Finite Group Databasebeta

← #10689290

Subgroups — #10689290

Order 768.

464 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 93 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#78604C211yesmaximal, characteristic
2C3×#1851C211yesmaximal, characteristic, complemented
3#72001C211yesmaximal, characteristic, complemented
4#71984C211yesmaximal, characteristic
5#69443C211yesmaximal, characteristic
6#67677C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#8874—31nomaximal, Sylow
9C22×C3×M16C2211yescharacteristic
10#5791C2211yescharacteristic
11C2×C3×C4⋊C8—21no—
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14C4×#217C2211yescharacteristic
15#5207C2211yescharacteristic
16C2×#830C2211yescharacteristic
17C2×#825—21no—
18C2×#822—21no—
19#5111—21no—
20C8×#67—21no—
21#4729—21no—
22#4728—21no—
23#4727—21no—
24#4720—21no—
25C3×#336—21no—
26#4558—21no—
27#4555—21no—
28#4452C411yescharacteristic, complemented
29#4452C411yescharacteristic, complemented
30#4451C411yescharacteristic, complemented
31#4451C411yescharacteristic, complemented
32#2976—31no—
33#1871—31no—
34#1855—31no—
35#1851D311yescharacteristic, complemented
36#1501—31no—
37#1297—31no—
38#1282—31no—
39C23×C3×C4C2311yescharacteristic
40C2×#235D411yescharacteristic
41C2×C3×M16—21no—
42C2×C3×M16—21no—
43C2×C3×M16—42no—
44C22×C3×C8—21no—
45C22×C3×C8D411yescharacteristic
46C22×C3×C8D411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C42—21no—
49#953—21no—
50C2×#217—21no—
51#930C2×C411yescharacteristic
52#930Q811yescharacteristic
53#929—21no—
54C4×#67—21no—
55C4×#67—21no—
56#926D411yescharacteristic
57#924C2×C411yescharacteristic
58#924Q811yescharacteristic
59C2×C4×Dic3—21no—
60C2×#208D411yescharacteristic
61C2×#208D411yescharacteristic
62C22×C3⋊C8—21no—
63C2×ΓL(1,𝔽25)—21no—
64C2×D3×C8—21no—
65C3×C4⋊C8—42no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95C2×C411yescharacteristic
69C3×#95C2×C411yescharacteristic
70C3⋊#95—21no—
71#830—21no—
72#830—21no—
73#830—21no—
74#830C2×C411yescharacteristic
75#830C2×C411yescharacteristic
76#825—42no—
77#822—42no—
78(C2×D3)⋊C8—21no—
79(C2×D3)⋊C8—21no—
80#815—21no—
81Dic3⋊C8—21no—
82Dic3⋊C8—21no—
83C8×Dic3—21no—
84C22×M16C2×D311yescharacteristic
85#525—31no—
86#436—61no—
87#435—61no—
88#435—61no—
89C8×D4—61no—
90#433—61no—
91C2×#108—61no—
92C2×C4⋊C8—21no—
93C2×#104—61no—
94#408—61no—
95#407—31no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8C2×D311yescharacteristic
98C4×C22⋊C4—31no—
99C4×C22⋊C4C2×D311yescharacteristic
100#341—61no—
101#339—61no—
102#336—21no—
103#328—31no—
104#328—31no—
105#327—31no—
106#327—31no—
107C24×C3#4111yescharacteristic
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×Q811yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C2×#67—21no—
119C2×#67—21no—
120C2×#67—21no—
121C22×Dic3—41no—
122#235—41no—
123#235—41no—
124C2×D12C4⋊C411yescharacteristic
125C2×C4×D3—21no—
126C2×Dic6C4⋊C411yescharacteristic
127C3×M16—22no—
128C3×M16—22no—
129C3×M16—41no—
130C3×M16—41no—
131C2×C3×C8—22no—
132C2×C3×C8—21no—
133C2×C3×C8—21no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C3×C22⋊C4—42no—
139C3×C42—41no—
140C3×C42—41no—
141#217—21no—
142#217—21no—
143#217—21no—
144#217—22no—
145#217—41no—
146(C2×D3)⋊C4—21no—
147(C2×D3)⋊C4—21no—
148C4⋊Dic3C4⋊C411yescharacteristic
149C4⋊Dic3C4⋊C411yescharacteristic
150Dic3⋊C4—21no—
151Dic3⋊C4—21no—
152C4×Dic3—21no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155#208—41no—
156#208—41no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163ΓL(1,𝔽25)—81no—
164D3×C8—81no—
165C2×#41—31no—
166C23×C4C22×D311yescharacteristic
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—31no—
170C2×M16—31no—
171C2×M16—42no—
172C2×M16—61no—
173C22×C8—21no—
174C22×C8—61no—
175C22×C8—61no—
176C22×C8#6711yescharacteristic
177C22×C8#6711yescharacteristic
178#123—61no—
179C22⋊Q8—31no—
180C22⋊Q8—31no—
181#121—31no—
182#121—31no—
183#120—61no—
184C4×D4—61no—
185C4×D4—61no—
186#117—31no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—61no—
189C2×C42—21no—
190C2×C42—61no—
191#108—122no—
192C4⋊C8—42no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195#104—122no—
196C22⋊C8—21no—
197C22⋊C8—21no—
198C22⋊C8—31no—
199C22⋊C8—31no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203C22⋊C8—61no—
204C22⋊C8—61no—
205#97—61no—
206C4×C8—61no—
207#95—61no—
208#95C4×D311yescharacteristic
209#95C4×D311yescharacteristic
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3C4×D411yescharacteristic
215C23×C3C22⋊Q811yescharacteristic
216C23×C3#12311yescharacteristic
217C22×D3—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4C2×C4⋊C411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4C22⋊Q811yescharacteristic
237#67—41no—
238#67—41no—
239#67—81no—
240#67—81no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—42no—
245C2×Dic3—81no—
246C2×Dic3—81no—
247D12—41no—
248C4×D3—41no—
249C4×D3—41no—
250Dic6—41no—
251C3×C8—41no—
252C3×C8—42no—
253C3×C8—41no—
254C3⋊C8—41no—
255C3⋊C8—41no—
256C3⋊C8—41no—
257C3⋊C8—41no—
258C24#23511yescharacteristic
259#41—121no—
260#41—121no—
261C2×Q8—31no—
262C2×D4—31no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—61no—
271C22×C4—121no—
272C22×C4C2×C4×D311yescharacteristic
273C22×C4D3×D411yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4D3×Q811yescharacteristic
276C22×C4#23911yescharacteristic
277C22×C4C2×#6711yescharacteristic
278M16—22no—
279M16—22no—
280M16—41no—
281M16—41no—
282M16—121no—
283M16—121no—
284M16—241no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—61no—
293C2×C8—61no—
294C2×C8—61no—
295C2×C8—61no—
296C2×C8—121no—
297C2×C8—121no—
298C2×C8—241no—
299C4⋊C4—31no—
300C4⋊C4—31no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C22⋊C4—42no—
304C22⋊C4—61no—
305C22⋊C4—61no—
306C22⋊C4—61no—
307C22⋊C4—61no—
308C22⋊C4—61no—
309C22⋊C4—62no—
310C22⋊C4—121no—
311C42—41no—
312C42—41no—
313C42—61no—
314C42—121no—
315C42—121no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3#38511yescharacteristic
327C22×C3#44311yescharacteristic
328C22×C3#44411yescharacteristic
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—41no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—42no—
336C3×C4—81no—
337C3×C4B(4,𝔽2)11yescharacteristic
338C3×C4#45811yescharacteristic
339Dic3—41no—
340Dic3—41no—
341Dic3—41no—
342Dic3—82no—
343C23—21no—
344C23—21no—
345C23—41no—
346C23—41no—
347C23—61no—
348C23#89411yescharacteristic
349C23#89611yescharacteristic
350C23C4×#6711yescharacteristic
351Q8—121no—
352D4—121no—
353D4—121no—
354D4—121no—
355D4—241no—
356D4—241no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—122no—
378C2×C4—241no—
379C2×C4—241no—
380C2×C4D3×C4⋊C411yescharacteristic
381C2×C4#89311yescharacteristic
382C2×C4#93711yescharacteristic
383C2×C4#94611yescharacteristic
384C8—42no—
385C8—41no—
386C8—41no—
387C8—121no—
388C8—121no—
389C8—121no—
390C8—121no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3#186111yescharacteristic
396C2×C3#186611yescharacteristic
397C2×C3#186711yescharacteristic
398D3—81no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—81no—
408C22—81no—
409C22—121no—
410C22—121no—
411C22—121no—
412C22#498911yescharacteristic
413C22#514411yescharacteristic
414C22#514511yescharacteristic
415C4—21no—
416C4—21no—
417C4—21no—
418C4—42no—
419C4—81no—
420C4—121no—
421C4—121no—
422C4—121no—
423C4—242no—
424C4#503211yescharacteristic
425C4#505311yescharacteristic
426C3#887411yescharacteristic, complemented, Sylow
427C2—21no—
428C2—21no—
429C2—41no—
430C2—41no—
431C2—241no—
432C2#6959511yescharacteristic
433C2#7053011yescharacteristic
434C2#7054511yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1851C2
socleC22×C3#385
prime radical#5791C22
cosocleC23×C3×C4C23
hypercenter#1851D3
hypocenterC3#8874
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
2. C3×#1851C2semidirect
3. #72001C2semidirect
6. #67677C2semidirect
7. #67618C2semidirect
28. #4452C4semidirect
29. #4452C4semidirect
30. #4451C4semidirect
31. #4451C4semidirect
35. #1851D3semidirect
426. C3#8874semidirect

Maximal subgroups

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

subgroupindex
#786042
C3×#18512
#720012
#719842
#694432
#676772
#676182
#88743