Finite Group Databasebeta

← #39176

Subgroups — #39176

Order 256.

747 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2864C211yesmaximal, characteristic
2#2806C212yesmaximal
3#2802C212yesmaximal
4#2799C212yesmaximal
5#2713C212yesmaximal
6#2647C211yesmaximal, characteristic
7#2647C211yesmaximal, characteristic
8#2588C212yesmaximal
9#2585C212yesmaximal
10#2570C211yesmaximal, characteristic
11#2570C211yesmaximal, characteristic
12#2552C211yesmaximal, characteristic
13#2551C211yesmaximal, characteristic
14#2550C211yesmaximal, characteristic
15#2550C211yesmaximal, characteristic
16#2462C212yesmaximal
17#2446C212yesmaximal
18#2355C211yesmaximal, characteristic
19#2355C211yesmaximal, characteristic
20#2331C211yesmaximal, characteristic
21#2330C211yesmaximal, characteristic
22C4×C4⋊Q8C211yesmaximal, characteristic
23#2295C211yesmaximal, characteristic
24C2×C4⋊Q8C2211yescharacteristic
25C2×C4⋊Q8C2211yescharacteristic
26C2×C4⋊Q8C2212yes—
27C2×#125C2211yescharacteristic
28C2×#125C2212yes—
29C2×C4×Q8C2212yes—
30C2×C4×Q8C2211yescharacteristic
31#400C2212yes—
32#400C2212yes—
33#400C2212yes—
34#400C2212yes—
35#400C2212yes—
36#400C2212yes—
37#398C2212yes—
38#398C2212yes—
39#398C2212yes—
40#398C2212yes—
41#398C2212yes—
42#398C2212yes—
43#398C2212yes—
44#398C2212yes—
45#398C2212yes—
46#398C2212yes—
47#395C2212yes—
48#395C2212yes—
49#395C2212yes—
50#395C2212yes—
51#391C2212yes—
52#391C2212yes—
53#391C2212yes—
54#391C2212yes—
55#391C2212yes—
56#391C2212yes—
57#391C2212yes—
58#391C2211yescharacteristic
59#391C2211yescharacteristic
60#389C2211yescharacteristic
61#389C2211yescharacteristic
62#389C2212yes—
63#389C2211yescharacteristic
64#389C2211yescharacteristic
65#389C2211yescharacteristic
66#389C2211yescharacteristic
67#389C2212yes—
68#389C2212yes—
69#389C2212yes—
70#389C2211yescharacteristic
71#389C2212yes—
72#389C2212yes—
73#389C2211yescharacteristic
74#389C2212yes—
75#389C2211yescharacteristic
76#389C2211yescharacteristic
77#389C2211yescharacteristic
78#389C2211yescharacteristic
79#387C2212yes—
80#387C2212yes—
81#387C2212yes—
82#387C2212yes—
83#387C2211yescharacteristic
84#387C2212yes—
85#387C2212yes—
86#387C2212yes—
87#387C2212yes—
88#387C2212yes—
89#387C2212yes—
90#387C2211yescharacteristic
91#387C2212yes—
92#387C2212yes—
93#387C2212yes—
94#387C2212yes—
95#387C2212yes—
96#387C2212yes—
97#387C2211yescharacteristic
98#387C2211yescharacteristic
99#384C2211yescharacteristic
100#384C2211yescharacteristic
101#384C2211yescharacteristic
102#384C2211yescharacteristic
103#382C2211yescharacteristic
104#382C2211yescharacteristic
105#382C2211yescharacteristic
106#382C2211yescharacteristic
107#382C2211yescharacteristic
108C4×C4⋊C4C2211yescharacteristic
109C4×C4⋊C4C2212yes—
110C4×C4⋊C4C2211yescharacteristic
111C4×C4⋊C4C2211yescharacteristic
112C4×C4⋊C4C2211yescharacteristic
113C4×C4⋊C4C2211yescharacteristic
114C4×C4⋊C4C2212yes—
115C4×C4⋊C4C2211yescharacteristic
116C4×C4⋊C4C2211yescharacteristic
117C4×C4⋊C4C2211yescharacteristic
118C4×C4⋊C4C2212yes—
119C4×C4⋊C4C2211yescharacteristic
120#376C2211yescharacteristic
121#376C2211yescharacteristic
122C43C2211yescharacteristic
123C22×Q8C2312yes—
124C22×Q8C2312yes—
125C22×Q8C2311yescharacteristic
126C4⋊Q8—24no—
127C4⋊Q8—24no—
128C4⋊Q8—44no—
129#125—24no—
130#125—44no—
131C4×Q8—44no—
132C4×Q8Q818yescomplemented
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2312yes—
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2312yes—
143C2×C4⋊C4C2312yes—
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2311yescharacteristic
146C2×C4⋊C4C2311yescharacteristic
147C2×C4⋊C4C2311yescharacteristic
148C2×C4⋊C4C2311yescharacteristic
149C2×C4⋊C4C2311yescharacteristic
150C2×C4⋊C4C2312yes—
151C2×C4⋊C4C2312yes—
152C2×C4⋊C4C2312yes—
153C2×C4⋊C4C2312yes—
154C2×C4⋊C4C2312yes—
155C2×C4⋊C4C2312yes—
156C2×C4⋊C4C2312yes—
157C2×C4⋊C4C2312yes—
158C2×C4⋊C4C2312yes—
159C2×C4⋊C4C2312yes—
160C2×C4⋊C4C2312yes—
161C2×C4⋊C4C2311yescharacteristic
162C2×C4⋊C4C2311yescharacteristic
163C2×C4⋊C4C2312yes—
164C2×C4⋊C4C2312yes—
165C2×C4⋊C4C2312yes—
166C2×C4⋊C4C2312yes—
167C2×C4⋊C4C2312yes—
168C2×C4⋊C4C2311yescharacteristic
169C2×C4⋊C4C2311yescharacteristic
170C2×C4⋊C4C2311yescharacteristic
171C2×C4⋊C4C2311yescharacteristic
172C2×C42C2311yescharacteristic
173C2×C42C2311yescharacteristic
174C2×C42C2311yescharacteristic
175C2×C42C2311yescharacteristic
176C2×C42C2311yescharacteristic
177C2×C42C2311yescharacteristic
178C2×C42C2312yes—
179C2×C42C2312yes—
180C2×C42C2312yes—
181C2×C42C2312yes—
182C2×C42C2311yescharacteristic
183C2×C42C2311yescharacteristic
184C2×C42C2312yes—
185C2×C42C2312yes—
186C2×C42C2311yescharacteristic
187C2×C42C2311yescharacteristic
188C2×C42C2311yescharacteristic
189#95C2311yescharacteristic
190#95C2311yescharacteristic
191#95C2311yescharacteristic
192#95C2311yescharacteristic
193#95C2311yescharacteristic
194#95C2311yescharacteristic
195#95C2311yescharacteristic
196#95C2311yescharacteristic
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2311yescharacteristic
202#95C2311yescharacteristic
203#95C2312yes—
204#95C2312yes—
205#95C2311yescharacteristic
206#95C2311yescharacteristic
207#95C2312yes—
208#95C2312yes—
209#95C2312yes—
210#95C2311yescharacteristic
211#95C2312yes—
212#95C2312yes—
213#95C2311yescharacteristic
214#95C2312yes—
215#95C2312yes—
216#95C2312yes—
217#95C2312yes—
218#95C2312yes—
219#95C2312yes—
220#95C2312yes—
221#95C2312yes—
222#95C2312yes—
223#95C2312yes—
224#95C2312yes—
225#95C2312yes—
226#95C2312yes—
227#95C2312yes—
228#95C2312yes—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8C2×Q814yes—
238C22×C4C2411yescharacteristic
239C22×C4C2412yes—
240C22×C4C2411yescharacteristic
241C22×C4C2411yescharacteristic
242C22×C4C2412yes—
243C22×C4C2411yescharacteristic
244C22×C4C2411yescharacteristic
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2411yescharacteristic
251C22×C4C2411yescharacteristic
252C22×C4C2411yescharacteristic
253C22×C4C2411yescharacteristic
254C22×C4C2411yescharacteristic
255C22×C4C2411yescharacteristic
256C22×C4C2411yescharacteristic
257C22×C4C2412yes—
258C22×C4C2411yescharacteristic
259C22×C4C2411yescharacteristic
260C22×C4C2411yescharacteristic
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4C2×Q814yes—
297C4⋊C4C2×Q814yes—
298C4⋊C4C2×Q814yes—
299C4⋊C4#4114yes—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—42no—
307C42—42no—
308C42—41no—
309C42—41no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42C2×Q814yes—
315C42C2×Q814yes—
316C42C2×Q814yes—
317C23C2511yescharacteristic
318Q8—44no—
319Q8—48no—
320Q8—84no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4C22×Q812yes—
379C2×C4C22×Q812yes—
380C2×C4C22×Q812yes—
381C2×C4C22×Q812yes—
382C2×C4C22×Q812yes—
383C2×C4C22×Q812yes—
384C2×C4C22×Q812yes—
385C2×C4C2×#4114yes—
386C2×C4C2×#4112yes—
387C2×C4H(5,𝔽2)12yes—
388C2×C42−1+412yes—
389C2×C42−1+412yes—
390C22C23×Q811yescharacteristic
391C22C22×#4111yescharacteristic
392C22C2×H(5,𝔽2)11yescharacteristic
393C22C2×2−1+411yescharacteristic
394C22C2×2−1+411yescharacteristic
395C22#58511yescharacteristic
396C22#58511yescharacteristic
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—22no—
403C4—24no—
404C4—22no—
405C4—22no—
406C4—41no—
407C4—42no—
408C4—42no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—42no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—41no—
418C4—42no—
419C4#55714yes—
420C2#345811yescharacteristic
421C2C2×#55711yescharacteristic
422C2#350311yescharacteristic
423C2#351311yescharacteristic
424C2#351611yescharacteristic
425C2#355411yescharacteristic
426C2#355511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
132. C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#28642
#28062
#28022
#27992
#27132
#26472
#25882
#25852
#25702
#25522
#25512
#25502
#24622
#24462
#23552
#23312
#23302
C4×C4⋊Q82
#22952