Finite Group Databasebeta

← #72341665

Subgroups — #72341665

Order 1536.

1698 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 1698 conjugacy classes of subgroups — too many to draw. Only its 54 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#11715659C212yesmaximal
2C2×#85666C211yesmaximal, characteristic
3#392195—31nomaximal, Sylow
4C2×#5950C2211yescharacteristic
5#85666C412yescomplemented
6#85564—22no—
7#73110—43nomaximal
8C2×#3573D311yescharacteristic
9C2×#1812—33no—
10#20662—33no—
11#20610—32no—
12#10742—36no—
13#5950C2×C411yescharacteristic
14#5950Q811yescharacteristic
15C2×#997D411yescharacteristic
16C2×#131⋊C3—81no—
17C2×C8×A4—43no—
18C4⋊SL(2,ℤ/4)—46no—
19C2×#585C2×D311yescharacteristic
20#3573—36no—
21#3573—62no—
22#3573Dic312yescomplemented
23C2×#568—33no—
24C2×#568—61no—
25C22×#131—33no—
26C22×#131—61no—
27C2×#427—33no—
28C22×#106—31no—
29C2×#410—36no—
30#2311—36no—
31#1812—66no—
32#1812—66no—
33#1809—63no—
34#1809—63no—
35#1807—63no—
36#1807—63no—
37#1790—66no—
38#1790—62no—
39#1736—36no—
40C2×H(3,ℤ/4)—36no—
41#997C4⋊C411yescharacteristic
42C2×#41⋊C3—81no—
43C2×C4×A4—43no—
44C2×SL(2,ℤ/4)—86no—
45#131⋊C3—82no—
46C8×A4—46no—
47#817—161no—
48#585—33no—
49#585—33no—
50#585—61no—
51#585—61no—
52#585Dic611yescharacteristic
53#585C2×Dic311yescharacteristic
54C2×2−1+4—61no—
55C2×H(5,𝔽2)—33no—
56C2×H(5,𝔽2)D1211yescharacteristic
57C22×#41—33no—
58C22×#41—61no—
59#568—36no—
60#568—36no—
61#568—62no—
62#568—63no—
63#568—612no—
64#568—66no—
65#568—66no—
66#568—63no—
67#568—62no—
68C2×#131—21no—
69C2×#131—36no—
70C2×#131—36no—
71C2×#131—36no—
72C2×#131—36no—
73C2×#131—62no—
74C2×#131—63no—
75C2×#131—66no—
76C2×#131—612no—
77C2×#131—63no—
78C2×#131—66no—
79C2×#131—66no—
80C2×#131—62no—
81C2×#131—61no—
82C22×M16—33no—
83C22×M16—33no—
84C22×M16—33no—
85C22×M16—63no—
86C23×C8—31no—
87C23×C8—33no—
88C23×C8S413yes—
89C2×#117—36no—
90C22×C4⋊C4—32no—
91#427—612no—
92C2×#107—63no—
93C2×#106—63no—
94C2×#106—63no—
95C2×#106—63no—
96#410—66no—
97#410—66no—
98#410—66no—
99#410—66no—
100C2×#99—66no—
101C2×#99—66no—
102#391—66no—
103#390—66no—
104#386—66no—
105#386—66no—
106#380—66no—
107#380—66no—
108#340—66no—
109#340—66no—
110H(3,ℤ/4)—612no—
111C22×A4—43no—
112#41⋊C3—81no—
113#41⋊C3—81no—
114C2×SL(2,𝔽3)—81no—
115C4×A4—43no—
116C4×A4—43no—
117SL(2,ℤ/4)—166no—
118C2×C3×C8—161no—
119C4⋊Dic3—162no—
1202−1+4—61no—
1212−1+4—61no—
1222−1+4—66no—
1232−1+4—63no—
1242−1+4—63no—
1252−1+4—62no—
126H(5,𝔽2)—33no—
127H(5,𝔽2)—33no—
128H(5,𝔽2)—33no—
129H(5,𝔽2)—36no—
130H(5,𝔽2)—61no—
131H(5,𝔽2)—63no—
132H(5,𝔽2)—66no—
133H(5,𝔽2)—66no—
134H(5,𝔽2)—63no—
135H(5,𝔽2)C4⋊Dic311yescharacteristic
136C2×#41—21no—
137C2×#41—33no—
138C2×#41—33no—
139C2×#41—33no—
140C2×#41—33no—
141C2×#41—33no—
142C2×#41—33no—
143C2×#41—33no—
144C2×#41—33no—
145C2×#41—61no—
146C2×#41—61no—
147C2×#41—61no—
148C2×#41—63no—
149C2×#41—66no—
150C2×#41—63no—
151C2×#41—66no—
152C2×#41—66no—
153C2×#41—63no—
154C2×#41—63no—
155C2×#41—63no—
156C2×#41—63no—
157C2×#41—61no—
158C2×#41—61no—
159C22×Q8—33no—
160C22×Q8—61no—
161C22×D4—33no—
162C22×D4—33no—
163C22×D4—33no—
164C22×D4—63no—
165C23×C4—31no—
166C23×C4—33no—
167C23×C4—62no—
168C23×C4C2×S413yes—
169#131—22no—
170#131—66no—
171#131—612no—
172#131—66no—
173#131—612no—
174#131—612no—
175#131—66no—
176#131—612no—
177#131—612no—
178#131—612no—
179#131—62no—
180#131—66no—
181#131—66no—
182#131—62no—
183C2×M16—36no—
184C2×M16—36no—
185C2×M16—36no—
186C2×M16—36no—
187C2×M16—36no—
188C2×M16—36no—
189C2×M16—63no—
190C2×M16—66no—
191C2×M16—66no—
192C2×M16—66no—
193C2×M16—66no—
194C2×M16—66no—
195C2×M16—66no—
196C2×M16—66no—
197C2×M16—66no—
198C2×M16—61no—
199C22×C8—33no—
200C22×C8—32no—
201C22×C8—36no—
202C22×C8—33no—
203C22×C8—33no—
204C22×C8—36no—
205C22×C8—33no—
206C22×C8—36no—
207C22×C8—36no—
208C22×C8—63no—
209C22×C8—66no—
210C22×C8—61no—
211C22×C8SL(2,ℤ/4)16yescomplemented
212#117—612no—
213#117—612no—
214C2×C4⋊C4—66no—
215C2×C4⋊C4—66no—
216C2×C4⋊C4—66no—
217C2×C22⋊C4—66no—
218C2×C22⋊C4—126no—
219C2×C22⋊C4—126no—
220C2×C42—66no—
221#107—126no—
222#106—123no—
223#106—126no—
224#106—121no—
225#99—126no—
226#99—126no—
227#99—126no—
228#99—126no—
229#95—126no—
230#95—126no—
231C2×A4—46no—
232C2×A4—43no—
233C2×C3×C4—161no—
234C2×Dic3—322no—
235SL(2,𝔽3)—81no—
236C3×C8—162no—
237C24—31no—
238C24—33no—
239C24C4⋊S413yes—
240#41—21no—
241#41—21no—
242#41—61no—
243#41—63no—
244#41—66no—
245#41—63no—
246#41—63no—
247#41—66no—
248#41—63no—
249#41—66no—
250#41—66no—
251#41—66no—
252#41—66no—
253#41—63no—
254#41—66no—
255#41—66no—
256#41—66no—
257#41—66no—
258#41—63no—
259#41—63no—
260#41—66no—
261#41—63no—
262#41—61no—
263#41—63no—
264#41—63no—
265#41—66no—
266#41—61no—
267#41—61no—
268C2×Q8—21no—
269C2×Q8—33no—
270C2×Q8—33no—
271C2×Q8—33no—
272C2×Q8—33no—
273C2×Q8—63no—
274C2×Q8—66no—
275C2×Q8—61no—
276C2×Q8—61no—
277C2×Q8—63no—
278C2×Q8—63no—
279C2×D4—33no—
280C2×D4—33no—
281C2×D4—33no—
282C2×D4—33no—
283C2×D4—33no—
284C2×D4—33no—
285C2×D4—33no—
286C2×D4—33no—
287C2×D4—33no—
288C2×D4—33no—
289C2×D4—33no—
290C2×D4—33no—
291C2×D4—66no—
292C2×D4—66no—
293C2×D4—63no—
294C2×D4—66no—
295C2×D4—63no—
296C2×D4—63no—
297C2×D4—66no—
298C2×D4—66no—
299C2×D4—66no—
300C2×D4—61no—
301C2×D4—63no—
302C2×D4—63no—
303C22×C4—33no—
304C22×C4—33no—
305C22×C4—33no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—33no—
309C22×C4—33no—
310C22×C4—33no—
311C22×C4—33no—
312C22×C4—33no—
313C22×C4—33no—
314C22×C4—33no—
315C22×C4—33no—
316C22×C4—33no—
317C22×C4—63no—
318C22×C4—63no—
319C22×C4—63no—
320C22×C4—61no—
321C22×C4—126no—
322C22×C4—126no—
323C22×C4—482no—
324C22×C4A4⋊Q813yes—
325C22×C4C2×SL(2,ℤ/4)13yes—
326M16—66no—
327M16—66no—
328M16—612no—
329M16—612no—
330M16—62no—
331M16—62no—
332C2×C8—36no—
333C2×C8—36no—
334C2×C8—33no—
335C2×C8—36no—
336C2×C8—36no—
337C2×C8—36no—
338C2×C8—36no—
339C2×C8—63no—
340C2×C8—66no—
341C2×C8—66no—
342C2×C8—63no—
343C2×C8—61no—
344C2×C8—62no—
345C2×C8—62no—
346C2×C8#102011yescharacteristic
347C4⋊C4—126no—
348C4⋊C4—126no—
349C4⋊C4—126no—
350C4⋊C4—122no—
351C22⋊C4—1212no—
352C22⋊C4—486no—
353C22⋊C4—486no—
354C42—1212no—
355C22×C3—161no—
356A4—43no—
357C3×C4—161no—
358C3×C4—161no—
359Dic3—642no—
360C23—33no—
361C23—31no—
362C23—33no—
363C23—33no—
364C23—33no—
365C23—33no—
366C23—46no—
367C23—63no—
368C23—61no—
369C23—122no—
370C23—126no—
371C23C4⋊SL(2,ℤ/4)13yes—
372Q8—21no—
373Q8—63no—
374Q8—63no—
375Q8—66no—
376Q8—61no—
377D4—66no—
378D4—66no—
379D4—66no—
380D4—66no—
381D4—63no—
382D4—63no—
383D4—66no—
384D4—61no—
385D4—61no—
386D4—62no—
387C2×C4—36no—
388C2×C4—33no—
389C2×C4—33no—
390C2×C4—33no—
391C2×C4—33no—
392C2×C4—33no—
393C2×C4—33no—
394C2×C4—33no—
395C2×C4—33no—
396C2×C4—33no—
397C2×C4—33no—
398C2×C4—33no—
399C2×C4—63no—
400C2×C4—63no—
401C2×C4—63no—
402C2×C4—63no—
403C2×C4—63no—
404C2×C4—63no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—246no—
411C2×C4—246no—
412C2×C4—246no—
413C2×C4—246no—
414C2×C4—246no—
415C2×C4—242no—
416C2×C4C2×#102011yescharacteristic
417C8—36no—
418C8—62no—
419C8#593612yes—
420C2×C3—162no—
421C2×C3—161no—
422C22—33no—
423C22—33no—
424C22—43no—
425C22—66no—
426C22—61no—
427C22—61no—
428C22—123no—
429C22—123no—
430C22—121no—
431C22—123no—
432C22—123no—
433C22—122no—
434C22—243no—
435C22C4⋊#102011yescharacteristic
436C4—33no—
437C4—33no—
438C4—61no—
439C4—61no—
440C4—486no—
441C4—482no—
442C4#8750211yescharacteristic
443C4C2×#593611yescharacteristic
444C3—161noSylow
445C2—63no—
446C2—63no—
447C2—121no—
448C2—121no—
449C2C4⋊#593611yescharacteristic
450C2#1171573512yes—
subgroupG/H
center, socleC22C4⋊#1020
commutator#5950C2×C4
FrattiniC2×C4C2×#1020
FittingC2×#3573D3
prime radical, cosocleC2×#5950C22
hypercenterC2×C8#1020
hypocenter#997C4⋊C4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
5. #85666C4semidirect
22. #3573Dic3semidirect
211. C22×C8SL(2,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
#117156592
C2×#856662
#3921953
#731104