yesmaximal, characteristic, complemented
3#26012129C211yesmaximal, characteristic, complemented
4#26011876C211yesmaximal, characteristic, complemented
5#26011746C211yesmaximal, characteristic, complemented
6#26011744C211yesmaximal, characteristic, complemented
7#26011715C211yesmaximal, characteristic, complemented
8#26011708C211yesmaximal, characteristic, complemented
9#26011652C211yesmaximal, characteristic, complemented
10#26011640C211yesmaximal, characteristic, complemented
11#26011636C211yesmaximal, characteristic, complemented
12#26011630C211yesmaximal, characteristic, complemented
13SD16×Dic173C211yesmaximal, characteristic, complemented
14#26011567C211yesmaximal, characteristic, complemented
15C2×#137×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18#19149706—24no—
19C2×#14583813C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21C2×#14583813C2211yescharacteristic, complemented
22C2×D1384C2211yescharacteristic
23C2×#14583811C2211yescharacteristic
24C2×#14583809C2211yescharacteristic
25#19149668—21no—
26#19149668C2211yescharacteristic
27#19149668C2211yescharacteristic, complemented
28#19149662—21no—
29#19149655—21no—
30#19149655C2211yescharacteristic
31#19149655C2211yescharacteristic
32#19149653—21no—
33#19149653C2211yescharacteristic
34#19149653C2211yescharacteristic, complemented
35#19149651—21no—
36#19149647—21no—
37Q8×Dic173C2211yescharacteristic, complemented
38D4×Dic173C2211yescharacteristic
39C4×C22⋊D173—21no—
40C4×C22⋊D173C2211yescharacteristic
41C2×#14583803—21no—
42#19149611C2211yescharacteristic, complemented
43#19149608C2211yescharacteristic, complemented
44#19149603C2211yescharacteristic, complemented
45#19149603C2211yescharacteristic, complemented
46#19149603C2211yescharacteristic, complemented
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic, complemented
49#19149600C2211yescharacteristic
50C22⋊#13063436C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic
52C8×Dic173C2211yescharacteristic, complemented
53C2×#41×C173C2211yescharacteristic
54C22×Q8×C173C2211yescharacteristic, complemented
55#137×C173—24no—
56C2×Q16×C173C2211yescharacteristic, complemented
57C2×Q16×C173C2211yescharacteristic, complemented
58C2×SD16×C173C2211yescharacteristic
59C2×SD16×C173C2211yescharacteristic
60C2×M16×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583827—22no—
63#14583827—41no—
64C2×D692—21no—
65C2×D692C2311yescharacteristic
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173D411yescharacteristic
71D4×D173—22no—
72D4×D173—41no—
73C22×Dic173—21no—
74C2×C4×D173—21no—
75C2×C4×D173C2311yescharacteristic
76#14583813—22no—
77#14583813—41no—
78#14583813—41no—
79D1384—22no—
80#14583811—22no—
81#14583809—22no—
82C2×#13063436C2311yescharacteristic
83C2×#13063436C2311yescharacteristic
84C4⋊Dic173—21no—
85C4⋊Dic173C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87#14583805—21no—
88#14583805—21no—
89#14583805D411yescharacteristic
90C22⋊Dic173—21no—
91C22⋊Dic173D411yescharacteristic
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803D411yescharacteristic
101C4×Dic173—21no—
102C4×Dic173C2311yescharacteristic
103#41×C173—22no—
104#41×C173—41no—
105C2×Q8×C173—21no—
106C2×Q8×C173—22no—
107C2×Q8×C173—41no—
108C2×Q8×C173C2311yescharacteristic
109C2×Q8×C173C2311yescharacteristic
110C2×Q8×C173C2311yescharacteristic
111C2×D4×C173—21no—
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173—21no—
114C22×C4×C173—21no—
115C22×C4×C173C2311yescharacteristic
116Q16×C173—22no—
117Q16×C173—22no—
118SD16×C173—22no—
119SD16×C173D414yescomplemented
120M16×C173—22no—
121C2×C8×C173C2311yescharacteristic
122C2×C8×C173C2311yescharacteristic
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—42no—
126C22×D173—42no—
127C22×D173—81no—
128C22×D173C2×D411yescharacteristic
129D692—21no—
130D692—21no—
131D692—21no—
132D692—21no—
133D692—42no—
134C22⋊D173—41no—
135C22⋊D173—42no—
136C22⋊D173—42no—
137C22⋊D173—81no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C2×Dic173—81no—
141C2×Dic173C2×D411yescharacteristic
142C2×Dic173C2×D411yescharacteristic
143C2×Dic173C2×D411yescharacteristic
144C4×D173—21no—
145C4×D173—21no—
146C4×D173—42no—
147#13063436—21no—
148#13063436—21no—
149C23×C173—21no—
150C23×C173C2×D411yescharacteristic
151Q8×C173—21no—
152Q8×C173—21no—
153Q8×C173—21no—
154Q8×C173—21no—
155Q8×C173—21no—
156Q8×C173—42no—
157Q8×C173C2×D412yes—
158D4×C173—21no—
159D4×C173—42no—
160D4×C173C2×D412yes—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—42no—
167C2×C4×C173—42no—
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2411yescharacteristic
170C8×C173—21no—
171C8×C173C2×D412yes—
172C22×C173—21no—
173C22×C173—22no—
174C22×C173—21no—
175C22×C173—21no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173C22×D411yescharacteristic
179C2×D173—21no—
180C2×D173—21no—
181C2×D173—21no—
182C2×D173—41no—
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—42no—
201C2×C173—41no—
202C2×C173D4211yescharacteristic
203C2×C173#57511yescharacteristic
204C2×C173#57711yescharacteristic
205D173—41no—
206D173—81no—
207D173—81no—
208C173#327511yescharacteristic, complemented, Sylow
209#3275—1731nomaximal, Sylow
210C2×#137D17311yescharacteristic, complemented
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#448—1731no—
223C4×SD16—1731no—
224#436—1731no—
225C2×#41—1731no—
226C2×#41C2×D17311yescharacteristic
227C22×Q8C2×D17311yescharacteristic, complemented
228C22×D4—1731no—
229#137—24no—
230AGL(1,ℤ/8)—3464no—
231C2×Q16C2×D17311yescharacteristic, complemented
232C2×Q16C2×D17311yescharacteristic, complemented
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×M16C2×D17311yescharacteristic, complemented
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#121—1731no—
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#103—1731no—
263#103—1731no—
264#102—1731no—
265#102—1731no—
266#102—1731no—
267C22⋊C8—1731no—
268C22⋊C8—1731no—
269C4×C8—1731no—
270C24—3461no—
271#41—22no—
272#41—41no—
273#41—3462no—
274#41—6921no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8C22×D17311yescharacteristic
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—21no—
294C22×C4—3461no—
295C22×C4—3461no—
296C22×C4—1731no—
297C22×C4C22×D17311yescharacteristic
298Q16—22no—
299Q16—22no—
300SD16—22no—
301SD16—3462no—
302SD16—6921no—
303SD16—6921no—
304SD16C22⋊D17314yescomplemented
305D8—3462no—
306D8—3462no—
307M16—22no—
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⋊C4—1731no—
319C22⋊C4—3461no—
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—21no—
344Q8—21no—
345Q8—42no—
346Q8C2×C22⋊D17312yes—
347D4—21no—
348D4—42no—
349D4—3461no—
350D4—3461no—
351D4—3461no—
352D4—3461no—
353D4—13841no—
354D4—6922no—
355D4—6921no—
356D4—6922no—
357D4—6922no—
358D4C2×C22⋊D17312yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—3461no—
367C2×C4—3461no—
368C2×C4—3461no—
369C2×C4—3461no—
370C2×C4—13841no—
371C2×C4—1731no—
372C2×C4—1731no—
373C2×C4—1731no—
374C2×C4—6922no—
375C2×C4D4×D17311yescharacteristic
376C2×C4C23×D17311yescharacteristic
377C8—21no—
378C8—3461no—
379C8—3461no—
380C8C2×C22⋊D17312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—3461no—
388C22—3461no—
389C22—3461no—
390C22—13844no—
391C22—6921no—
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—
407C4C22×C22⋊D17311yescharacteristic
408C4#1914972411yescharacteristic
409C2—41no—
410C2—42no—
411C2—13841no—
412C2—13841no—
413C2—6921no—
414C2D4×C22⋊D17311yescharacteristic
415C2#2601229011yescharacteristic
416C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#137×C173C2
socleC22×C173C22×D4
hypercenterC2×#137D173
hypocenterC173#3275
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. #26012152C2semidirect
3. #26012129C2semidirect
4. #26011876C2semidirect
5. #26011746C2semidirect
6. #26011744C2semidirect
7. #26011715C2semidirect
8. #26011708C2semidirect
9. #26011652C2semidirect
10. #26011640C2semidirect
11. #26011636C2semidirect
12. #26011630C2semidirect
13. SD16×Dic173C2semidirect
14. #26011567C2semidirect
15. C2×#137×C173C2semidirect
21. C2×#14583813C22semidirect
27. #19149668C22semidirect
34. #19149653C22semidirect
37. Q8×Dic173C22semidirect
42. #19149611C22semidirect
43. #19149608C22semidirect
44. #19149603C22semidirect
45. #19149603C22semidirect
46. #19149603C22semidirect
48. #19149601C22semidirect
50. C22⋊#13063436C22semidirect
52. C8×Dic173C22semidirect
54. C22×Q8×C173C22semidirect
56. C2×Q16×C173C22semidirect
57. C2×Q16×C173C22semidirect
60. C2×M16×C173C22semidirect
119. SD16×C173D4semidirect
208. C173#3275semidirect
210. C2×#137D173semidirect
227. C22×Q8C2×D173semidirect
231. C2×Q16C2×D173semidirect
232. C2×Q16C2×D173semidirect
241. C2×M16C2×D173semidirect
304. SD16C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
#260121522
#260121292
#260118762
#260117462
#260117442
#260117152
#260117082
#260116522
#260116402
#260116362
#260116302
SD16×Dic1732
#260115672
C2×#137×C1732
#3275173