Finite Group Databasebeta

← #10695530

Subgroups — #10695530

Order 768.

750 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×#2616C211yesmaximal, characteristic
2#84084C211yesmaximal, characteristic
3#82742C211yesmaximal, characteristic, complemented
4#72138C211yesmaximal, characteristic
5#71688C211yesmaximal, characteristic, complemented
6#69529C211yesmaximal, characteristic, complemented
7#67626C211yesmaximal, characteristic, complemented
8#9935—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×C22⋊Q8—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×#951—21no—
13C2×#924C2211yescharacteristic
14C3×#395—21no—
15C3×#394—21no—
16C3×#394—22no—
17C3×#380—21no—
18C3×#380C2211yescharacteristic
19#5239—22no—
20C2×#837—22no—
21#5232—21no—
22#5228—21no—
23#5228—21no—
24#5218—21no—
25C4×#217C2211yescharacteristic
26#5207C2211yescharacteristic
27#5207C2211yescharacteristic
28#5052—22no—
29#5050—21no—
30#5043—21no—
31#4971—21no—
32#4538—22no—
33#4527—22no—
34#3195—31no—
35#3007—31no—
36#2616D311yescharacteristic
37#1996—31no—
38#1789—31no—
39#1508—31no—
40#1305—31no—
41C2×C3×#41—21no—
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—44no—
46C22×C3×Q8D411yescharacteristic
47C22×C3×Q8D411yescharacteristic
48C22×C3×D4—21no—
49C22×C3×D4D411yescharacteristic
50C23×C3×C4—21no—
51C23×C3×C4C2311yescharacteristic
52C22×Dic6D411yescharacteristic
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—44no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4D412yes—
66C2×#217—21no—
67C2×#217—21no—
68#951—44no—
69C2×#216—22no—
70C2×#216—22no—
71C2×#214—22no—
72C2×C4⋊Dic3D411yescharacteristic
73#924—21no—
74#924—21no—
75#924—21no—
76#924—21no—
77C2×Dic3⋊C4—21no—
78C2×C4×Dic3—21no—
79C2×#208—21no—
80C2×#208D412yes—
81C3×#95—22no—
82C3×#95—21no—
83#837—44no—
84#835—22no—
85#835—22no—
86#832—22no—
87C3⋊#95—21no—
88#830—21no—
89#830—21no—
90#830—21no—
91#830D412yes—
92#810—22no—
93#808—22no—
94C22×#41C2×D311yescharacteristic
95C2×#137—61no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—31no—
98C2×C22⋊Q8C2×D311yescharacteristic
99#483—61no—
100#483—61no—
101#470—62no—
102#452—62no—
103#451—61no—
104#451—61no—
105#451—61no—
106#450—61no—
107C2×#104—62no—
108#417—61no—
109#407—31no—
110#407—31no—
111#395—21no—
112#394—21no—
113#394—22no—
114#380—21no—
115#380C2×D311yescharacteristic
116C4×C22⋊C4—31no—
117#342—62no—
118#328—62no—
119C24×C3—21no—
120C24×C3C2×D411yescharacteristic
121C3×#41—24no—
122C3×#41—24no—
123C3×#41—42no—
124C3×#41—42no—
125C3×#41—42no—
126C3×#41—42no—
127C3×#41—42no—
128C3×#41—42no—
129C2×C3×Q8—21no—
130C2×C3×Q8—21no—
131C2×C3×Q8—21no—
132C2×C3×Q8—21no—
133C2×C3×Q8—42no—
134C2×C3×Q8—42no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4—42no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—41no—
141C2×C3×D4—41no—
142C22×C3×C4—24no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4—41no—
150C22×C3×C4—41no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4#4111yescharacteristic
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—41no—
160C2×Dic6—21no—
161C2×Dic6—21no—
162C2×Dic6—42no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4—44no—
167C3×C4⋊C4—41no—
168C3×C4⋊C4—41no—
169C3×C22⋊C4—44no—
170C3×C22⋊C4—41no—
171C3×C22⋊C4—41no—
172C3×C22⋊C4—42no—
173C3×C22⋊C4—82no—
174#217—22no—
175#217—41no—
176#217—41no—
177#217—41no—
178#216—44no—
179#216—82no—
180#214—44no—
181C4⋊Dic3—21no—
182C4⋊Dic3—21no—
183Dic3⋊C4—41no—
184Dic3⋊C4—41no—
185C4×Dic3—42no—
186#208—42no—
187#208—42no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—22no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—44no—
195C22×Q8—31no—
196C22×Q8#6711yescharacteristic
197C22×Q8#6711yescharacteristic
198C22×D4—21no—
199C22×D4#6711yescharacteristic
200C23×C4—21no—
201C23×C4C22×D311yescharacteristic
202#137—124no—
203C2×Q16—62no—
204C2×Q16—62no—
205C2×SD16—62no—
206C2×M16—32no—
207C2×M16—61no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—44no—
213C22⋊Q8—61no—
214C22⋊Q8—61no—
215C22⋊Q8—61no—
216C22⋊Q8—61no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4#6711yescharacteristic
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4#6712yes—
229C2×C42—61no—
230#106—62no—
231#104—124no—
232#103—62no—
233#103—62no—
234#103—62no—
235#102—62no—
236C22⋊C8—32no—
237C22⋊C8—61no—
238C22⋊C8—61no—
239C22⋊C8—61no—
240#95—22no—
241#95—21no—
242#95—61no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—22no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3#12011yescharacteristic
255C23×C3#12111yescharacteristic
256C23×C3#12711yescharacteristic
257C3×Q8—22no—
258C3×Q8—22no—
259C3×Q8—41no—
260C3×Q8—41no—
261C3×Q8—42no—
262C3×Q8—81no—
263C3×D4—22no—
264C3×D4—22no—
265C3×D4—41no—
266C3×D4—41no—
267C3×D4—44no—
268C3×D4—44no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—44no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—44no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4#12011yescharacteristic
289C2×C3×C4#12011yescharacteristic
290C2×C3×C4#12111yescharacteristic
291C2×C3×C4#12111yescharacteristic
292C2×Dic3—42no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—81no—
296C2×Dic3—81no—
297C2×Dic3—81no—
298C2×Dic3—81no—
299Dic6—42no—
300Dic6—42no—
301C3⋊C8—42no—
302C3⋊C8—42no—
303C24—21no—
304C24C2×#6711yescharacteristic
305#41—24no—
306#41—24no—
307#41—42no—
308#41—42no—
309#41—42no—
310#41—42no—
311#41—42no—
312#41—42no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—61no—
320C2×Q8—61no—
321C2×Q8—122no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—41no—
328C2×D4—41no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—121no—
341C22×C4D3×D411yescharacteristic
342C22×C4D3×D411yescharacteristic
343C22×C4D3×D411yescharacteristic
344C22×C4#23711yescharacteristic
345C22×C4C2×#6711yescharacteristic
346C22×C4C2×#6711yescharacteristic
347Q16—124no—
348Q16—242no—
349SD16—124no—
350M16—122no—
351M16—122no—
352C2×C8—62no—
353C2×C8—62no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—42no—
357C4⋊C4—44no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—61no—
361C4⋊C4—61no—
362C4⋊C4—121no—
363C4⋊C4—121no—
364C22⋊C4—44no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—42no—
368C22⋊C4—62no—
369C22⋊C4—82no—
370C22⋊C4—121no—
371C22⋊C4—121no—
372C22⋊C4—121no—
373C42—122no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3#39211yescharacteristic
392C22×C3#45511yescharacteristic
393C22×C3#45611yescharacteristic
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—42no—
398C3×C4—82no—
399C3×C4—81no—
400C3×C4—81no—
401C3×C4B(4,𝔽2)12yes—
402Dic3—82no—
403Dic3—81no—
404Dic3—81no—
405C23—21no—
406C23—21no—
407C23—22no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23#93811yescharacteristic
417C23#94011yescharacteristic
418C23#95311yescharacteristic
419Q8—22no—
420Q8—22no—
421Q8—41no—
422Q8—41no—
423Q8—42no—
424Q8—81no—
425Q8—122no—
426Q8—122no—
427D4—22no—
428D4—22no—
429D4—41no—
430D4—41no—
431D4—44no—
432D4—44no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—122no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—241no—
456C2×C4—241no—
457C2×C4—241no—
458C2×C4—241no—
459C2×C4#93711yescharacteristic
460C2×C4#93711yescharacteristic
461C2×C4#93911yescharacteristic
462C2×C4#93911yescharacteristic
463C8—122no—
464C8—122no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—41no—
468C2×C3—41no—
469C2×C3—82no—
470C2×C3#200711yescharacteristic
471C2×C3#200911yescharacteristic
472C2×C3#202111yescharacteristic
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22#522311yescharacteristic
491C22#524211yescharacteristic
492C22#524611yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—42no—
497C4—82no—
498C4—81no—
499C4—81no—
500C4—242no—
501C4—241no—
502C4—241no—
503C4#503212yes—
504C3#993511yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—41no—
508C2—41no—
509C2—82no—
510C2#7167111yescharacteristic
511C2#7186911yescharacteristic
512C2#7188111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2616C2
socleC22×C3#392
prime radicalC22×C3×#41C22
hypercenter#2616D3
hypocenterC3#9935
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2616

Complements

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

NH ≅ G/Ntype
3. #82742C2semidirect
5. #71688C2semidirect
6. #69529C2semidirect
7. #67626C2semidirect
504. C3#9935semidirect

Maximal subgroups

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

subgroupindex
C3×#26162
#840842
#827422
#721382
#716882
#695292
#676262
#99353