Finite Group Databasebeta

← #34056351

Subgroups — #34056351

Order 19072.

482 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
1C2×#18011943C211yesmaximal, characteristic, complemented
2#23750294C211yesmaximal, characteristic
3Q8×C22⋊D149C211yesmaximal, characteristic, complemented
4#23749996C211yesmaximal, characteristic, complemented
5#23749871C211yesmaximal, characteristic, complemented
6#23749868C211yesmaximal, characteristic, complemented
7#23749833C211yesmaximal, characteristic, complemented
8#23749823C211yesmaximal, characteristic, complemented
9#23749787C211yesmaximal, characteristic, complemented
10#23749786C211yesmaximal, characteristic, complemented
11#23749764C211yesmaximal, characteristic, complemented
12#23749762C211yesmaximal, characteristic, complemented
13Q16×Dic149C211yesmaximal, characteristic, complemented
14#23749701C211yesmaximal, characteristic, complemented
15C2×#137×C149C211yesmaximal, characteristic, complemented
16C2×#14558737C2211yescharacteristic
17C2×Q8×D149C2211yescharacteristic
18#18011943—24no—
19C2×#14558726C2211yescharacteristic
20C2×#14558725C2211yescharacteristic
21C2×#14558724C2211yescharacteristic
22C2×#14558723C2211yescharacteristic
23C2×#14558723C2211yescharacteristic
24C2×#14558719C2211yescharacteristic
25#18011909—21no—
26#18011909C2211yescharacteristic
27#18011907—21no—
28#18011907C2211yescharacteristic
29#18011890—21no—
30#18011890C2211yescharacteristic
31#18011887—21no—
32#18011887C2211yescharacteristic
33#18011885—21no—
34#18011885C2211yescharacteristic
35#18011882—21no—
36#18011882C2211yescharacteristic
37Q8×Dic149C2211yescharacteristic
38Q8×Dic149C2211yescharacteristic
39C4×C22⋊D149—21no—
40C4×C22⋊D149—21no—
41C4×C22⋊D149C2211yescharacteristic
42#18011845C2211yescharacteristic
43#18011840C2211yescharacteristic
44#18011836C2211yescharacteristic
45#18011835C2211yescharacteristic
46#18011835C2211yescharacteristic
47#18011835C2211yescharacteristic
48#18011833C2211yescharacteristic
49#18011832C2211yescharacteristic
50C22⋊#11924746C2211yescharacteristic
51#18011823C2211yescharacteristic
52C8×Dic149C2211yescharacteristic
53C2×#41×C149C2211yescharacteristic
54C22×Q8×C149C2211yescharacteristic
55#137×C149—24no—
56C2×Q16×C149C2211yescharacteristic
57C2×Q16×C149C2211yescharacteristic
58C2×SD16×C149C2211yescharacteristic
59C2×SD16×C149C2211yescharacteristic
60C2×M16×C149C2211yescharacteristic
61#14558737—22no—
62#14558737—41no—
63C2×Dic298—21no—
64C2×Dic298C2311yescharacteristic
65Q8×D149—22no—
66Q8×D149—41no—
67C2×D596—21no—
68C2×D596C2311yescharacteristic
69C2×C22⋊D149—21no—
70C2×C22⋊D149—21no—
71C2×C22⋊D149—21no—
72C2×C22⋊D149—21no—
73C2×C22⋊D149D411yescharacteristic
74C2×C4×D149—21no—
75C2×C4×D149—21no—
76C2×C4×D149C2311yescharacteristic
77#14558726—22no—
78#14558725—22no—
79#14558724—41no—
80#14558723—22no—
81#14558723—41no—
82#14558719—22no—
83C2×#11924746C2311yescharacteristic
84C2×#11924746C2311yescharacteristic
85C4⋊Dic149—21no—
86C4⋊Dic149—21no—
87C4⋊Dic149C2311yescharacteristic
88C4⋊Dic149C2311yescharacteristic
89#14558715—21no—
90#14558715—21no—
91#14558715—21no—
92#14558715—21no—
93#14558715D411yescharacteristic
94C22⋊Dic149—21no—
95C22⋊Dic149—21no—
96C22⋊Dic149D411yescharacteristic
97#14558713—21no—
98#14558713—21no—
99#14558713D411yescharacteristic
100C4×Dic149—21no—
101C4×Dic149—21no—
102C4×Dic149C2311yescharacteristic
103#41×C149—22no—
104#41×C149—41no—
105C2×Q8×C149—21no—
106C2×Q8×C149—22no—
107C2×Q8×C149—41no—
108C2×Q8×C149C2311yescharacteristic
109C2×Q8×C149C2311yescharacteristic
110C2×Q8×C149C2311yescharacteristic
111C2×D4×C149—21no—
112C2×D4×C149C2311yescharacteristic
113C22×C4×C149—21no—
114C22×C4×C149—21no—
115C22×C4×C149C2311yescharacteristic
116Q16×C149—22no—
117Q16×C149D414yescomplemented
118SD16×C149—22no—
119SD16×C149—22no—
120M16×C149—22no—
121C2×C8×C149C2311yescharacteristic
122C2×C8×C149C2311yescharacteristic
123C22×D149—21no—
124C22×D149C2×D411yescharacteristic
125Dic298—21no—
126Dic298—21no—
127Dic298—42no—
128D596—21no—
129D596—21no—
130D596—42no—
131C22⋊D149—22no—
132C22⋊D149—42no—
133C22⋊D149—81no—
134C22⋊D149—81no—
135C22⋊D149—81no—
136C2×Dic149—21no—
137C2×Dic149—21no—
138C2×Dic149—21no—
139C2×Dic149C2×D411yescharacteristic
140C2×Dic149C2×D411yescharacteristic
141C2×Dic149C2×D411yescharacteristic
142C4×D149—21no—
143C4×D149—21no—
144C4×D149—42no—
145C4×D149—42no—
146#11924746—21no—
147#11924746—21no—
148C23×C149—21no—
149C23×C149C2×D411yescharacteristic
150Q8×C149—21no—
151Q8×C149—21no—
152Q8×C149—21no—
153Q8×C149—21no—
154Q8×C149—42no—
155Q8×C149C2×D412yes—
156Q8×C149C2×D412yes—
157D4×C149—21no—
158D4×C149—21no—
159D4×C149—42no—
160C2×C4×C149—21no—
161C2×C4×C149—21no—
162C2×C4×C149—21no—
163C2×C4×C149—21no—
164C2×C4×C149—21no—
165C2×C4×C149—42no—
166C2×C4×C149—42no—
167C2×C4×C149C2×D411yescharacteristic
168C2×C4×C149C2411yescharacteristic
169C8×C149—21no—
170C8×C149C2×D412yes—
171C22×C149—21no—
172C22×C149—21no—
173C22×C149—21no—
174C22×C149—41no—
175C22×C149—41no—
176C22×C149—41no—
177C22×C149C22×D411yescharacteristic
178C2×D149—21no—
179C2×D149—21no—
180C2×D149—21no—
181C2×D149—41no—
182C2×D149—41no—
183C2×D149—41no—
184C4×C149—21no—
185C4×C149—22no—
186C4×C149—22no—
187C4×C149—41no—
188C4×C149C22×D411yescharacteristic
189C4×C149H(5,𝔽2)11yescharacteristic
190Dic149—21no—
191Dic149—21no—
192Dic149—41no—
193Dic149—41no—
194Dic149—41no—
195Dic149#13712yescomplemented
196C2×C149—41no—
197C2×C149—81no—
198C2×C149D4211yescharacteristic
199C2×C149C2×#13711yescharacteristic
200C2×C149#57711yescharacteristic
201D149—41no—
202D149—81no—
203C149#327811yescharacteristic, complemented, Sylow
204#3278—1491nomaximal, Sylow
205C2×#137—1491no—
206C2×#137D14911yescharacteristic, complemented
207#550—1491no—
208D4×Q8—1491no—
209#492—1491no—
210#468—1491no—
211#468—1491no—
212#462—1491no—
213#460—1491no—
214#452—1491no—
215#452—1491no—
216#448—1491no—
217#448—1491no—
218C4×Q16—1491no—
219#436—1491no—
220C2×#41—1491no—
221C2×#41C2×D14911yescharacteristic
222C22×Q8—1491no—
223C22×Q8C2×D14911yescharacteristic
224#137—24no—
225#137—2984no—
226C2×Q16—1491no—
227C2×Q16C2×D14911yescharacteristic
228C2×Q16C2×D14911yescharacteristic
229C2×SD16—1491no—
230C2×SD16—1491no—
231C2×SD16—1491no—
232C2×SD16—1491no—
233C2×SD16C2×D14911yescharacteristic
234C2×SD16C2×D14911yescharacteristic
235C2×M16—1491no—
236C2×M16C2×D14911yescharacteristic
237C4⋊Q8—2981no—
238C4⋊Q8—1491no—
239#127—2981no—
240#127—1491no—
241C22⋊Q8—2981no—
242C22⋊Q8—2981no—
243C22⋊Q8—1491no—
244C22⋊Q8—1491no—
245#121—2981no—
246#121—2981no—
247#121—1491no—
248#121—1491no—
249C4×Q8—1491no—
250C4×Q8—1491no—
251C4×D4—2981no—
252C4×D4—2981no—
253C4×D4—1491no—
254#107—1491no—
255C4⋊C8—1491no—
256#103—1491no—
257#103—1491no—
258#103—1491no—
259#103—1491no—
260#102—1491no—
261#102—1491no—
262C22⋊C8—1491no—
263C22⋊C8—1491no—
264C4×C8—1491no—
265#41—22no—
266#41—41no—
267#41—2982no—
268#41—5961no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—2981no—
273C2×Q8—2982no—
274C2×Q8—5961no—
275C2×Q8—1491no—
276C2×Q8C22×D14911yescharacteristic
277C2×Q8C22×D14911yescharacteristic
278C2×Q8C22×D14911yescharacteristic
279C2×D4—21no—
280C2×D4—2981no—
281C2×D4—2981no—
282C2×D4—2981no—
283C2×D4—2981no—
284C2×D4—2981no—
285C2×D4—1491no—
286C2×D4—1491no—
287C2×D4C22×D14911yescharacteristic
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—2981no—
291C22×C4—2981no—
292C22×C4—1491no—
293C22×C4C22×D14911yescharacteristic
294Q16—22no—
295Q16—2982no—
296Q16C22⋊D14914yescomplemented
297SD16—22no—
298SD16—22no—
299SD16—2982no—
300SD16—2982no—
301SD16—5961no—
302SD16—5961no—
303M16—22no—
304M16—2982no—
305C2×C8—1491no—
306C2×C8—1491no—
307C2×C8C22×D14911yescharacteristic
308C2×C8C22×D14911yescharacteristic
309C4⋊C4—2981no—
310C4⋊C4—2981no—
311C4⋊C4—2981no—
312C4⋊C4—2981no—
313C4⋊C4—2981no—
314C4⋊C4—2981no—
315C4⋊C4—1491no—
316C4⋊C4—1491no—
317C4⋊C4—1491no—
318C22⋊C4—2981no—
319C22⋊C4—2981no—
320C22⋊C4—2981no—
321C22⋊C4—2981no—
322C22⋊C4—1491no—
323C22⋊C4—1491no—
324C42—2981no—
325C42—2981no—
326C42—1491no—
327C23—21no—
328C23—2981no—
329C23—1491no—
330C23D4×D14911yescharacteristic
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—21no—
335Q8—42no—
336Q8—2981no—
337Q8—2981no—
338Q8—5962no—
339Q8C2×C22⋊D14912yes—
340Q8C2×C22⋊D14912yes—
341D4—21no—
342D4—21no—
343D4—42no—
344D4—2982no—
345D4—2981no—
346D4—2981no—
347D4—5962no—
348D4—5962no—
349D4—11921no—
350D4—11921no—
351D4—11921no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—2981no—
360C2×C4—2981no—
361C2×C4—2981no—
362C2×C4—2981no—
363C2×C4—2981no—
364C2×C4—5962no—
365C2×C4—5962no—
366C2×C4—1491no—
367C2×C4—1491no—
368C2×C4—1491no—
369C2×C4D4×D14911yescharacteristic
370C2×C4C23×D14911yescharacteristic
371C8—21no—
372C8—2981no—
373C8—2981no—
374C8C2×C22⋊D14912yes—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—2981no—
382C22—2981no—
383C22—2981no—
384C22—5961no—
385C22—5961no—
386C22—5961no—
387C22C2×D4×D14911yescharacteristic
388C4—21no—
389C4—22no—
390C4—22no—
391C4—41no—
392C4—2981no—
393C4—2981no—
394C4—5961no—
395C4—5961no—
396C4—5961no—
397C4—1492no—
398C4C22×C22⋊D14911yescharacteristic
399C4#1801195611yescharacteristic
400C2—41no—
401C2—81no—
402C2—5961no—
403C2—11921no—
404C2D4×C22⋊D14911yescharacteristic
405C2#137×D14911yescharacteristic
406C2#2375043411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC2×#137×C149C2
socleC22×C149C22×D4
prime radical#23750294C2
hypercenterC2×#137D149
hypocenterC149#3278
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#18011943C2semidirect
3. Q8×C22⋊D149C2semidirect
4. #23749996C2semidirect
5. #23749871C2semidirect
6. #23749868C2semidirect
7. #23749833C2semidirect
8. #23749823C2semidirect
9. #23749787C2semidirect
10. #23749786C2semidirect
11. #23749764C2semidirect
12. #23749762C2semidirect
13. Q16×Dic149C2semidirect
14. #23749701C2semidirect
15. C2×#137×C149C2semidirect
117. Q16×C149D4semidirect
195. Dic149#137semidirect
203. C149#3278semidirect
206. C2×#137D149semidirect
296. Q16C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119432
#237502942
Q8×C22⋊D1492
#237499962
#237498712
#237498682
#237498332
#237498232
#237497872
#237497862
#237497642
#237497622
Q16×Dic1492
#237497012
C2×#137×C1492
#3278149