Finite Group Databasebeta

← #34082760

Subgroups — #34082760

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×#18012787C211yesmaximal, characteristic, complemented
2C22×Q8×D151C211yesmaximal, characteristic, complemented
3#23752877C213yesmaximal, complemented
4Q8×C22⋊D151C211yesmaximal, characteristic, complemented
5#23752875C213yesmaximal, complemented
6#23752874C211yesmaximal, characteristic, complemented
7#23752873C211yesmaximal, characteristic, complemented
8#23752872C213yesmaximal, complemented
9#23752871C213yesmaximal, complemented
10#23752787C213yesmaximal, complemented
11#23752785C213yesmaximal, complemented
12#23752725C213yesmaximal, complemented
13#23752721C213yesmaximal, complemented
14#23752688C211yesmaximal, characteristic, complemented
15D4×Q8×C151C211yesmaximal, characteristic, complemented
16#18012787—28no—
17C2×#14559005C2213yes—
18C2×#14559005C2211yescharacteristic, complemented
19C2×#14559004C2213yes—
20C2×#14559004C2213yescomplemented
21C22×Dic302C2213yes—
22C2×Q8×D151—28no—
23C2×Q8×D151C2213yes—
24C2×Q8×D151C2213yes—
25C2×Q8×D151C2211yescharacteristic
26C2×Q8×D151C2211yescharacteristic
27C2×Q8×D151C2213yescomplemented
28C22×C4×D151C2213yes—
29#18012738C2216yescomplemented
30#18012737C2213yescomplemented
31#18012728C2213yescomplemented
32#18012727C2213yescomplemented
33#18012725C2216yescomplemented
34#18012724C2213yes—
35#18012724C2216yes—
36#18012721C2213yes—
37#18012719C2213yescomplemented
38#18012718C2213yescomplemented
39#18012717C2213yes—
40#18012717C2216yescomplemented
41#18012717C2216yescomplemented
42#18012716C2213yes—
43#18012715C2213yes—
44#18012715C2213yescomplemented
45#18012714C2213yes—
46#18012713C2213yes—
47#18012712C2213yes—
48#18012711C2213yes—
49#18012709C2213yes—
50#18012707C2211yescharacteristic
51Q8×Dic151C2211yescharacteristic, complemented
52C4×Dic302C2213yescomplemented
53#18012704C2213yescomplemented
54C4×D604C2213yes—
55#18012701C2216yes—
56C4×C22⋊D151C2213yes—
57C4×C22⋊D151C2213yes—
58(C22⋊D151)⋊C4C2216yes—
59#18012697C2213yes—
60#18012696C2213yes—
61C22×Q8×C151C2211yescharacteristic
62C22×Q8×C151C2211yescharacteristic, complemented
63C4⋊Q8×C151C2213yescomplemented
64C22⋊Q8×C151C2213yes—
65C22⋊Q8×C151C2213yescomplemented
66C4×Q8×C151C2211yescharacteristic, complemented
67C4×D4×C151C2213yes—
68C23×D151C2311yescharacteristic
69#14559005—212no—
70#14559005—24no—
71#14559004—212no—
72#14559004—212no—
73C2×Dic302—212no—
74C2×Dic302C2313yes—
75C2×Dic302C2316yes—
76C2×Dic302C2313yes—
77C2×Dic302C2313yes—
78Q8×D151—212no—
79Q8×D151—212no—
80Q8×D151—212no—
81Q8×D151—24no—
82Q8×D151D418yescomplemented
83C2×D604C2313yes—
84C2×D604C2313yes—
85C2×D604C2313yes—
86C2×C22⋊D151C2313yes—
87C2×C22⋊D151C2311yescharacteristic
88C2×C22⋊D151C2311yescharacteristic
89C2×C22⋊D151C2313yes—
90C22×Dic151C2313yes—
91C2×C4×D151—212no—
92C2×C4×D151C2316yes—
93C2×C4×D151C2313yes—
94C2×C4×D151C2313yes—
95C2×C4×D151C2313yes—
96C2×C4×D151C2313yes—
97C2×C4×D151C2313yes—
98C4⋊Dic151C2313yes—
99C4⋊Dic151C2313yes—
100Dic151⋊C4C2313yes—
101Dic151⋊C4C2313yes—
102Dic151⋊C4C2316yes—
103Dic151⋊C4C2313yes—
104Dic151⋊C4C2313yes—
105C22⋊Dic151C2311yescharacteristic
106C22⋊Dic151C2313yes—
107(C2×D151)⋊C4C2311yescharacteristic
108(C2×D151)⋊C4C2313yes—
109(C2×D151)⋊C4C2316yes—
110(C2×D151)⋊C4C2313yes—
111(C2×D151)⋊C4C2313yes—
112(C2×D151)⋊C4C2316yes—
113(C2×D151)⋊C4C2311yescharacteristic
114(C2×D151)⋊C4C2313yes—
115C4×Dic151C2313yes—
116C4×Dic151C2316yes—
117C2×Q8×C151—24no—
118C2×Q8×C151—24no—
119C2×Q8×C151C2313yes—
120C2×Q8×C151C2313yes—
121C2×Q8×C151C2311yescharacteristic
122C2×D4×C151C2311yescharacteristic
123C22×C4×C151C2313yes—
124C22×C4×C151C2313yes—
125C4⋊C4×C151C2313yes—
126C4⋊C4×C151C2316yes—
127C4⋊C4×C151C2313yes—
128C22⋊C4×C151C2313yes—
129C22⋊C4×C151C2313yes—
130C42×C151C2313yes—
131C22×D151—22no—
132C22×D151—22no—
133C22×D151—22no—
134C22×D151C2413yes—
135C22×D151C2411yescharacteristic
136C22×D151C2411yescharacteristic
137Dic302—26no—
138Dic302—212no—
139Dic302—26no—
140Dic302C2×D4112yes—
141D604—26no—
142D604—26no—
143D604—26no—
144C22⋊D151—26no—
145C22⋊D151—22no—
146C22⋊D151—22no—
147C22⋊D151—26no—
148C2×Dic151—26no—
149C2×Dic151C2411yescharacteristic
150C2×Dic151C2413yes—
151C2×Dic151C2413yes—
152C2×Dic151C2411yescharacteristic
153C2×Dic151C2413yes—
154C4×D151—26no—
155C4×D151—26no—
156C4×D151—26no—
157C4×D151—26no—
158C4×D151—212no—
159C4×D151C2×D4112yes—
160C23×C151C2411yescharacteristic
161C23×C151C2411yescharacteristic
162Q8×C151—26no—
163Q8×C151—26no—
164Q8×C151C2×D414yes—
165D4×C151—22no—
166C2×C4×C151—26no—
167C2×C4×C151—26no—
168C2×C4×C151C2413yes—
169C2×C4×C151C2413yes—
170C2×C4×C151C2413yes—
171C2×C4×C151C2411yescharacteristic
172C2×C4×C151C2413yes—
173C22×C151—21no—
174C22×C151—21no—
175C22×C151—21no—
176C22×C151—21no—
177C22×C151—21no—
178C22×C1512−1+412yescomplemented
179C22×C151C2511yescharacteristic
180C2×D151—28no—
181C2×D151—21no—
182C2×D151—21no—
183C2×D151—21no—
184C2×D151—21no—
185C2×D151—23no—
186C2×D151—23no—
187C2×D151—23no—
188C2×D151C22×D412yes—
189C2×D1512−1+412yescomplemented
190C4×C151—23no—
191C4×C151—23no—
192C4×C151—21no—
193C4×C151—23no—
194C4×C151C22×D416yes—
195Dic151—21no—
196Dic151—23no—
197Dic151—21no—
198Dic151—23no—
199Dic151C22×D416yes—
200C2×C151—22no—
201C2×C151—41no—
202C2×C151C23×D411yescharacteristic
203C2×C151C2×2−1+411yescharacteristic
204C2×C151C2×2−1+411yescharacteristic
205D151—22no—
206D151—22no—
207D151—43no—
208C151#348611yescharacteristic, complemented, Sylow
209#3486—1511nomaximal, Sylow
210C2×2−1+4—1511no—
211C23×Q8—1511no—
212D4×Q8—1511no—
213D4×Q8—1513no—
214D4×Q8—1513no—
215D4×Q8D15111yescharacteristic, complemented
216#548—1511no—
217#548—1513no—
218#548—1511no—
219#548—1513no—
220#536—1513no—
221#536—1513no—
222#525—1513no—
223#525—1513no—
224#519—1511no—
2252−1+4—3028no—
226C2×#41—1513no—
227C2×#41—1511no—
228C2×#41—1513no—
229C2×#41—1513no—
230C22×Q8—3028no—
231C22×Q8—1513no—
232C22×Q8—1513no—
233C22×Q8—1511no—
234C22×Q8—1513no—
235C22×Q8—1511no—
236C22×Q8—1513no—
237C22×Q8C2×D15111yescharacteristic
238C22×Q8C2×D15111yescharacteristic, complemented
239C23×C4—1513no—
240C4⋊Q8—1513no—
241C4⋊Q8—1516no—
242C4⋊Q8C2×D15113yescomplemented
243#124—1513no—
244#124—1516no—
245#124—1513no—
246#123—1516no—
247#123—1513no—
248#123—1513no—
249C22⋊Q8—1513no—
250C22⋊Q8—1513no—
251C22⋊Q8—1513no—
252C22⋊Q8—1513no—
253C22⋊Q8—1516no—
254C22⋊Q8—1516no—
255C22⋊Q8—1513no—
256C22⋊Q8—1513no—
257C22⋊Q8C2×D15113yes—
258C22⋊Q8C2×D15113yescomplemented
259#121—1513no—
260#121—1513no—
261#121—1513no—
262#121—1513no—
263#120—1511no—
264#120—1513no—
265C4×Q8—1511no—
266C4×Q8—1513no—
267C4×Q8—1513no—
268C4×Q8C2×D15111yescharacteristic, complemented
269C4×D4—1513no—
270C4×D4—1513no—
271C4×D4—1516no—
272C4×D4—1516no—
273C4×D4—1513no—
274C4×D4C2×D15113yes—
275#117—1513no—
276#117—1513no—
277C24—1511no—
278#41—30212no—
279#41—30212no—
280#41—3024no—
281#41—30212no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—30212no—
285C2×Q8—30212no—
286C2×Q8—30212no—
287C2×Q8—3024no—
288C2×Q8—30212no—
289C2×Q8—1518no—
290C2×Q8—1513no—
291C2×Q8—1516no—
292C2×Q8—1513no—
293C2×Q8—1513no—
294C2×Q8C22×D15113yes—
295C2×Q8C22×D15113yes—
296C2×Q8C22×D15111yescharacteristic
297C2×D4—1511no—
298C2×D4—1513no—
299C2×D4—1511no—
300C2×D4—1513no—
301C2×D4—1513no—
302C2×D4—1513no—
303C2×D4—1513no—
304C2×D4C22×D15111yescharacteristic
305C22×C4—30212no—
306C22×C4—1513no—
307C22×C4—1513no—
308C22×C4—1513no—
309C22×C4—1516no—
310C22×C4—1513no—
311C22×C4—1513no—
312C22×C4—1513no—
313C22×C4C22×D15113yes—
314C22×C4C22×D15113yes—
315C4⋊C4—1513no—
316C4⋊C4—1513no—
317C4⋊C4—1516no—
318C4⋊C4—1513no—
319C4⋊C4—1513no—
320C4⋊C4—1513no—
321C4⋊C4—1513no—
322C4⋊C4C22×D15113yes—
323C4⋊C4C22×D15116yes—
324C4⋊C4C22×D15113yes—
325C22⋊C4—1511no—
326C22⋊C4—1513no—
327C22⋊C4—1513no—
328C22⋊C4—1511no—
329C22⋊C4—1513no—
330C22⋊C4—1511no—
331C22⋊C4—1513no—
332C22⋊C4—1516no—
333C22⋊C4—1516no—
334C22⋊C4—1513no—
335C22⋊C4C22×D15113yes—
336C22⋊C4C22×D15113yes—
337C42—1513no—
338C42—1516no—
339C42C22×D15113yes—
340C23—3022no—
341C23—3022no—
342C23—3022no—
343C23—1513no—
344C23—1511no—
345C23—1511no—
346C23C23×D15111yescharacteristic
347C23C23×D15111yescharacteristic
348Q8—26no—
349Q8—26no—
350Q8—30212no—
351Q8—3026no—
352Q8—3026no—
353Q8—15112no—
354Q8D4×D15114yes—
355D4—22no—
356D4—3026no—
357D4—3026no—
358D4—3022no—
359D4—3022no—
360D4—3026no—
361D4—3026no—
362D4—3026no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—3026no—
366C2×C4—3026no—
367C2×C4—30212no—
368C2×C4—3026no—
369C2×C4—3026no—
370C2×C4—3026no—
371C2×C4—1511no—
372C2×C4—1513no—
373C2×C4—1513no—
374C2×C4—1511no—
375C2×C4—1513no—
376C2×C4—15112no—
377C2×C4C23×D15113yes—
378C2×C4C23×D15111yescharacteristic
379C2×C4C23×D15113yes—
380C2×C4C23×D15113yes—
381C2×C4C23×D15113yes—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—3028no—
388C22—3021no—
389C22—3021no—
390C22—3021no—
391C22—3021no—
392C22—3023no—
393C22—3023no—
394C22—3023no—
395C22—1512no—
396C22—1512no—
397C22#1801278712yescomplemented
398C22C24×D15111yescharacteristic
399C4—23no—
400C4—23no—
401C4—23no—
402C4—21no—
403C4—3021no—
404C4—3023no—
405C4—3021no—
406C4—3023no—
407C4—1516no—
408C4C2×D4×D15116yes—
409C2—22no—
410C2—41no—
411C2—3022no—
412C2—3022no—
413C2—6043no—
414C2C22×D4×D15111yescharacteristic
415C22−1+4×D15111yescharacteristic
416C2C2×#1801278711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D151
commutator, socle, cosocleC22×C151C25
FittingD4×Q8×C151C2
hypercenterD4×Q8D151
hypocenterC151#3486
LCSG ▷ C22×C151 ▷ C151
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#18012787C2semidirect
2. C22×Q8×D151C2semidirect
3. #23752877C2semidirect
4. Q8×C22⋊D151C2semidirect
5. #23752875C2semidirect
6. #23752874C2semidirect
7. #23752873C2semidirect
8. #23752872C2semidirect
9. #23752871C2semidirect
10. #23752787C2semidirect
11. #23752785C2semidirect
12. #23752725C2semidirect
13. #23752721C2semidirect
14. #23752688C2semidirect
15. D4×Q8×C151C2semidirect
18. C2×#14559005C22semidirect
20. C2×#14559004C22semidirect
27. C2×Q8×D151C22semidirect
29. #18012738C22semidirect
30. #18012737C22semidirect
31. #18012728C22semidirect
32. #18012727C22semidirect
33. #18012725C22semidirect
37. #18012719C22semidirect
38. #18012718C22semidirect
40. #18012717C22semidirect
41. #18012717C22semidirect
44. #18012715C22semidirect
51. Q8×Dic151C22semidirect
52. C4×Dic302C22semidirect
53. #18012704C22semidirect
62. C22×Q8×C151C22semidirect
63. C4⋊Q8×C151C22semidirect
65. C22⋊Q8×C151C22semidirect
66. C4×Q8×C151C22semidirect
82. Q8×D151D4semidirect
178. C22×C1512−1+4semidirect
189. C2×D1512−1+4semidirect
208. C151#3486semidirect
215. D4×Q8D151semidirect
238. C22×Q8C2×D151semidirect
242. C4⋊Q8C2×D151semidirect
258. C22⋊Q8C2×D151semidirect
268. C4×Q8C2×D151semidirect
397. C22#18012787semidirect

Maximal subgroups

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

subgroupindex
C2×#180127872
C22×Q8×D1512
#237528772
Q8×C22⋊D1512
#237528752
#237528742
#237528732
#237528722
#237528712
#237527872
#237527852
#237527252
#237527212
#237526882
D4×Q8×C1512
#3486151