Finite Group Databasebeta

← #34078400

Subgroups — #34078400

Order 19328.

700 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18012787C211yesmaximal, characteristic
2C2×#18012771C211yesmaximal, characteristic, complemented
3C2×Q16×D151C211yesmaximal, characteristic, complemented
4#23752785C211yesmaximal, characteristic, complemented
5#23752377C211yesmaximal, characteristic, complemented
6#23752376C211yesmaximal, characteristic, complemented
7#23752373C211yesmaximal, characteristic, complemented
8#23752372C211yesmaximal, characteristic, complemented
9#23752369C211yesmaximal, characteristic, complemented
10#23752368C211yesmaximal, characteristic, complemented
11#23752366C211yesmaximal, characteristic, complemented
12#23752219C211yesmaximal, characteristic, complemented
13#23752207C211yesmaximal, characteristic, complemented
14#23752177C211yesmaximal, characteristic, complemented
15#451×C151C211yesmaximal, characteristic, complemented
16#18012787—24no—
17#18012787—24no—
18C2×#14559005—21no—
19C2×#14559005C2211yescharacteristic
20C2×#14559005C2211yescharacteristic
21C2×#14559004—21no—
22C2×#14559004—21no—
23C2×#14559004C2211yescharacteristic
24C2×#14559004C2211yescharacteristic
25C2×Q8×D151—21no—
26C2×Q8×D151C2211yescharacteristic
27C2×Q8×D151C2211yescharacteristic
28C2×Q8×D151C2211yescharacteristic
29#18012771—24no—
30C2×Dic604C2211yescharacteristic
31C2×#14558994C2211yescharacteristic
32C2×#14558994C2211yescharacteristic
33C2×#14558994C2211yescharacteristic
34Q16×D151—24no—
35C2×#14558993C2211yescharacteristic
36C2×#14558991C2211yescharacteristic
37C2×#14558987C2211yescharacteristic
38C2×C8×D151C2211yescharacteristic
39#18012738—21no—
40#18012725—21no—
41#18012724—21no—
42#18012721—21no—
43#18012719C2211yescharacteristic
44#18012718C2211yescharacteristic
45#18012715C2211yescharacteristic
46#18012697C2211yescharacteristic
47#18012664C2211yescharacteristic
48#18012663C2211yescharacteristic
49#18012663C2211yescharacteristic
50#18012662C2211yescharacteristic
51#18012661C2211yescharacteristic
52#18012659C2211yescharacteristic
53C22⋊(C151⋊C8)C2211yescharacteristic
54(C2×D151)⋊C8C2211yescharacteristic
55(C2×D151)⋊C8C2211yescharacteristic
56C22×Q8×C151C2211yescharacteristic
57C2×Q16×C151C2211yescharacteristic
58C2×Q16×C151C2211yescharacteristic
59C22⋊Q8×C151C2211yescharacteristic
60#103×C151C2211yescharacteristic
61#103×C151C2211yescharacteristic
62C22⋊C8×C151C2211yescharacteristic
63#14559005—28no—
64#14559005—22no—
65#14559005—22no—
66#14559005—22no—
67#14559005D414yes—
68#14559004—28no—
69#14559004—28no—
70#14559004—22no—
71#14559004—22no—
72#14559004—22no—
73#14559004—22no—
74C2×Dic302—21no—
75C2×Dic302—21no—
76C2×Dic302—21no—
77C2×Dic302D411yescharacteristic
78C2×Dic302C2311yescharacteristic
79C2×Dic302C2311yescharacteristic
80Q8×D151—28no—
81Q8×D151—22no—
82Q8×D151—22no—
83Q8×D151—22no—
84Q8×D151—22no—
85Q8×D151—22no—
86Q8×D151D414yes—
87C2×D604—21no—
88C2×D604—21no—
89C2×D604D411yescharacteristic
90C2×D604C2311yescharacteristic
91C2×C22⋊D151—21no—
92C2×C22⋊D151D411yescharacteristic
93C2×C22⋊D151D411yescharacteristic
94C2×C4×D151—21no—
95C2×C4×D151—21no—
96C2×C4×D151—21no—
97C2×C4×D151—21no—
98C2×C4×D151—21no—
99C2×C4×D151—21no—
100C2×C4×D151C2311yescharacteristic
101C2×C4×D151C2311yescharacteristic
102Dic604—22no—
103#14558994—22no—
104#14558994—22no—
105#14558994—22no—
106#14558993—22no—
107#14558991—22no—
108#14558987—22no—
109C2×C151⋊C8C2311yescharacteristic
110C2×C151⋊C8C2311yescharacteristic
111C8×D151—22no—
112C4⋊Dic151—21no—
113C4⋊Dic151C2311yescharacteristic
114Dic151⋊C4—21no—
115Dic151⋊C4—21no—
116C22⋊Dic151—21no—
117(C2×D151)⋊C4—21no—
118(C2×D151)⋊C4—21no—
119(C2×D151)⋊C4—21no—
120C4×Dic151—21no—
121C2×Q8×C151—22no—
122C2×Q8×C151—22no—
123C2×Q8×C151—21no—
124C2×Q8×C151C2311yescharacteristic
125C2×Q8×C151C2311yescharacteristic
126C2×Q8×C151C2311yescharacteristic
127C22×C4×C151—21no—
128C22×C4×C151C2311yescharacteristic
129Q16×C151—22no—
130Q16×C151—22no—
131C2×C8×C151C2311yescharacteristic
132C2×C8×C151C2311yescharacteristic
133C4⋊C4×C151—21no—
134C4⋊C4×C151C2311yescharacteristic
135C22⋊C4×C151—21no—
136C22×D151—21no—
137C22×D151C2×D411yescharacteristic
138C22×D151C2×D411yescharacteristic
139Dic302—24no—
140Dic302—21no—
141Dic302—22no—
142Dic302—24no—
143Dic302—21no—
144Dic302—21no—
145Dic302—42no—
146Dic302C2×D412yes—
147D604—24no—
148D604—24no—
149D604—22no—
150D604—21no—
151D604C2×D412yes—
152C22⋊D151—24no—
153C22⋊D151—22no—
154C22⋊D151—22no—
155C2×Dic151—21no—
156C2×Dic151—21no—
157C2×Dic151—21no—
158C2×Dic151C2×D411yescharacteristic
159C2×Dic151C2×D411yescharacteristic
160C4×D151—24no—
161C4×D151—24no—
162C4×D151—24no—
163C4×D151—24no—
164C4×D151—21no—
165C4×D151—21no—
166C4×D151—24no—
167C4×D151—42no—
168C4×D151C2×D412yes—
169C4×D151C2×D412yes—
170C151⋊C8—21no—
171C151⋊C8—21no—
172C23×C151C2×D411yescharacteristic
173Q8×C151—24no—
174Q8×C151—21no—
175Q8×C151—21no—
176Q8×C151—21no—
177Q8×C151—21no—
178Q8×C151C2×D412yes—
179Q8×C151C2×D412yes—
180C2×C4×C151—24no—
181C2×C4×C151—21no—
182C2×C4×C151—21no—
183C2×C4×C151—21no—
184C2×C4×C151—21no—
185C2×C4×C151—21no—
186C2×C4×C151—21no—
187C2×C4×C151C2×D411yescharacteristic
188C2×C4×C151C2411yescharacteristic
189C8×C151—21no—
190C8×C151—21no—
191C22×C151—21no—
192C22×C151—21no—
193C22×C151—21no—
194C22×C151C22×D411yescharacteristic
195C2×D151—21no—
196C2×D151—21no—
197C2×D151—21no—
198C2×D151—22no—
199C2×D151—21no—
200C2×D151—21no—
201C2×D151—41no—
202C2×D151—41no—
203C2×D151#12012yes—
204C4×C151—22no—
205C4×C151—22no—
206C4×C151—21no—
207C4×C151—41no—
208C4×C151—41no—
209C4×C151C22×D411yescharacteristic
210C4×C151C22×D411yescharacteristic
211Dic151—22no—
212Dic151—21no—
213Dic151—41no—
214Dic151—41no—
215Dic151#12012yes—
216C2×C151—41no—
217C2×C151C2×#12011yescharacteristic
218C2×C151#57811yescharacteristic
219C2×C151#57811yescharacteristic
220D151—22no—
221D151—41no—
222D151—42no—
223C151#301111yescharacteristic, complemented, Sylow
224#3011—1511nomaximal, Sylow
225C2×2−1+4—1511no—
226C2×#137—1511no—
227C22×Q16—1511no—
228#536—1511no—
229#452—1511no—
230#452—1511no—
231#452—1511no—
232#452—1511no—
233#451—1511no—
234#451—1511no—
235#451—1511no—
236#451D15111yescharacteristic, complemented
237#419—1511no—
238#416—1511no—
239#408—1511no—
2402−1+4—3024no—
2412−1+4—3024no—
242C2×#41—3021no—
243C2×#41—3021no—
244C2×#41—3021no—
245C2×#41—1511no—
246C2×#41—1511no—
247C2×#41—1511no—
248C2×#41—1511no—
249C22×Q8—3021no—
250C22×Q8—1511no—
251C22×Q8—1511no—
252C22×Q8—1511no—
253C22×Q8C2×D15111yescharacteristic
254#137—3024no—
255C2×Q16—3024no—
256C2×Q16—1511no—
257C2×Q16—1511no—
258C2×Q16—1511no—
259C2×Q16—1511no—
260C2×Q16C2×D15111yescharacteristic
261C2×Q16C2×D15111yescharacteristic
262C2×SD16—1511no—
263C2×SD16—1511no—
264C2×M16—1511no—
265C22×C8—1511no—
266C4⋊Q8—3021no—
267#124—3021no—
268#123—3021no—
269#123—3021no—
270C22⋊Q8—1511no—
271C22⋊Q8—1511no—
272C22⋊Q8—1511no—
273C22⋊Q8C2×D15111yescharacteristic
274#117—1511no—
275#103—1511no—
276#103—1511no—
277#103—1511no—
278#103—1511no—
279#103C2×D15111yescharacteristic
280#103C2×D15111yescharacteristic
281#102—1511no—
282#102—1511no—
283C22⋊C8—1511no—
284C22⋊C8—1511no—
285C22⋊C8—1511no—
286C22⋊C8C2×D15111yescharacteristic
287#41—3028no—
288#41—3028no—
289#41—3022no—
290#41—3022no—
291#41—3022no—
292#41—3022no—
293#41—3028no—
294#41—3022no—
295#41—3022no—
296#41—3022no—
297#41—1514no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—21no—
301C2×Q8—3021no—
302C2×Q8—3021no—
303C2×Q8—3021no—
304C2×Q8—3028no—
305C2×Q8—3022no—
306C2×Q8—3022no—
307C2×Q8—3022no—
308C2×Q8—3022no—
309C2×Q8—3022no—
310C2×Q8—1514no—
311C2×Q8—1511no—
312C2×Q8—1511no—
313C2×Q8—1511no—
314C2×Q8C22×D15111yescharacteristic
315C2×Q8C22×D15111yescharacteristic
316C2×Q8C22×D15111yescharacteristic
317C2×D4—3021no—
318C2×D4—3021no—
319C2×D4—3021no—
320C2×D4—1511no—
321C2×D4—1511no—
322C2×D4—1511no—
323C2×D4—1511no—
324C22×C4—21no—
325C22×C4—3021no—
326C22×C4—3021no—
327C22×C4—3021no—
328C22×C4—3021no—
329C22×C4—3021no—
330C22×C4—3021no—
331C22×C4—1511no—
332C22×C4—1511no—
333C22×C4C22×D15111yescharacteristic
334Q16—22no—
335Q16—22no—
336Q16—3022no—
337Q16—3022no—
338Q16—3022no—
339Q16—3022no—
340SD16—3022no—
341SD16—3022no—
342M16—3022no—
343C2×C8—3022no—
344C2×C8—1511no—
345C2×C8—1511no—
346C2×C8C22×D15111yescharacteristic
347C2×C8C22×D15111yescharacteristic
348C4⋊C4—21no—
349C4⋊C4—3021no—
350C4⋊C4—3021no—
351C4⋊C4—3021no—
352C4⋊C4—1511no—
353C4⋊C4C22×D15111yescharacteristic
354C22⋊C4—21no—
355C22⋊C4—3021no—
356C22⋊C4—3021no—
357C22⋊C4—3021no—
358C22⋊C4—3021no—
359C42—3021no—
360C23—3021no—
361C23—1511no—
362C23—1511no—
363C23D4×D15111yescharacteristic
364Q8—24no—
365Q8—21no—
366Q8—21no—
367Q8—21no—
368Q8—21no—
369Q8—3024no—
370Q8—3024no—
371Q8—3022no—
372Q8—3021no—
373Q8—3021no—
374Q8—3021no—
375Q8—6042no—
376Q8—1512no—
377Q8D4×D15112yes—
378Q8D4×D15112yes—
379D4—3024no—
380D4—3024no—
381D4—3022no—
382D4—3024no—
383D4—3022no—
384D4—3022no—
385D4—3021no—
386D4—1512no—
387C2×C4—24no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—3024no—
395C2×C4—3024no—
396C2×C4—3024no—
397C2×C4—3024no—
398C2×C4—3024no—
399C2×C4—3021no—
400C2×C4—3021no—
401C2×C4—3021no—
402C2×C4—3021no—
403C2×C4—3021no—
404C2×C4—6042no—
405C2×C4—1511no—
406C2×C4—1511no—
407C2×C4—1512no—
408C2×C4—1512no—
409C2×C4D4×D15111yescharacteristic
410C2×C4C23×D15111yescharacteristic
411C8—21no—
412C8—21no—
413C8—3021no—
414C8—3021no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—3021no—
419C22—3021no—
420C22—3021no—
421C22—3021no—
422C22—3021no—
423C22—3022no—
424C22—6041no—
425C22—6041no—
426C22—1512no—
427C22C2×D4×D15111yescharacteristic
428C4—22no—
429C4—22no—
430C4—21no—
431C4—41no—
432C4—41no—
433C4—3022no—
434C4—3021no—
435C4—6041no—
436C4—6041no—
437C4—1512no—
438C4C2×D4×D15111yescharacteristic
439C4C2×D4×D15111yescharacteristic
440C2—41no—
441C2—3022no—
442C2—6041no—
443C2—6042no—
444C2#120×D15111yescharacteristic
445C2#2375302511yescharacteristic
446C2#2375302811yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#451×C151C2
socleC22×C151C22×D4
prime radicalC2×#18012787C2
hypercenter#451D151
hypocenterC151#3011
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#18012771C2semidirect
3. C2×Q16×D151C2semidirect
4. #23752785C2semidirect
5. #23752377C2semidirect
6. #23752376C2semidirect
7. #23752373C2semidirect
8. #23752372C2semidirect
9. #23752369C2semidirect
10. #23752368C2semidirect
11. #23752366C2semidirect
12. #23752219C2semidirect
13. #23752207C2semidirect
14. #23752177C2semidirect
15. #451×C151C2semidirect
223. C151#3011semidirect
236. #451D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127872
C2×#180127712
C2×Q16×D1512
#237527852
#237523772
#237523762
#237523732
#237523722
#237523692
#237523682
#237523662
#237522192
#237522072
#237521772
#451×C1512
#3011151