Finite Group Databasebeta

← #100219

Subgroups — #100219

Order 512.

610 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22329C211yesmaximal, characteristic
2#21454C211yesmaximal, characteristic
3#8311C211yesmaximal, characteristic, complemented
4#8303C211yesmaximal, characteristic, complemented
5#7599C211yesmaximal, characteristic
6#7033C211yesmaximal, characteristic, complemented
7#7021C211yesmaximal, characteristic, complemented
8C22×C4×Q8C2211yescharacteristic
9C2×#448C411yescharacteristic
10C2×#440—21no—
11C4×#121C2211yescharacteristic
12#1954—21no—
13#1923—21no—
14#1916—21no—
15#1880—21no—
16#1874C411yescharacteristic
17#1869—21no—
18#1869—21no—
19#1863—21no—
20#1852C411yescharacteristic
21#1817C2211yescharacteristic
22#1817C2211yescharacteristic
23#1797C2211yescharacteristic
24#1787C411yescharacteristic
25#1763—21no—
26C8×C22⋊C4—21no—
27#1749C2211yescharacteristic
28C4×C22⋊C8C2211yescharacteristic
29C2×Q8⋊C8—21no—
30C2×Q8⋊C8—21no—
31C2×#324—21no—
32C2×#324—21no—
33#1271—21no—
34#1270—21no—
35C23×Q8C2×C411yescharacteristic
36C22×SD16—21no—
37C23×C8—21no—
38C2×#121C2×C411yescharacteristic
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—42no—
44C2×C4×Q8—42no—
45C2×C4×D4—21no—
46C2×C4×D4D411yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C42—21no—
50C22×C42C2311yescharacteristic
51#448—22no—
52#448C814yescomplemented
53#440—42no—
54#440—42no—
55C2×#107—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×#103—21no—
59C2×#103—21no—
60C2×#103—21no—
61C2×#102—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8C2×C411yescharacteristic
69C2×C22⋊C8C2×C411yescharacteristic
70C2×C22⋊C8D411yescharacteristic
71C2×C22⋊C8D411yescharacteristic
72C2×#97—21no—
73C2×C4×C8—21no—
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76#390D411yescharacteristic
77#389D411yescharacteristic
78#388—21no—
79#386—21no—
80#380C2×C411yescharacteristic
81C4×C4⋊C4D411yescharacteristic
82C4×C22⋊C4—21no—
83#340—21no—
84#340—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90#336—21no—
91#336—21no—
92#336—21no—
93#336—21no—
94Q8⋊C8—42no—
95Q8⋊C8—42no—
96#324—42no—
97#324—42no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—42no—
104C22×Q8—42no—
105C22×Q8C2×C812yescomplemented
106C22×D4—21no—
107C22×D4C22⋊C411yescharacteristic
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4#4111yescharacteristic
114C2×SD16—24no—
115C2×SD16—42no—
116C2×SD16—42no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—42no—
124C22×C8—41no—
125C22×C8—41no—
126C22×C8—81no—
127C22×C8—81no—
128#121—21no—
129#121—21no—
130#121C2×C812yes—
131#121M1612yes—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—42no—
141C4×Q8—42no—
142C4×Q8—42no—
143C4×Q8—42no—
144C4×D4—41no—
145C4×D4—41no—
146C4×D4—81no—
147C4×D4—81no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C42—21no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42#4111yescharacteristic
177#107—42no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—81no—
181#103—24no—
182#103—42no—
183#103—42no—
184#102—42no—
185C22⋊C8—21no—
186C22⋊C8—21no—
187C22⋊C8—21no—
188C22⋊C8—42no—
189C22⋊C8—42no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—81no—
195C22⋊C8—81no—
196C22⋊C8—81no—
197C22⋊C8C2×C812yescomplemented
198#97—41no—
199#97—41no—
200C4×C8—41no—
201C4×C8—41no—
202C4×C8—81no—
203#95—21no—
204#95—21no—
205#95—21no—
206C24—21no—
207C24C4×D411yescharacteristic
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—44no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—21no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—81no—
230C2×D4—81no—
231C2×D4C22⋊C812yes—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12311yescharacteristic
269C22×C4C22×C811yescharacteristic
270C22×C4C2×M1611yescharacteristic
271C22×C4#13111yescharacteristic
272C22×C4#13111yescharacteristic
273SD16—48no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—84no—
293C2×C8—81no—
294C2×C8—81no—
295C2×C8—81no—
296C2×C8—81no—
297C2×C8—81no—
298C2×C8—81no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—81no—
311C4⋊C4C22⋊C812yes—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—81no—
317C22⋊C4—81no—
318C22⋊C4—81no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—42no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23#38611yescharacteristic
345C23C8×D411yescharacteristic
346C23#43511yescharacteristic
347Q8—24no—
348Q8—24no—
349Q8—42no—
350Q8—42no—
351Q8—42no—
352Q8—42no—
353Q8—42no—
354Q8—42no—
355D4—42no—
356D4—81no—
357D4—82no—
358D4—82no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—82no—
406C2×C4C2×C22⋊C811yescharacteristic
407C2×C4#40811yescharacteristic
408C2×C4C8×D411yescharacteristic
409C2×C4#43511yescharacteristic
410C2×C4B(4,𝔽2)11yescharacteristic
411C2×C4B(4,𝔽2)11yescharacteristic
412C2×C4#47111yescharacteristic
413C2×C4#47311yescharacteristic
414C8—42no—
415C8—42no—
416C8—81no—
417C8—81no—
418C8—81no—
419C8—81no—
420C8—161no—
421C8—161no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22#186911yescharacteristic
440C22#188911yescharacteristic
441C22#189111yescharacteristic
442C22#190211yescharacteristic
443C22#190411yescharacteristic
444C22#190611yescharacteristic
445C22#190811yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—44no—
455C4—41no—
456C4—41no—
457C4—42no—
458C4—81no—
459C4—82no—
460C4—81no—
461C2—41no—
462C2—41no—
463C2—82no—
464C2#896511yescharacteristic
465C2#908711yescharacteristic
466C2#909611yescharacteristic
467C2#951911yescharacteristic
468C2#952311yescharacteristic
469C2#952711yescharacteristic
470C2#953111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8311C2semidirect
4. #8303C2semidirect
6. #7033C2semidirect
7. #7021C2semidirect
52. #448C8semidirect
105. C22×Q8C2×C8semidirect
197. C22⋊C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#223292
#214542
#83112
#83032
#75992
#70332
#70212