Finite Group Databasebeta

← #10691167

Subgroups — #10691167

Order 768.

560 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 91 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#83002C211yesmaximal, characteristic
2#79335C211yesmaximal, characteristic, complemented
3C3×#1527C211yesmaximal, characteristic, complemented
4#71718C211yesmaximal, characteristic, complemented
5C4×#823C211yesmaximal, characteristic, complemented
6#67752C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#9187—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5231C411yescharacteristic
14#5231C411yescharacteristic
15#5223C411yescharacteristic
16#5223C411yescharacteristic
17C2×#823—21no—
18#4960—21no—
19#4956—21no—
20#4956—21no—
21#4953—21no—
22#4937C2211yescharacteristic
23#4926C2211yescharacteristic
24C4×#208C2211yescharacteristic
25#4836—22no—
26#4726—21no—
27D3×C22⋊C8—21no—
28#4706—22no—
29C3×#329—22no—
30#4554—21no—
31#4540—22no—
32#2973—31no—
33#2631—31no—
34C4×#100—31no—
35#1527—31no—
36#1527D311yescharacteristic, complemented
37#1288—31no—
38#1288—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C22×C3×D4C2×C411yescharacteristic
41C22×#67—21no—
42C22×#67—21no—
43C22×D12D411yescharacteristic
44C22×C4×D3D411yescharacteristic
45C22×C4×D3D411yescharacteristic
46C2×C3×M16—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#217—21no—
55C2×#217—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4C2×C411yescharacteristic
59C2×(C2×D3)⋊C4C2×C411yescharacteristic
60C2×C4⋊Dic3D411yescharacteristic
61C2×#208—21no—
62C2×ΓL(1,𝔽25)—22no—
63C2×D3×C8—22no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#97D412yescomplemented
72C3×C4×C8—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95C2×C411yescharacteristic
78C3⋊#95C2×C411yescharacteristic
79#830—21no—
80#823—24no—
81(C2×D3)⋊C8—22no—
82(C2×D3)⋊C8—22no—
83C4⋊(C3⋊C8)—21no—
84#803—21no—
85#803D412yescomplemented
86C4×C3⋊C8—21no—
87C2×#124C2×D311yescharacteristic
88C2×C4×D4—31no—
89#435—62no—
90#435—62no—
91#432—31no—
92#432C2×D311yescharacteristic
93C2×#100—61no—
94#407—61no—
95C2×C22⋊C8—61no—
96C4×M16—31no—
97C4×M16C2×D311yescharacteristic
98#397—61no—
99#393—31no—
100#393—31no—
101#392—31no—
102#392—31no—
103#392—61no—
104#386—61no—
105#386—61no—
106#383—31no—
107#343—61no—
108#329—22no—
109#329—62no—
110C24×C3—21no—
111C23×D3C22⋊C411yescharacteristic
112C23×D3C22⋊C411yescharacteristic
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—81no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—81no—
119C22×C3×C4—21no—
120C22×C3×C4C22×C411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C2×#67—42no—
124C2×#67—42no—
125C2×#67—81no—
126C2×#67—81no—
127C2×#67—81no—
128C2×#67—81no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3C22⋊C411yescharacteristic
132C22×Dic3C22⋊C411yescharacteristic
133C2×D12—22no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—41no—
137C2×C4×D3—21no—
138C2×C4×D3—21no—
139C2×C4×D3—21no—
140C2×C4×D3—21no—
141C2×C4×D3—81no—
142C2×C4×D3—81no—
143C3×M16—22no—
144C3×M16—41no—
145C2×C3×C8—22no—
146C2×C3×C8—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154C3×C42#4111yescharacteristic
155#217—81no—
156#217—81no—
157(C2×D3)⋊C4—21no—
158(C2×D3)⋊C4—21no—
159(C2×D3)⋊C4—21no—
160(C2×D3)⋊C4—21no—
161(C2×D3)⋊C4—81no—
162(C2×D3)⋊C4—81no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165#208—22no—
166#208—41no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—41no—
169ΓL(1,𝔽25)—82no—
170D3×C8—82no—
171C22×Q8C4×D311yescharacteristic
172C22×D4—31no—
173C22×D4—61no—
174C22×D4—61no—
175C22×D4C4×D311yescharacteristic
176C23×C4—31no—
177C23×C4—31no—
178C2×M16—21no—
179C2×M16—62no—
180C2×M16—61no—
181C22×C8—62no—
182#124—21no—
183#124—21no—
184#124—21no—
185#124—21no—
186C4×D4—61no—
187C4×D4—61no—
188C4×D4—61no—
189C4×D4—61no—
190C2×C4⋊C4—31no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—31no—
194C2×C22⋊C4—31no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C42C22×D311yescharacteristic
200C4⋊C8—21no—
201C4⋊C8—61no—
202#100—64no—
203C22⋊C8—21no—
204C22⋊C8—62no—
205C22⋊C8—62no—
206C22⋊C8—61no—
207#97—21no—
208#97—32no—
209#97—61no—
210#97#6712yescomplemented
211C4×C8—21no—
212C4×C8—61no—
213#95—31no—
214#95—31no—
215#95—61no—
216#95—61no—
217#95—61no—
218#95—61no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3C2×C22⋊C411yescharacteristic
227C22×D3—21no—
228C22×D3—21no—
229C22×D3—21no—
230C22×D3—21no—
231C22×D3—21no—
232C22×D3—41no—
233C22×D3—41no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3#10412yes—
237C3×Q8—81no—
238C3×Q8—81no—
239C3×D4—81no—
240C3×D4—81no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4C4×D411yescharacteristic
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12311yescharacteristic
254#67—84no—
255#67—84no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266D12—42no—
267D12—42no—
268C4×D3—42no—
269C4×D3—42no—
270C4×D3—81no—
271C4×D3—81no—
272C3×C8—42no—
273C3×C8—42no—
274C3⋊C8—42no—
275C3⋊C8—42no—
276C24—21no—
277C24—31no—
278C24—31no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—81no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—62no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—81no—
288C2×D4—121no—
289C2×D4—122no—
290C2×D4—122no—
291C2×D4—241no—
292C2×D4—241no—
293C2×D4—241no—
294C2×D4—241no—
295C22×C4—21no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—241no—
305C22×C4—241no—
306C22×C4C2×C4×D311yescharacteristic
307C22×C4D3×D411yescharacteristic
308C22×C4D3×D411yescharacteristic
309M16—22no—
310M16—41no—
311M16—62no—
312M16—121no—
313M16—242no—
314C2×C8—22no—
315C2×C8—41no—
316C2×C8—62no—
317C2×C8—121no—
318C2×C8—242no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—61no—
326C22⋊C4—61no—
327C22⋊C4—61no—
328C22⋊C4—61no—
329C22⋊C4—241no—
330C22⋊C4—241no—
331C22⋊C4—241no—
332C22⋊C4—241no—
333C42#23511yescharacteristic
334C42D3×D411yescharacteristic
335C42#23711yescharacteristic
336C42C2×#6711yescharacteristic
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3#38611yescharacteristic
347C22×C3C2×#10411yescharacteristic
348C22×C3#42111yescharacteristic
349C2×D3—22no—
350C2×D3—41no—
351C2×D3—41no—
352C2×D3—42no—
353C2×D3—42no—
354C2×D3—41no—
355C2×D3—41no—
356C2×D3—41no—
357C2×D3—81no—
358C2×D3—81no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—81no—
366Dic3—41no—
367Dic3—41no—
368Dic3—81no—
369Dic3—81no—
370C23—32no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—61no—
379C23—61no—
380C23—61no—
381C23—61no—
382C23—61no—
383C23—121no—
384C23—121no—
385C23—121no—
386C23—121no—
387C23D3×C22⋊C411yescharacteristic
388Q8—81no—
389Q8—81no—
390D4—81no—
391D4—81no—
392D4—122no—
393D4—122no—
394D4—244no—
395D4—244no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—121no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—241no—
416C2×C4—241no—
417C2×C4#67⋊C411yescharacteristic
418C2×C4#88311yescharacteristic
419C2×C4#88411yescharacteristic
420C2×C4C4×#6711yescharacteristic
421C2×C4#93711yescharacteristic
422C2×C4#93811yescharacteristic
423C8—42no—
424C8—42no—
425C8—122no—
426C8—122no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—161no—
430C2×C3#187111yescharacteristic
431C2×C3#191111yescharacteristic
432C2×C3#191411yescharacteristic
433D3—42no—
434D3—81no—
435C22—21no—
436C22—21no—
437C22—62no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—122no—
446C22—122no—
447C22—121no—
448C22—121no—
449C22—121no—
450C22—121no—
451C22—121no—
452C22—241no—
453C22—241no—
454C22D3×#10411yescharacteristic
455C22#482111yescharacteristic
456C22#495611yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—81no—
464C4—121no—
465C4—121no—
466C4—241no—
467C4—241no—
468C3#918711yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—122no—
472C2—161no—
473C2—241no—
474C2#7008911yescharacteristic
475C2#7038111yescharacteristic
476C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1527C2
socleC22×C3#386
prime radical#83002C2
cosocleC2×C3×C42C23
hypercenter#1527D3
hypocenterC3#9187
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1527

Complements

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

NH ≅ G/Ntype
2. #79335C2semidirect
3. C3×#1527C2semidirect
4. #71718C2semidirect
5. C4×#823C2semidirect
6. #67752C2semidirect
7. #67666C2semidirect
36. #1527D3semidirect
71. C3×#97D4semidirect
85. #803D4semidirect
210. #97#67semidirect
468. C3#9187semidirect

Maximal subgroups

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

subgroupindex
#830022
#793352
C3×#15272
#717182
C4×#8232
#677522
#676662
#91873