Finite Group Databasebeta

← #10692963

Subgroups — #10692963

Order 768.

716 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 109 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#84068C211yesmaximal, characteristic
2#78602C211yesmaximal, characteristic
3C3×#1787C211yesmaximal, characteristic, complemented
4#72132C211yesmaximal, characteristic, complemented
5#69530C211yesmaximal, characteristic
6#67676C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9471—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×#946—21no—
11C2×C4×#67C2211yescharacteristic
12C2×C3×#103—22no—
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15#5231C412yescomplemented
16#5230C412yescomplemented
17#5229—21no—
18C2×#835—22no—
19#5213—21no—
20#5212C2211yescharacteristic
21#5212C2211yescharacteristic
22#5210—21no—
23C2×#830C2211yescharacteristic
24#5111—22no—
25C2×#806—21no—
26#4730—22no—
27#4726—21no—
28#4719—21no—
29#4527—21no—
30#2973—31no—
31#2409—31no—
32#1799—31no—
33#1787—31no—
34#1787D311yescharacteristic, complemented
35#1289—31no—
36#1289—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—44no—
43C22×C3×Q8C2×C412yes—
44C23×C3×C4—21no—
45C23×C3×C4C2311yescharacteristic
46C22×#67D411yescharacteristic
47C22×C4×D3D411yescharacteristic
48C22×C3×C8D412yes—
49C2×C3×C4⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217D411yescharacteristic
54#946—44no—
55C4×#67—21no—
56C4×#67—21no—
57C4×#67—21no—
58C4×#67—21no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4C2×C411yescharacteristic
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×Dic3⋊C4—22no—
64C2×Dic3⋊C4C2×C411yescharacteristic
65C2×C4×Dic3D411yescharacteristic
66C22×C3⋊C8D412yes—
67C2×ΓL(1,𝔽25)—22no—
68C3×#103—44no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#95—21no—
72#835—44no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75C3⋊#95—21no—
76C3⋊#95C2×C412yes—
77#830—21no—
78#830—21no—
79(C2×D3)⋊C8—22no—
80#815—22no—
81Dic3⋊C8—22no—
82#806—42no—
83C23×Q8C2×D311yescharacteristic
84C2×C22⋊Q8—61no—
85C2×C4×D4—31no—
86#435—62no—
87#435—62no—
88#432—61no—
89C2×#103—22no—
90C2×#103—62no—
91C2×#99—61no—
92#407—61no—
93C2×C22⋊C8—31no—
94C2×C22⋊C8C2×D311yescharacteristic
95#395—32no—
96#393—32no—
97#391—61no—
98#386—61no—
99#385—61no—
100#380—31no—
101#380—31no—
102#380C2×D311yescharacteristic
103#342—61no—
104C24×C3C2×D411yescharacteristic
105C23×D3C22⋊C411yescharacteristic
106C2×C3×Q8—24no—
107C2×C3×Q8—24no—
108C2×C3×Q8—24no—
109C2×C3×Q8—22no—
110C2×C3×Q8—22no—
111C2×C3×Q8—44no—
112C2×C3×Q8—42no—
113C2×C3×Q8—42no—
114C2×C3×Q8—42no—
115C2×C3×Q8—44no—
116C2×C3×Q8—44no—
117C2×C3×Q8—41no—
118C2×C3×Q8—41no—
119C22×C3×C4—22no—
120C22×C3×C4—42no—
121C22×C3×C4—44no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4C22×C411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C2×#67—21no—
132C2×#67—21no—
133C2×#67—21no—
134C2×#67—21no—
135C2×#67—21no—
136C2×#67—21no—
137C22×Dic3—22no—
138C22×Dic3—41no—
139C22×Dic3C22⋊C411yescharacteristic
140C22×Dic3C22⋊C411yescharacteristic
141C22×Dic3C22⋊C411yescharacteristic
142C2×C4×D3—21no—
143C2×C4×D3—21no—
144C2×C4×D3—42no—
145C2×C3×C8—22no—
146C2×C3×C8—42no—
147C3×C4⋊C4—42no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150#217—21no—
151#217—21no—
152#217—41no—
153#217—41no—
154#217—41no—
155#217—41no—
156(C2×D3)⋊C4—21no—
157(C2×D3)⋊C4—21no—
158(C2×D3)⋊C4—44no—
159C4⋊Dic3—42no—
160C4⋊Dic3—42no—
161Dic3⋊C4—21no—
162Dic3⋊C4—21no—
163Dic3⋊C4—44no—
164C4×Dic3—21no—
165C4×Dic3—21no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—42no—
168ΓL(1,𝔽25)—82no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—44no—
175C22×Q8C4×D312yes—
176C22×D4—31no—
177C23×C4—21no—
178C23×C4—31no—
179C23×C4C22×D311yescharacteristic
180C2×M16—62no—
181C22×C8—32no—
182C22×C8#6712yes—
183C22⋊Q8—124no—
184C4×D4—61no—
185C4×D4—61no—
186C4×D4—61no—
187C4×D4—61no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—61no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—31no—
195C2×C22⋊C4—31no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C42—31no—
200C4⋊C8—62no—
201#103—44no—
202#103—124no—
203#99—122no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—62no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209#97—62no—
210#95—21no—
211#95—32no—
212#95—62no—
213#95—62no—
214#95—61no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3C2×C22⋊C411yescharacteristic
218C23×C3#12111yescharacteristic
219C23×C3#12111yescharacteristic
220C23×C3AGL(1,ℤ/8)11yescharacteristic
221C23×C3#13711yescharacteristic
222C23×C3#13711yescharacteristic
223C23×C3#13711yescharacteristic
224C22×D3—21no—
225C22×D3—21no—
226C22×D3—21no—
227C22×D3—41no—
228C22×D3—41no—
229C3×Q8—24no—
230C3×Q8—24no—
231C3×Q8—44no—
232C3×Q8—42no—
233C3×Q8—42no—
234C3×Q8—44no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—44no—
244C2×C3×C4—44no—
245C2×C3×C4—42no—
246C2×C3×C4—81no—
247C2×C3×C4—81no—
248C2×C3×C4—81no—
249C2×C3×C4C4×D411yescharacteristic
250C2×C3×C4C4×D411yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12311yescharacteristic
253#67—41no—
254#67—41no—
255#67—41no—
256#67—41no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259C2×Dic3—21no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—81no—
268C2×Dic3—81no—
269C2×Dic3—81no—
270C4×D3—42no—
271C4×D3—81no—
272C3×C8—42no—
273C3⋊C8—42no—
274C24—31no—
275C24D3×D411yescharacteristic
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—44no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—61no—
295C22×C4—22no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—62no—
307C22×C4—122no—
308C22×C4—121no—
309C22×C4C2×C4×D311yescharacteristic
310C22×C4#23511yescharacteristic
311C22×C4D3×D411yescharacteristic
312C22×C4D3×D411yescharacteristic
313C22×C4#23711yescharacteristic
314C22×C4C2×#6711yescharacteristic
315M16—242no—
316C2×C8—22no—
317C2×C8—42no—
318C2×C8—62no—
319C2×C8—122no—
320C4⋊C4—42no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C4⋊C4—124no—
324C4⋊C4—122no—
325C4⋊C4—122no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—61no—
329C22⋊C4—61no—
330C22⋊C4—61no—
331C22⋊C4—61no—
332C22⋊C4—124no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C42—61no—
338C42—61no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3#38611yescharacteristic
350C22×C3#41711yescharacteristic
351C22×C3#41911yescharacteristic
352C22×C3#46811yescharacteristic
353C22×C3#46811yescharacteristic
354C22×C3#47011yescharacteristic
355C22×C3#47011yescharacteristic
356C2×D3—42no—
357C2×D3—41no—
358C2×D3—41no—
359C2×D3—41no—
360C2×D3—81no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—44no—
365C3×C4—161no—
366C3×C4B(4,𝔽2)12yes—
367Dic3—41no—
368Dic3—41no—
369Dic3—41no—
370Dic3—82no—
371Dic3—161no—
372C23—41no—
373C23—41no—
374C23—61no—
375C23—61no—
376C23—61no—
377C23—121no—
378C23—121no—
379C23D3×C22⋊C411yescharacteristic
380C23#88411yescharacteristic
381C23#91411yescharacteristic
382C23#91511yescharacteristic
383C23#91811yescharacteristic
384C23#91811yescharacteristic
385C23#93811yescharacteristic
386Q8—24no—
387Q8—24no—
388Q8—44no—
389Q8—42no—
390Q8—42no—
391Q8—44no—
392D4—121no—
393D4—121no—
394D4—121no—
395D4—121no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—122no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—241no—
422C2×C4—241no—
423C2×C4—241no—
424C2×C4—241no—
425C2×C4#67⋊C411yescharacteristic
426C2×C4#88311yescharacteristic
427C2×C4C4×#6711yescharacteristic
428C2×C4#93711yescharacteristic
429C8—42no—
430C8—122no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3#188911yescharacteristic
438C2×C3#189411yescharacteristic
439C2×C3#190911yescharacteristic
440D3—81no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—122no—
452C22—121no—
453C22—121no—
454C22—121no—
455C22—241no—
456C22#480211yescharacteristic
457C22#480311yescharacteristic
458C22#481211yescharacteristic
459C22#481611yescharacteristic
460C22#495611yescharacteristic
461C22#517411yescharacteristic
462C22#518911yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—44no—
467C4—121no—
468C4—121no—
469C4—121no—
470C4—161no—
471C4—242no—
472C4—481no—
473C4#503212yes—
474C3#947111yescharacteristic, complemented, Sylow
475C2—21no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—241no—
482C2#6949911yescharacteristic
483C2#7027111yescharacteristic
484C2#7035011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1787C2
socleC22×C3#386
prime radicalC22×#67D4
cosocleC23×C3×C4C23
hypercenter#1787D3
hypocenterC3#9471
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
3. C3×#1787C2semidirect
4. #72132C2semidirect
6. #67676C2semidirect
7. #67625C2semidirect
15. #5231C4semidirect
16. #5230C4semidirect
34. #1787D3semidirect
474. C3#9471semidirect

Maximal subgroups

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

subgroupindex
#840682
#786022
C3×#17872
#721322
#695302
#676762
#676252
#94713