Finite Group Databasebeta

← #10692222

Subgroups — #10692222

Order 768.

576 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2377C211yesmaximal, characteristic, complemented
2#82627C211yesmaximal, characteristic
3#71673C211yesmaximal, characteristic, complemented
4#69531C211yesmaximal, characteristic, complemented
5#69528C211yesmaximal, characteristic, complemented
6#67628C211yesmaximal, characteristic, complemented
7#67626C211yesmaximal, characteristic, complemented
8#9343—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C3×#390—21no—
12C3×#388—21no—
13C3×#387—21no—
14C3×#386—21no—
15C3×#386—21no—
16C3×#385—21no—
17C3×#381C2211yescharacteristic
18C3×#380C2211yescharacteristic
19#5217—21no—
20C4×#217C2211yescharacteristic
21#5207C2211yescharacteristic
22#5207C2211yescharacteristic
23#5039—22no—
24#5036C412yescomplemented
25#5034C412yescomplemented
26#4968—21no—
27C2×#806—21no—
28C2×#806—21no—
29#4537—22no—
30#4451—21no—
31#4451—21no—
32#3192—31no—
33#2377D311yescharacteristic, complemented
34#1964—31no—
35#1510—31no—
36#1507—31no—
37#1305—31no—
38#1284—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×D12C2×C411yescharacteristic, complemented
45C3×#121—21no—
46C3×#121—21no—
47C3×#121C2×C412yes—
48C3×#121D412yes—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4D411yescharacteristic
58C2×C3×C22⋊C4D411yescharacteristic
59C2×C3×C42—21no—
60C2×#217—21no—
61C2×#213—22no—
62#930—21no—
63#930—21no—
64#930—21no—
65#930—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×C4⋊Dic3C2×C411yescharacteristic, complemented
68C2×C4×Dic3—21no—
69C2×#208—21no—
70C2×#208—21no—
71C3×#95—21no—
72C3×#95—21no—
73C3×#95—21no—
74C3×#95—21no—
75C3×#95D411yescharacteristic
76C3×#95D411yescharacteristic
77#832—22no—
78C3⋊#95—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830C2×C412yescomplemented
83#809—22no—
84#808—22no—
85#806—42no—
86#806—42no—
87C2×#121—31no—
88C2×#121C2×D311yescharacteristic
89#481—32no—
90#481—61no—
91#469—62no—
92#447—32no—
93#447—61no—
94C2×#99—61no—
95C2×#99—61no—
96#407—31no—
97#407—31no—
98#390—21no—
99#388—21no—
100#387—21no—
101#386—21no—
102#386—21no—
103#385—21no—
104#381C2×D311yescharacteristic
105#380C2×D311yescharacteristic
106C4×C22⋊C4—31no—
107#327—62no—
108#327—61no—
109#327—61no—
110C24×C3—21no—
111C24×C3C2×D411yescharacteristic
112C23×D3—21no—
113C2×C3×D4—22no—
114C2×C3×D4—22no—
115C2×C3×D4—21no—
116C2×C3×D4—81no—
117C2×C3×D4—81no—
118C2×C3×D4—81no—
119C2×C3×D4C22⋊C412yes—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—81no—
126C22×C3×C4—81no—
127C22×C3×C4—81no—
128C22×C3×C4—81no—
129C22×C3×C4C22×C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C2×#67—41no—
136C2×#67—41no—
137C2×#67—41no—
138C2×#67—41no—
139C2×#67—41no—
140C2×#67—41no—
141C22×Dic3—21no—
142C22×Dic3—41no—
143C2×D12—21no—
144C2×D12—21no—
145C2×D12—81no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—81no—
148C3×C4⋊C4—81no—
149C3×C4⋊C4C22⋊C412yes—
150C3×C22⋊C4—22no—
151C3×C22⋊C4—22no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—81no—
155C3×C22⋊C4—81no—
156C3×C22⋊C4—81no—
157C3×C22⋊C4—81no—
158C3×C42—81no—
159#217—22no—
160#217—41no—
161#213—82no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—41no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4×Dic3—81no—
167#208—81no—
168#208—81no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—41no—
171C22×D4—21no—
172C22×D4—31no—
173C22×D4—61no—
174C22×D4#6711yescharacteristic
175C23×C4—21no—
176C23×C4C22×D311yescharacteristic
177C2×D8—62no—
178C2×M16—61no—
179C2×M16—61no—
180#121—21no—
181#121—21no—
182#121—61no—
183#121—61no—
184#121—61no—
185#121—61no—
186#121C4×D312yes—
187#121D1212yes—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4#6711yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4#6711yescharacteristic
200C2×C22⋊C4#6711yescharacteristic
201C2×C42—21no—
202C2×C42—61no—
203#106—62no—
204#102—62no—
205#102—62no—
206#99—122no—
207#99—122no—
208C22⋊C8—32no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—61no—
217#95#6711yescharacteristic
218#95#6711yescharacteristic
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—22no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3C4×D411yescharacteristic
231C23×C3#12011yescharacteristic
232C23×C3#12111yescharacteristic
233C22×D3—41no—
234C22×D3—41no—
235C22×D3—41no—
236C22×D3—81no—
237C22×D3—81no—
238C3×D4—42no—
239C3×D4—81no—
240C3×D4—82no—
241C3×D4—82no—
242C2×C3×C4—22no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—84no—
264C2×C3×C4C2×C22⋊C411yescharacteristic
265C2×C3×C4C4×D411yescharacteristic
266C2×C3×C4#12111yescharacteristic
267C2×C3×C4#12311yescharacteristic
268#67—81no—
269#67—81no—
270#67—81no—
271#67—81no—
272C2×Dic3—42no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—41no—
276C2×Dic3—81no—
277C2×Dic3—81no—
278D12—42no—
279D12—81no—
280C3⋊C8—42no—
281C3⋊C8—81no—
282C24—21no—
283C24—61no—
284C24C2×#6711yescharacteristic
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—21no—
288C2×D4—61no—
289C2×D4—61no—
290C2×D4—81no—
291C2×D4—81no—
292C2×D4—81no—
293C2×D4—121no—
294C2×D4—121no—
295C2×D4—121no—
296C2×D4—121no—
297C2×D4—121no—
298C2×D4—121no—
299C2×D4—241no—
300C2×D4(C2×D3)⋊C412yes—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—61no—
307C22×C4—81no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4—121no—
312C22×C4C2×C4×D311yescharacteristic
313C22×C4C2×D1211yescharacteristic
314C22×C4#23511yescharacteristic
315C22×C4#23511yescharacteristic
316C22×C4C2×#6711yescharacteristic
317C22×C4C2×#6711yescharacteristic
318D8—242no—
319M16—241no—
320M16—241no—
321C2×C8—62no—
322C2×C8—121no—
323C4⋊C4—21no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—81no—
327C4⋊C4—81no—
328C4⋊C4(C2×D3)⋊C412yes—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—62no—
334C22⋊C4—81no—
335C22⋊C4—81no—
336C22⋊C4—81no—
337C22⋊C4—81no—
338C22⋊C4—121no—
339C22⋊C4—121no—
340C22⋊C4—121no—
341C42—81no—
342C42—241no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3#38611yescharacteristic
361C22×C3#45311yescharacteristic
362C22×C3#45411yescharacteristic
363C2×D3—81no—
364C2×D3—81no—
365C2×D3—81no—
366C2×D3—161no—
367C2×D3—161no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—42no—
373C3×C4—81no—
374C3×C4—81no—
375C3×C4—81no—
376C3×C4—81no—
377Dic3—82no—
378Dic3—81no—
379C23—21no—
380C23—21no—
381C23—22no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—121no—
391C23—121no—
392C23—121no—
393C23—241no—
394C23—241no—
395C23C4×#6711yescharacteristic
396C23#93011yescharacteristic
397C23#95311yescharacteristic
398D4—42no—
399D4—81no—
400D4—82no—
401D4—82no—
402D4—122no—
403D4—241no—
404D4—241no—
405D4—241no—
406D4—241no—
407D4—241no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—84no—
430C2×C4—122no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—241no—
435C2×C4—241no—
436C2×C4C2×(C2×D3)⋊C411yescharacteristic
437C2×C4C4×#6711yescharacteristic
438C2×C4#92911yescharacteristic
439C2×C4#93011yescharacteristic
440C8—122no—
441C8—241no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—82no—
447C2×C3#188311yescharacteristic
448C2×C3#190411yescharacteristic
449C2×C3#191311yescharacteristic
450D3—161no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—241no—
469C22—241no—
470C22—241no—
471C22—481no—
472C22—481no—
473C22#521311yescharacteristic
474C22#524111yescharacteristic
475C22#524511yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—42no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—242no—
486C4—241no—
487C3#934311yescharacteristic, complemented, Sylow
488C2—21no—
489C2—21no—
490C2—41no—
491C2—41no—
492C2—82no—
493C2—481no—
494C2#7165111yescharacteristic
495C2#7165611yescharacteristic
496C2#7166711yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2377C2
socleC22×C3#386
prime radical#82627C2
cosocleC23×C3×C4C23
hypercenter#2377D3
hypocenterC3#9343
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2377

Complements

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

NH ≅ G/Ntype
1. C3×#2377C2semidirect
3. #71673C2semidirect
4. #69531C2semidirect
5. #69528C2semidirect
6. #67628C2semidirect
7. #67626C2semidirect
24. #5036C4semidirect
25. #5034C4semidirect
33. #2377D3semidirect
44. C22×D12C2×C4semidirect
67. C2×C4⋊Dic3C2×C4semidirect
82. #830C2×C4semidirect
487. C3#9343semidirect

Maximal subgroups

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

subgroupindex
C3×#23772
#826272
#716732
#695312
#695282
#676282
#676262
#93433