Finite Group Databasebeta

← #36365073

Subgroups — #36365073

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×#19149708C211yesmaximal, characteristic, complemented
2Q8×C22⋊D173C211yesmaximal, characteristic, complemented
3#26012135C211yesmaximal, characteristic
4#26011882C211yesmaximal, characteristic, complemented
5#26011734C211yesmaximal, characteristic, complemented
6#26011732C211yesmaximal, characteristic, complemented
7#26011711C211yesmaximal, characteristic, complemented
8#26011694C211yesmaximal, characteristic, complemented
9#26011650C211yesmaximal, characteristic, complemented
10#26011647C211yesmaximal, characteristic, complemented
11#26011642C211yesmaximal, characteristic, complemented
12#26011628C211yesmaximal, characteristic, complemented
13#26011592C211yesmaximal, characteristic, complemented
14#26011562C211yesmaximal, characteristic, complemented
15C22×SD16×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic, complemented
18#19149708—24no—
19C2×Dic692C2211yescharacteristic, complemented
20C2×#14583816C2211yescharacteristic, complemented
21C2×#14583816C2211yescharacteristic, complemented
22C2×#14583814C2211yescharacteristic
23C2×#14583814C2211yescharacteristic
24C2×#14583809C2211yescharacteristic, complemented
25#19149677—21no—
26#19149677C2211yescharacteristic, complemented
27#19149666C2211yescharacteristic
28#19149660—21no—
29#19149658—21no—
30#19149658C2211yescharacteristic
31#19149658C2211yescharacteristic
32#19149655—21no—
33#19149655C2211yescharacteristic, complemented
34#19149651—21no—
35#19149650C2211yescharacteristic
36#19149648—21no—
37Q8×Dic173C2211yescharacteristic, complemented
38D4×Dic173C2211yescharacteristic
39C4×C22⋊D173—21no—
40C4×C22⋊D173C2211yescharacteristic
41C2×C22⋊Dic173—21no—
42#19149613C2211yescharacteristic, complemented
43#19149608C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149604C2211yescharacteristic, complemented
46#19149603C2211yescharacteristic, complemented
47#19149603C2211yescharacteristic, complemented
48#19149600C2211yescharacteristic
49#19149600C2211yescharacteristic
50C22⋊#13063436C2211yescharacteristic
51#19149591C2211yescharacteristic, complemented
52#19149589C2211yescharacteristic, complemented
53C22×Q8×C173C2211yescharacteristic
54C22×D4×C173C2211yescharacteristic
55C2×SD16×C173—24no—
56C2×SD16×C173C2211yescharacteristic
57C2×SD16×C173C2211yescharacteristic
58C2×SD16×C173C2211yescharacteristic
59C2×SD16×C173C2211yescharacteristic
60C22×C8×C173C2211yescharacteristic
61#14583825—22no—
62#14583825—41no—
63C2×Dic346—21no—
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66Q8×D173—22no—
67Q8×D173—41no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173D411yescharacteristic
71C22×Dic173—21no—
72C22×Dic173—21no—
73C2×C4×D173—21no—
74C2×C4×D173C2311yescharacteristic
75Dic692—22no—
76#14583816—22no—
77#14583816—41no—
78#14583814—22no—
79#14583814—41no—
80#14583809—22no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
83C4⋊Dic173—21no—
84C4⋊Dic173C2311yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805—21no—
90#14583805—21no—
91#14583805D411yescharacteristic
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95C22⋊Dic173—21no—
96C22⋊Dic173—21no—
97C22⋊Dic173—21no—
98C22⋊Dic173D411yescharacteristic
99#14583803—21no—
100#14583803D411yescharacteristic
101C4×Dic173—21no—
102C4×Dic173C2311yescharacteristic
103C24×C173—21no—
104C2×Q8×C173—21no—
105C2×Q8×C173—22no—
106C2×Q8×C173—41no—
107C2×Q8×C173C2311yescharacteristic
108C2×Q8×C173C2311yescharacteristic
109C2×D4×C173—22no—
110C2×D4×C173—21no—
111C2×D4×C173—41no—
112C2×D4×C173C2311yescharacteristic
113C2×D4×C173C2311yescharacteristic
114C22×C4×C173—21no—
115C22×C4×C173C2311yescharacteristic
116SD16×C173—22no—
117SD16×C173—22no—
118SD16×C173—22no—
119SD16×C173D414yescomplemented
120C2×C8×C173—22no—
121C2×C8×C173C2311yescharacteristic
122C2×C8×C173C2311yescharacteristic
123C22×D173C2×D411yescharacteristic
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—42no—
129C22⋊D173—22no—
130C22⋊D173—42no—
131C22⋊D173—81no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—42no—
137C2×Dic173—81no—
138C2×Dic173C2×D411yescharacteristic
139C2×Dic173C2×D411yescharacteristic
140C2×Dic173C2×D411yescharacteristic
141C4×D173—21no—
142C4×D173—21no—
143C4×D173—42no—
144#13063436—21no—
145#13063436—21no—
146C23×C173—21no—
147C23×C173—21no—
148C23×C173—42no—
149C23×C173—42no—
150C23×C173—81no—
151C23×C173C2×D411yescharacteristic
152Q8×C173—21no—
153Q8×C173—21no—
154Q8×C173—21no—
155Q8×C173—42no—
156Q8×C173C2×D412yes—
157D4×C173—21no—
158D4×C173—21no—
159D4×C173—21no—
160D4×C173—42no—
161D4×C173C2×D412yes—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—42no—
167C2×C4×C173C2×D411yescharacteristic
168C2×C4×C173C2411yescharacteristic
169C8×C173—21no—
170C8×C173C2×D412yes—
171C22×C173—21no—
172C22×C173—21no—
173C22×C173—22no—
174C22×C173—41no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173—41no—
179C22×C173—84no—
180C22×C173#13712yescomplemented
181C22×C173C22×D411yescharacteristic
182C2×D173—21no—
183C2×D173—21no—
184C2×D173—21no—
185C4×C173—21no—
186C4×C173—22no—
187C4×C173—41no—
188C4×C173C22×D411yescharacteristic
189C4×C173H(5,𝔽2)11yescharacteristic
190Dic173—21no—
191Dic173—21no—
192Dic173—21no—
193Dic173—41no—
194Dic173—41no—
195Dic173—41no—
196Dic173—41no—
197C2×C173—22no—
198C2×C173—42no—
199C2×C173—81no—
200C2×C173D4211yescharacteristic
201C2×C173C2×#13711yescharacteristic
202C2×C173#57711yescharacteristic
203D173—41no—
204C173#326611yescharacteristic, complemented, Sylow
205#3266—1731nomaximal, Sylow
206C2×#137—1731no—
207C22×SD16D17311yescharacteristic, complemented
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#435—1731no—
221C2×#41—1731no—
222C22×Q8—1731no—
223C22×Q8C2×D17311yescharacteristic
224C22×D4C2×D17311yescharacteristic
225#137—3464no—
226C2×Q16—1731no—
227C2×Q16—1731no—
228C2×Q16—1731no—
229C2×SD16—24no—
230C2×SD16—1731no—
231C2×SD16—1731no—
232C2×SD16C2×D17311yescharacteristic
233C2×SD16C2×D17311yescharacteristic
234C2×SD16C2×D17311yescharacteristic
235C2×SD16C2×D17311yescharacteristic
236C2×M16—1731no—
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×D4—1731no—
254C2×C22⋊C4—3461no—
255#107—1731no—
256C4⋊C8—1731no—
257#103—1731no—
258#103—1731no—
259#103—1731no—
260#103—1731no—
261#102—1731no—
262#102—1731no—
263C22⋊C8—1731no—
264C22⋊C8—1731no—
265#97—1731no—
266C24—21no—
267#41—3462no—
268#41—6921no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—3461no—
273C2×Q8—3462no—
274C2×Q8—1731no—
275C2×Q8—1731no—
276C2×Q8—6921no—
277C2×Q8C22×D17311yescharacteristic
278C2×Q8C22×D17311yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—3461no—
283C2×D4—3461no—
284C2×D4—1731no—
285C2×D4C22×D17311yescharacteristic
286C2×D4C22×D17311yescharacteristic
287C22×C4—21no—
288C22×C4—3461no—
289C22×C4—3461no—
290C22×C4—3461no—
291C22×C4—1731no—
292C22×C4C22×D17311yescharacteristic
293Q16—3462no—
294Q16—3462no—
295Q16—6921no—
296SD16—22no—
297SD16—22no—
298SD16—22no—
299SD16—3462no—
300SD16—6921no—
301SD16C22⋊D17314yescomplemented
302M16—3462no—
303C2×C8—22no—
304C2×C8—1731no—
305C2×C8—1731no—
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⋊C4—1731no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—3461no—
320C22⋊C4—3461no—
321C22⋊C4—3461no—
322C22⋊C4—3461no—
323C22⋊C4—3461no—
324C22⋊C4—1731no—
325C22⋊C4—1731no—
326C42—3461no—
327C42—1731no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1731no—
334C23D4×D17311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—3461no—
340Q8—3461no—
341Q8—3461no—
342Q8—3461no—
343Q8—6922no—
344Q8C2×C22⋊D17312yes—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—3462no—
350D4—13841no—
351D4—6922no—
352D4C2×C22⋊D17312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—3461no—
359C2×C4—3461no—
360C2×C4—3461no—
361C2×C4—3461no—
362C2×C4—3461no—
363C2×C4—3461no—
364C2×C4—13841no—
365C2×C4—1731no—
366C2×C4—1731no—
367C2×C4—1731no—
368C2×C4—6922no—
369C2×C4—6922no—
370C2×C4D4×D17311yescharacteristic
371C2×C4C23×D17311yescharacteristic
372C8—21no—
373C8—3461no—
374C8—3461no—
375C8C2×C22⋊D17312yes—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—3461no—
386C22—3461no—
387C22—3461no—
388C22#1914970812yescomplemented
389C22C2×D4×D17311yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—3461no—
394C4—3461no—
395C4—3461no—
396C4—6921no—
397C4—6921no—
398C4—6921no—
399C4—6921no—
400C4C22×C22⋊D17311yescharacteristic
401C4#1914972411yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—6921no—
406C2D4×C22⋊D17311yescharacteristic
407C2C2×#1914970811yescharacteristic
408C2#2601229711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC22×SD16×C173C2
socleC22×C173C22×D4
prime radical#26012135C2
hypercenterC22×SD16D173
hypocenterC173#3266
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#19149708C2semidirect
2. Q8×C22⋊D173C2semidirect
4. #26011882C2semidirect
5. #26011734C2semidirect
6. #26011732C2semidirect
7. #26011711C2semidirect
8. #26011694C2semidirect
9. #26011650C2semidirect
10. #26011647C2semidirect
11. #26011642C2semidirect
12. #26011628C2semidirect
13. #26011592C2semidirect
14. #26011562C2semidirect
15. C22×SD16×C173C2semidirect
17. C2×Q8×D173C22semidirect
19. C2×Dic692C22semidirect
20. C2×#14583816C22semidirect
21. C2×#14583816C22semidirect
24. C2×#14583809C22semidirect
26. #19149677C22semidirect
33. #19149655C22semidirect
37. Q8×Dic173C22semidirect
42. #19149613C22semidirect
43. #19149608C22semidirect
44. #19149604C22semidirect
45. #19149604C22semidirect
46. #19149603C22semidirect
47. #19149603C22semidirect
51. #19149591C22semidirect
52. #19149589C22semidirect
119. SD16×C173D4semidirect
180. C22×C173#137semidirect
204. C173#3266semidirect
207. C22×SD16D173semidirect
301. SD16C22⋊D173semidirect
388. C22#19149708semidirect

Maximal subgroups

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

subgroupindex
C2×#191497082
Q8×C22⋊D1732
#260121352
#260118822
#260117342
#260117322
#260117112
#260116942
#260116502
#260116472
#260116422
#260116282
#260115922
#260115622
C22×SD16×C1732
#3266173