Finite Group Databasebeta

← #34082756

Subgroups — #34082756

Order 19328.

1492 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×#18012786C211yesmaximal, characteristic, complemented
2C22×Q8×D151C211yesmaximal, characteristic, complemented
3Q8×D604C211yesmaximal, characteristic, complemented
4D4×Dic302C213yesmaximal, complemented
5Q8×C22⋊D151C211yesmaximal, characteristic, complemented
6#23752875C213yesmaximal, complemented
7#23752870C213yesmaximal, complemented
8#23752869C211yesmaximal, characteristic, complemented
9#23752868C213yesmaximal, complemented
10#23752785C213yesmaximal, complemented
11#23752782C213yesmaximal, complemented
12#23752722C213yesmaximal, complemented
13#23752721C213yesmaximal, complemented
14#23752688C211yesmaximal, characteristic, complemented
15#548×C151C211yesmaximal, characteristic, complemented
16#18012786—28no—
17C2×#14559004C2213yes—
18C2×#14559003C2213yes—
19C2×#14559003C2213yescomplemented
20C22×Dic302C2213yes—
21C22×Dic302C2213yescomplemented
22C2×Q8×D151—28no—
23C2×Q8×D151C2213yes—
24C2×Q8×D151C2211yescharacteristic
25C2×Q8×D151C2213yes—
26C2×Q8×D151C2211yescharacteristic
27C2×Q8×D151C2211yescharacteristic, complemented
28C22×C4×D151C2213yes—
29#18012739C2213yescomplemented
30#18012738C2216yescomplemented
31#18012737C2213yescomplemented
32#18012726C2213yescomplemented
33#18012725C2216yescomplemented
34#18012724C2213yes—
35#18012721C2213yes—
36#18012721C2216yes—
37#18012719C2213yes—
38#18012719C2213yescomplemented
39#18012718C2213yes—
40#18012718C2213yescomplemented
41#18012717C2213yes—
42#18012716C2213yes—
43#18012716C2216yescomplemented
44#18012716C2216yescomplemented
45#18012715C2213yescomplemented
46#18012712C2213yes—
47#18012711C2213yes—
48#18012710C2213yes—
49#18012709C2211yescharacteristic
50#18012707C2213yes—
51Q8×Dic151C2211yescharacteristic, complemented
52C4×Dic302C2213yescomplemented
53#18012704C2213yescomplemented
54C4×D604C2213yes—
55D4×Dic151C2216yes—
56C4×C22⋊D151C2213yes—
57C4×C22⋊D151C2213yes—
58(C22⋊D151)⋊C4C2216yes—
59#18012697C2213yes—
60#18012696C2213yes—
61C2×#41×C151C2211yescharacteristic, complemented
62C22×Q8×C151C2211yescharacteristic
63#124×C151C2213yescomplemented
64C22⋊Q8×C151C2213yescomplemented
65#121×C151C2213yes—
66C4×Q8×C151C2211yescharacteristic, complemented
67C4×D4×C151C2213yes—
68C23×D151C2311yescharacteristic
69#14559004—212no—
70#14559003—212no—
71#14559003—212no—
72C2×Dic302—212no—
73C2×Dic302—212no—
74C2×Dic302C2313yes—
75C2×Dic302C2316yes—
76C2×Dic302C2313yes—
77C2×Dic302C2313yes—
78C2×Dic302C2313yes—
79Q8×D151—212no—
80Q8×D151—212no—
81Q8×D151—24no—
82Q8×D151—24no—
83Q8×D151D418yescomplemented
84C2×D604C2311yescharacteristic
85C2×D604C2311yescharacteristic
86C2×C22⋊D151C2313yes—
87C2×C22⋊D151C2311yescharacteristic
88C2×C22⋊D151C2313yes—
89C2×C22⋊D151C2313yes—
90C22×Dic151C2316yes—
91C22×Dic151C2313yes—
92C2×C4×D151—212no—
93C2×C4×D151C2313yes—
94C2×C4×D151C2313yes—
95C2×C4×D151C2313yes—
96C2×C4×D151C2313yes—
97C2×C4×D151C2313yes—
98C4⋊Dic151C2313yes—
99C4⋊Dic151C2316yes—
100C4⋊Dic151C2313yes—
101Dic151⋊C4C2313yes—
102Dic151⋊C4C2313yes—
103Dic151⋊C4C2316yes—
104Dic151⋊C4C2313yes—
105Dic151⋊C4C2313yes—
106C22⋊Dic151C2313yes—
107C22⋊Dic151C2313yes—
108C22⋊Dic151C2316yes—
109(C2×D151)⋊C4C2311yescharacteristic
110(C2×D151)⋊C4C2313yes—
111(C2×D151)⋊C4C2313yes—
112(C2×D151)⋊C4C2311yescharacteristic
113(C2×D151)⋊C4C2313yes—
114(C2×D151)⋊C4C2313yes—
115C4×Dic151C2313yes—
116C4×Dic151C2316yes—
117#41×C151—24no—
118C2×Q8×C151—24no—
119C2×Q8×C151C2313yes—
120C2×Q8×C151C2311yescharacteristic
121C2×D4×C151C2313yes—
122C2×D4×C151C2313yes—
123C22×C4×C151C2313yes—
124C22×C4×C151C2313yes—
125C4⋊C4×C151C2313yes—
126C4⋊C4×C151C2313yes—
127C22⋊C4×C151C2316yes—
128C22⋊C4×C151C2311yescharacteristic
129C22⋊C4×C151C2313yes—
130C42×C151C2313yes—
131C22×D151—22no—
132C22×D151—22no—
133C22×D151—22no—
134C22×D151C2411yescharacteristic
135C22×D151C2411yescharacteristic
136C22×D151C2411yescharacteristic
137Dic302—26no—
138Dic302—26no—
139Dic302—212no—
140Dic302—26no—
141Dic302C2×D4112yes—
142D604—22no—
143D604—22no—
144C22⋊D151—22no—
145C22⋊D151—26no—
146C22⋊D151—26no—
147C22⋊D151—26no—
148C2×Dic151—26no—
149C2×Dic151—212no—
150C2×Dic151C2411yescharacteristic
151C2×Dic151C2413yes—
152C2×Dic151C2413yes—
153C2×Dic151C2413yes—
154C2×Dic151C2413yes—
155C4×D151—26no—
156C4×D151—26no—
157C4×D151—26no—
158C4×D151—26no—
159C4×D151C2×D4112yes—
160C23×C151C2413yes—
161C23×C151C2411yescharacteristic
162Q8×C151—26no—
163Q8×C151C2×D414yes—
164D4×C151—26no—
165D4×C151—26no—
166C2×C4×C151—26no—
167C2×C4×C151—26no—
168C2×C4×C151C2413yes—
169C2×C4×C151C2413yes—
170C2×C4×C151C2411yescharacteristic
171C2×C4×C151C2411yescharacteristic
172C2×C4×C151C2413yes—
173C22×C151—21no—
174C22×C151—21no—
175C22×C151—23no—
176C22×C151—23no—
177C22×C151—23no—
178C22×C1512−1+412yescomplemented
179C22×C151C2511yescharacteristic
180C2×D151—28no—
181C2×D151—21no—
182C2×D151—21no—
183C2×D151—21no—
184C2×D151—21no—
185C2×D151—21no—
186C2×D151—21no—
187C2×D151—21no—
188C2×D151C22×D412yes—
189C2×D1512−1+412yescomplemented
190C4×C151—21no—
191C4×C151—23no—
192C4×C151—23no—
193C4×C151—21no—
194C4×C151C22×D416yes—
195Dic151—23no—
196Dic151—23no—
197Dic151—23no—
198Dic151—21no—
199Dic151C22×D416yes—
200C2×C151—22no—
201C2×C151—43no—
202C2×C151C23×D411yescharacteristic
203C2×C151C2×2−1+411yescharacteristic
204C2×C151C2×2−1+411yescharacteristic
205D151—22no—
206D151—22no—
207D151—41no—
208C151#348611yescharacteristic, complemented, Sylow
209#3486—1511nomaximal, Sylow
210C2×2−1+4—1511no—
211C23×Q8—1511no—
212D4×Q8—1511no—
213D4×Q8—1513no—
214D4×Q8—1511no—
215D4×Q8—1513no—
216#548—1511no—
217#548—1513no—
218#548—1513no—
219#548D15111yescharacteristic, complemented
220#536—1513no—
221#536—1513no—
222#525—1513no—
223#525—1513no—
224#519—1511no—
2252−1+4—3028no—
226C2×#41—1513no—
227C2×#41—1513no—
228C2×#41—1513no—
229C2×#41C2×D15111yescharacteristic, complemented
230C22×Q8—3028no—
231C22×Q8—1513no—
232C22×Q8—1513no—
233C22×Q8—1511no—
234C22×Q8—1511no—
235C22×Q8—1511no—
236C22×Q8—1513no—
237C22×Q8—1513no—
238C22×Q8C2×D15111yescharacteristic
239C23×C4—1513no—
240C4⋊Q8—1513no—
241C4⋊Q8—1516no—
242C4⋊Q8—1513no—
243#124—1513no—
244#124—1516no—
245#124C2×D15113yescomplemented
246#123—1513no—
247#123—1516no—
248#123—1513no—
249C22⋊Q8—1513no—
250C22⋊Q8—1513no—
251C22⋊Q8—1513no—
252C22⋊Q8—1513no—
253C22⋊Q8—1513no—
254C22⋊Q8—1513no—
255C22⋊Q8—1513no—
256C22⋊Q8—1516no—
257C22⋊Q8—1516no—
258C22⋊Q8C2×D15113yescomplemented
259#121—1513no—
260#121—1513no—
261#121—1513no—
262#121C2×D15113yes—
263#120—1513no—
264#120—1511no—
265C4×Q8—1511no—
266C4×Q8—1513no—
267C4×Q8—1513no—
268C4×Q8C2×D15111yescharacteristic, complemented
269C4×D4—1516no—
270C4×D4—1513no—
271C4×D4—1513no—
272C4×D4—1516no—
273C4×D4—1513no—
274C4×D4C2×D15113yes—
275#117—1513no—
276#117—1513no—
277C24—1511no—
278#41—24no—
279#41—30212no—
280#41—30212no—
281#41—30212no—
282C2×Q8—24no—
283C2×Q8—30212no—
284C2×Q8—30212no—
285C2×Q8—30212no—
286C2×Q8—3024no—
287C2×Q8—3024no—
288C2×Q8—30212no—
289C2×Q8—1518no—
290C2×Q8—1513no—
291C2×Q8—1516no—
292C2×Q8—1513no—
293C2×Q8—1513no—
294C2×Q8—1513no—
295C2×Q8C22×D15113yes—
296C2×Q8C22×D15111yescharacteristic
297C2×D4—1513no—
298C2×D4—1511no—
299C2×D4—1513no—
300C2×D4—1513no—
301C2×D4—1511no—
302C2×D4—1511no—
303C2×D4C22×D15113yes—
304C2×D4C22×D15113yes—
305C22×C4—30212no—
306C22×C4—1513no—
307C22×C4—1516no—
308C22×C4—1513no—
309C22×C4—1513no—
310C22×C4—1513no—
311C22×C4—1513no—
312C22×C4—1513no—
313C22×C4C22×D15113yes—
314C22×C4C22×D15113yes—
315C4⋊C4—1513no—
316C4⋊C4—1513no—
317C4⋊C4—1513no—
318C4⋊C4—1516no—
319C4⋊C4—1513no—
320C4⋊C4—1513no—
321C4⋊C4—1516no—
322C4⋊C4—1513no—
323C4⋊C4C22×D15113yes—
324C4⋊C4C22×D15113yes—
325C22⋊C4—1513no—
326C22⋊C4—1513no—
327C22⋊C4—1516no—
328C22⋊C4—1513no—
329C22⋊C4—1511no—
330C22⋊C4—1511no—
331C22⋊C4—1513no—
332C22⋊C4—1513no—
333C22⋊C4—1513no—
334C22⋊C4C22×D15116yes—
335C22⋊C4C22×D15111yescharacteristic
336C22⋊C4C22×D15113yes—
337C42—1513no—
338C42—1516no—
339C42C22×D15113yes—
340C23—3022no—
341C23—3022no—
342C23—3022no—
343C23—1511no—
344C23—1511no—
345C23—1511no—
346C23C23×D15113yes—
347C23C23×D15111yescharacteristic
348Q8—26no—
349Q8—30212no—
350Q8—3026no—
351Q8—3026no—
352Q8—3026no—
353Q8—15112no—
354Q8D4×D15114yes—
355D4—26no—
356D4—26no—
357D4—3022no—
358D4—3022no—
359D4—3026no—
360D4—3026no—
361D4—3022no—
362D4—3026no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—3026no—
366C2×C4—3026no—
367C2×C4—3026no—
368C2×C4—30212no—
369C2×C4—3026no—
370C2×C4—3026no—
371C2×C4—1513no—
372C2×C4—1511no—
373C2×C4—1513no—
374C2×C4—1513no—
375C2×C4—1513no—
376C2×C4—15112no—
377C2×C4C23×D15113yes—
378C2×C4C23×D15111yescharacteristic
379C2×C4C23×D15113yes—
380C2×C4C23×D15111yescharacteristic
381C2×C4C23×D15113yes—
382C22—21no—
383C22—21no—
384C22—23no—
385C22—23no—
386C22—23no—
387C22—3028no—
388C22—3021no—
389C22—3021no—
390C22—3021no—
391C22—3021no—
392C22—3021no—
393C22—3021no—
394C22—3021no—
395C22—1512no—
396C22—1512no—
397C22#1801278612yescomplemented
398C22C24×D15111yescharacteristic
399C4—21no—
400C4—23no—
401C4—21no—
402C4—23no—
403C4—3023no—
404C4—3023no—
405C4—3023no—
406C4—3021no—
407C4—1516no—
408C4C2×D4×D15116yes—
409C2—22no—
410C2—43no—
411C2—3022no—
412C2—3022no—
413C2—6041no—
414C2C22×D4×D15111yescharacteristic
415C22−1+4×D15111yescharacteristic
416C2C2×#1801278611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D151
commutator, socle, cosocleC22×C151C25
Fitting#548×C151C2
hypercenter#548D151
hypocenterC151#3486
LCSG ▷ C22×C151 ▷ C151
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#18012786C2semidirect
2. C22×Q8×D151C2semidirect
3. Q8×D604C2semidirect
4. D4×Dic302C2semidirect
5. Q8×C22⋊D151C2semidirect
6. #23752875C2semidirect
7. #23752870C2semidirect
8. #23752869C2semidirect
9. #23752868C2semidirect
10. #23752785C2semidirect
11. #23752782C2semidirect
12. #23752722C2semidirect
13. #23752721C2semidirect
14. #23752688C2semidirect
15. #548×C151C2semidirect
19. C2×#14559003C22semidirect
21. C22×Dic302C22semidirect
27. C2×Q8×D151C22semidirect
29. #18012739C22semidirect
30. #18012738C22semidirect
31. #18012737C22semidirect
32. #18012726C22semidirect
33. #18012725C22semidirect
38. #18012719C22semidirect
40. #18012718C22semidirect
43. #18012716C22semidirect
44. #18012716C22semidirect
45. #18012715C22semidirect
51. Q8×Dic151C22semidirect
52. C4×Dic302C22semidirect
53. #18012704C22semidirect
61. C2×#41×C151C22semidirect
63. #124×C151C22semidirect
64. C22⋊Q8×C151C22semidirect
66. C4×Q8×C151C22semidirect
83. Q8×D151D4semidirect
178. C22×C1512−1+4semidirect
189. C2×D1512−1+4semidirect
208. C151#3486semidirect
219. #548D151semidirect
229. C2×#41C2×D151semidirect
245. #124C2×D151semidirect
258. C22⋊Q8C2×D151semidirect
268. C4×Q8C2×D151semidirect
397. C22#18012786semidirect

Maximal subgroups

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

subgroupindex
C2×#180127862
C22×Q8×D1512
Q8×D6042
D4×Dic3022
Q8×C22⋊D1512
#237528752
#237528702
#237528692
#237528682
#237527852
#237527822
#237527222
#237527212
#237526882
#548×C1512
#3486151