Finite Group Databasebeta

← #17598

Subgroups — #17598

Order 256.

498 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3520C211yesmaximal, characteristic
2#2974C211yesmaximal, characteristic, complemented
3#2318C211yesmaximal, characteristic, complemented
4#1521C211yesmaximal, characteristic, complemented
5#1520C211yesmaximal, characteristic, complemented
6#1518C211yesmaximal, characteristic, complemented
7#1516C211yesmaximal, characteristic, complemented
8#1515C211yesmaximal, characteristic, complemented
9#1513C211yesmaximal, characteristic, complemented
10#1512C211yesmaximal, characteristic, complemented
11#1511C211yesmaximal, characteristic, complemented
12#1508C211yesmaximal, characteristic, complemented
13#1504C211yesmaximal, characteristic, complemented
14#1501C211yesmaximal, characteristic, complemented
15C2×#328C211yesmaximal, characteristic, complemented
16#556C411yescharacteristic, complemented
17#554—21no—
18#552C411yescharacteristic, complemented
19#552C411yescharacteristic, complemented
20#551—21no—
21#551—21no—
22D4×Q8C411yescharacteristic, complemented
23#548C411yescharacteristic, complemented
24#548C411yescharacteristic, complemented
25#547C411yescharacteristic, complemented
26#547C411yescharacteristic, complemented
27#546—21no—
28#542—21no—
29#539—21no—
30#539—21no—
31#533C2211yescharacteristic
32#529C2211yescharacteristic
33#525C2211yescharacteristic
34#525C2211yescharacteristic
35C2×C22⋊Q8C2211yescharacteristic
36C2×C22⋊Q8C2211yescharacteristic
37#520—21no—
38C2×C4×D4C2211yescharacteristic
39#436—21no—
40#435—21no—
41#435—21no—
42C8×D4—21no—
43#432C2211yescharacteristic
44#431C2211yescharacteristic
45#407C2211yescharacteristic
46#407C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49#388—21no—
50#387—21no—
51#386—21no—
52#385—21no—
53#383C2211yescharacteristic
54#382C2211yescharacteristic
55#381C2211yescharacteristic
56#380C2211yescharacteristic
57C4×C22⋊C4C2211yescharacteristic
58C2×#95C2211yescharacteristic
59#328C2211yescharacteristic
60#328C2211yescharacteristic
61#328C2211yescharacteristic
62#328C2211yescharacteristic
63#328C2211yescharacteristic
64#328C2211yescharacteristic
65#328C2211yescharacteristic
66#328C2211yescharacteristic
67#328C2211yescharacteristic
68#328C2211yescharacteristic
69#328C2211yescharacteristic
70#328C2211yescharacteristic
71#327C2211yescharacteristic
72#327C2211yescharacteristic
73#327C2211yescharacteristic
74#327C2211yescharacteristic
75C2×#41—21no—
76C2×#41C2×C411yescharacteristic
77C2×#41C2×C411yescharacteristic, complemented
78C22×Q8C2×C411yescharacteristic
79C22×Q8C2×C411yescharacteristic, complemented
80C22×D4D411yescharacteristic
81C23×C4—21no—
82C23×C4C2311yescharacteristic
83C23×C4C2311yescharacteristic
84C2×M16—21no—
85C2×M16—21no—
86C22×C8—21no—
87C22×C8—21no—
88C4⋊Q8—21no—
89C4⋊Q8C2×C411yescharacteristic, complemented
90C4⋊Q8C2×C411yescharacteristic, complemented
91#126—21no—
92#126—21no—
93#126—21no—
94#126—21no—
95#126—21no—
96#125—21no—
97#125—21no—
98#125—21no—
99#125C2×C411yescharacteristic, complemented
100#124—21no—
101#124—21no—
102#124—21no—
103#124C2×C411yescharacteristic, complemented
104#123—21no—
105#123—21no—
106#123—21no—
107#123—21no—
108#123—21no—
109#123—21no—
110#123—21no—
111#123—21no—
112#123—21no—
113C22⋊Q8—21no—
114C22⋊Q8—21no—
115C22⋊Q8—21no—
116C22⋊Q8—21no—
117C22⋊Q8—21no—
118C22⋊Q8—21no—
119C22⋊Q8—21no—
120C22⋊Q8—21no—
121C22⋊Q8—21no—
122C22⋊Q8—21no—
123C22⋊Q8C2×C411yescharacteristic
124C22⋊Q8C2×C411yescharacteristic
125C22⋊Q8C2×C411yescharacteristic
126C22⋊Q8C2×C411yescharacteristic, complemented
127C22⋊Q8C2×C411yescharacteristic, complemented
128C22⋊Q8C2×C411yescharacteristic, complemented
129C22⋊Q8C2311yescharacteristic
130C22⋊Q8C2311yescharacteristic
131C22⋊Q8C2311yescharacteristic
132#121—21no—
133#121—21no—
134#121—21no—
135#121C2×C411yescharacteristic
136#121C2×C411yescharacteristic, complemented
137#121C2311yescharacteristic
138#120—21no—
139#120—21no—
140C4×Q8—21no—
141C4×Q8—21no—
142C4×Q8C2×C411yescharacteristic, complemented
143C4×Q8C2×C411yescharacteristic, complemented
144C4×Q8C2×C411yescharacteristic, complemented
145C4×D4—21no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—21no—
149C4×D4—21no—
150C4×D4—21no—
151C4×D4—21no—
152C4×D4—21no—
153C4×D4C2×C411yescharacteristic
154C4×D4C2×C411yescharacteristic
155C4×D4C2×C411yescharacteristic
156C4×D4C2×C411yescharacteristic
157C4×D4C2×C411yescharacteristic, complemented
158C4×D4D411yescharacteristic
159C4×D4D411yescharacteristic
160C4×D4D411yescharacteristic
161C4×D4D411yescharacteristic
162#117—21no—
163#117—21no—
164#117—21no—
165#117—21no—
166#117C2×C411yescharacteristic
167#117C2×C411yescharacteristic, complemented
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4C2×C411yescharacteristic
174C2×C4⋊C4C2×C411yescharacteristic, complemented
175C2×C4⋊C4D411yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4D411yescharacteristic
182C2×C22⋊C4D411yescharacteristic
183C2×C42—21no—
184C2×C42C2311yescharacteristic
185C4⋊C8—21no—
186C4⋊C8—21no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8C2311yescharacteristic
192C22⋊C8C2311yescharacteristic
193C22⋊C8C2311yescharacteristic
194C22⋊C8C2311yescharacteristic
195#97—21no—
196C4×C8—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95C2311yescharacteristic
203#95C2311yescharacteristic
204#95C2311yescharacteristic
205#95C2311yescharacteristic
206C24C2×D411yescharacteristic
207C24C2×D411yescharacteristic
208#41—44no—
209#41—81no—
210#41—81no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—81no—
215C2×Q8—81no—
216C2×Q8C22×C411yescharacteristic
217C2×Q8C22×C411yescharacteristic
218C2×Q8C22×C411yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—42no—
225C2×D4C22⋊C411yescharacteristic
226C2×D4C22⋊C411yescharacteristic
227C2×D4C22⋊C411yescharacteristic
228C2×D4C22⋊C411yescharacteristic
229C2×D4C22×C411yescharacteristic
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4C22×C411yescharacteristic
247C22×C4C22×C411yescharacteristic
248C22×C4C22×C411yescharacteristic
249C22×C4C22×C411yescharacteristic
250C22×C4C2×D411yescharacteristic
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D411yescharacteristic
253C22×C4C2×D411yescharacteristic
254C22×C4C2×D411yescharacteristic
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D411yescharacteristic
258C22×C4C2411yescharacteristic
259M16—81no—
260M16—81no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—81no—
266C2×C8—81no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—81no—
287C4⋊C4—81no—
288C4⋊C4C22⋊C411yescharacteristic
289C4⋊C4C22⋊C411yescharacteristic
290C4⋊C4C22⋊C411yescharacteristic
291C4⋊C4C22⋊C411yescharacteristic
292C4⋊C4C22×C411yescharacteristic
293C4⋊C4C22×C411yescharacteristic
294C4⋊C4C22×C411yescharacteristic
295C4⋊C4C22×C411yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—42no—
314C22⋊C4—81no—
315C22⋊C4—81no—
316C22⋊C4C22⋊C411yescharacteristic
317C22⋊C4C22⋊C411yescharacteristic
318C22⋊C4C22⋊C411yescharacteristic
319C22⋊C4C22⋊C411yescharacteristic
320C22⋊C4C22⋊C411yescharacteristic
321C22⋊C4C22⋊C411yescharacteristic
322C22⋊C4C22⋊C411yescharacteristic
323C22⋊C4C22⋊C411yescharacteristic
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—81no—
330C42C22×C411yescharacteristic
331C42C22×C411yescharacteristic
332C42C2×D411yescharacteristic
333C42C2×D411yescharacteristic
334C23—21no—
335C23—21no—
336C23—22no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23C2×C22⋊C411yescharacteristic
344C23C2×C22⋊C411yescharacteristic
345C23C2×C22⋊C411yescharacteristic
346C23C2×C22⋊C411yescharacteristic
347C23C22×D411yescharacteristic
348Q8—41no—
349Q8—41no—
350Q8—41no—
351Q8—42no—
352Q8—81no—
353Q8—81no—
354D4—22no—
355D4—22no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—42no—
360D4—41no—
361D4—41no—
362D4—42no—
363D4—81no—
364D4—81no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—84no—
406C2×C4C2×C22⋊C411yescharacteristic
407C2×C4C2×C22⋊C411yescharacteristic
408C2×C4C2×C22⋊C411yescharacteristic
409C2×C4C2×C22⋊C411yescharacteristic
410C2×C4C2×C22⋊C411yescharacteristic
411C2×C4C2×C22⋊C411yescharacteristic
412C2×C4C2×C22⋊C411yescharacteristic
413C2×C4C2×C22⋊C411yescharacteristic
414C2×C4C23×C411yescharacteristic
415C2×C4C22×D411yescharacteristic
416C8—41no—
417C8—41no—
418C8—41no—
419C8—41no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—42no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—81no—
439C22—81no—
440C22#42112yes—
441C22C22×C22⋊C411yescharacteristic
442C22#57711yescharacteristic
443C22#57811yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—81no—
465C2—21no—
466C2—21no—
467C2—22no—
468C2—41no—
469C2—41no—
470C2—41no—
471C2—81no—
472C2#288311yescharacteristic
473C2#289811yescharacteristic
474C2C2×#42111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3520C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2974C2semidirect
3. #2318C2semidirect
4. #1521C2semidirect
5. #1520C2semidirect
6. #1518C2semidirect
7. #1516C2semidirect
8. #1515C2semidirect
9. #1513C2semidirect
10. #1512C2semidirect
11. #1511C2semidirect
12. #1508C2semidirect
13. #1504C2semidirect
14. #1501C2semidirect
15. C2×#328C2semidirect
16. #556C4semidirect
18. #552C4semidirect
19. #552C4semidirect
22. D4×Q8C4semidirect
23. #548C4semidirect
24. #548C4semidirect
25. #547C4semidirect
26. #547C4semidirect
77. C2×#41C2×C4semidirect
79. C22×Q8C2×C4semidirect
89. C4⋊Q8C2×C4semidirect
90. C4⋊Q8C2×C4semidirect
99. #125C2×C4semidirect
103. #124C2×C4semidirect
126. C22⋊Q8C2×C4semidirect
127. C22⋊Q8C2×C4semidirect
128. C22⋊Q8C2×C4semidirect
136. #121C2×C4semidirect
142. C4×Q8C2×C4semidirect
143. C4×Q8C2×C4semidirect
144. C4×Q8C2×C4semidirect
157. C4×D4C2×C4semidirect
167. #117C2×C4semidirect
174. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#35202
#29742
#23182
#15212
#15202
#15182
#15162
#15152
#15132
#15122
#15112
#15082
#15042
#15012
C2×#3282