Finite Group Databasebeta

← #224620

Subgroups — #224620

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36584C211yesmaximal, characteristic
2#21345C211yesmaximal, characteristic
3#20520C211yesmaximal, characteristic
4#9690C212yesmaximal
5#9686C211yesmaximal, characteristic
6#9686C211yesmaximal, characteristic
7#9686C212yesmaximal
8#8582C212yesmaximal
9#8580C211yesmaximal, characteristic
10#8227C212yesmaximal
11C4×#330C211yesmaximal, characteristic
12C2×#446—21no—
13#2859C2211yescharacteristic
14#2859C2212yes—
15#2571C411yescharacteristic
16#2571C411yescharacteristic
17#2571C412yes—
18#2571C412yes—
19#2509—22no—
20#2509—21no—
21#2509—21no—
22#2509—22no—
23#2351—21no—
24#2351—21no—
25Q8×C4⋊C4C411yescharacteristic
26Q8×C4⋊C4C411yescharacteristic
27C4×#125C2211yescharacteristic
28C4×#125C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1771—22no—
34C8×C4⋊C4—21no—
35#1767C2211yescharacteristic
36#1767C2212yes—
37#1767C2211yescharacteristic
38#1767C2212yes—
39#1745C2212yes—
40C4×#97C2211yescharacteristic
41C4×#97C2211yescharacteristic
42C4×#97C2212yes—
43C2×#330C2211yescharacteristic
44C2×#330C2211yescharacteristic
45C2×#330C2212yes—
46C2×#330C2212yes—
47C2×#330C2212yes—
48C2×#330C2212yes—
49C2×#330C2212yes—
50#1295C2211yescharacteristic
51#1295C2211yescharacteristic
52#1295C2212yes—
53C2×#125C2311yescharacteristic
54C2×#125C2312yes—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60#446—28no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—21no—
65C2×#97—22no—
66C2×#97—21no—
67C2×#97—21no—
68C2×#97C2311yescharacteristic
69C2×#97C2311yescharacteristic
70C2×#97C2312yes—
71C2×#97C2312yes—
72C2×#97C2311yescharacteristic
73C2×#97C2311yescharacteristic
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—22no—
77#400—22no—
78#400—22no—
79#400—22no—
80#400—22no—
81#400—22no—
82#400—22no—
83#389—22no—
84#389—21no—
85#389—21no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—21no—
97#389—21no—
98#389—22no—
99#389C2×C411yescharacteristic
100#389C2×C411yescharacteristic
101#389C2×C412yes—
102#389C2×C412yes—
103#387—22no—
104#387—21no—
105#387—21no—
106#387—22no—
107#384C2×C411yescharacteristic
108#384C2×C411yescharacteristic
109#384C2×C412yes—
110#384C2×C412yes—
111#382—22no—
112#382—21no—
113#382—21no—
114#382—22no—
115#382C2×C412yes—
116#382C2×C412yes—
117#382C2×C412yes—
118#382C2311yescharacteristic
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D412yes—
133C43C2311yescharacteristic
134C43C2312yes—
135#336—21no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—21no—
141#330—22no—
142#330—22no—
143#330—24no—
144#330—24no—
145#330—42no—
146#330—42no—
147#330—42no—
148C22×Q8C22⋊C411yescharacteristic
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155#125—21no—
156#125—21no—
157#125—22no—
158#125—22no—
159#125—21no—
160#125—22no—
161#125—21no—
162#125—22no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—42no—
166C4×Q8—44no—
167C4×Q8—44no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C412yes—
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2411yescharacteristic
213C4⋊C8—24no—
214C4⋊C8—28no—
215C4⋊C8—42no—
216C4⋊C8—44no—
217#97—28no—
218#97—21no—
219#97—21no—
220#97—22no—
221#97—22no—
222#97—22no—
223#97—21no—
224#97—21no—
225#97—22no—
226#97—21no—
227#97—21no—
228#97—42no—
229#97—42no—
230#97#4112yes—
231#97#4112yes—
232C4×C8—24no—
233C4×C8—42no—
234C4×C8—44no—
235#95—21no—
236#95—22no—
237#95—21no—
238#95—22no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—21no—
250#95—21no—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8#10414yes—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—44no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—81no—
323C4⋊C4—81no—
324C4⋊C4—82no—
325C4⋊C4—81no—
326C4⋊C4—81no—
327C4⋊C4#10114yes—
328C42—24no—
329C42—22no—
330C42—22no—
331C42—24no—
332C42—22no—
333C42—42no—
334C42—41no—
335C42—41no—
336C42—42no—
337C42—41no—
338C42—42no—
339C42—42no—
340C42—44no—
341C42—44no—
342C42—42no—
343C42—42no—
344C42C2×#4111yescharacteristic
345C42C2×#4111yescharacteristic
346C42H(5,𝔽2)11yescharacteristic
347C422−1+411yescharacteristic
348C23C22×C22⋊C411yescharacteristic
349Q8—28no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4C2×#10114yes—
394C2×C4C2×#10112yes—
395C2×C4C2×#10414yes—
396C2×C4C2×#10412yes—
397C2×C4#42112yes—
398C2×C4C4×#4111yescharacteristic
399C2×C4#52011yescharacteristic
400C2×C4#54811yescharacteristic
401C2×C4#54811yescharacteristic
402C2×C4#55011yescharacteristic
403C2×C4#55011yescharacteristic
404C8—24no—
405C8—42no—
406C8—44no—
407C8—42no—
408C8—42no—
409C8—82no—
410C22#234111yescharacteristic
411C22C22×#10111yescharacteristic
412C22#288911yescharacteristic
413C22C22×#10411yescharacteristic
414C22C2×#42111yescharacteristic
415C22#290111yescharacteristic
416C22#290111yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—24no—
424C4—24no—
425C4—41no—
426C4—42no—
427C4—41no—
428C4—41no—
429C4—42no—
430C4—81no—
431C4—82no—
432C4—81no—
433C4—82no—
434C4—81no—
435C4—81no—
436C4#152814yes—
437C2C2×#152811yescharacteristic
438C2#1765411yescharacteristic
439C2#1768311yescharacteristic
440C2#1768911yescharacteristic
441C2#2253811yescharacteristic
442C2#2255011yescharacteristic
443C2#2255111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#365842
#213452
#205202
#96902
#96862
#85822
#85802
#82272
C4×#3302