Finite Group Databasebeta

← #10692837

Subgroups — #10692837

Order 768.

770 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2431C211yesmaximal, characteristic
2#84074C211yesmaximal, characteristic, complemented
3#72139C211yesmaximal, characteristic, complemented
4#71674C212yesmaximal, complemented
5#67616C212yesmaximal, complemented
6#9445—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×C3×#127C411yescharacteristic, complemented
9C2×C3×#124C411yescharacteristic, complemented
10C2×C3×#121C2211yescharacteristic
11C22×#208—21no—
12C3×#392—21no—
13C3×#392—21no—
14C3×#391C411yescharacteristic, complemented
15C3×#390C411yescharacteristic, complemented
16C3×#386—21no—
17C3×#386—21no—
18C3×C4×C22⋊C4C2211yescharacteristic
19#5234—24no—
20C2×#834—21no—
21C2×#834—21no—
22#5211C2212yes—
23#5207—21no—
24#5207C2212yes—
25#4537—24no—
26#2879—31no—
27#2431D311yescharacteristic
28#1782—31no—
29#1510—32no—
30#1284—32no—
31C2×C3×#41—22no—
32C2×C3×#41—22no—
33C2×C3×#41—42no—
34C2×C3×#41—42no—
35C22×C3×Q8C2×C411yescharacteristic
36C22×C3×D4—21no—
37C22×C3×D4—21no—
38C22×C3×D4—21no—
39C22×C3×D4C2×C411yescharacteristic
40C22×C3×D4C2×C411yescharacteristic
41C22×C3×D4D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C3×#127—42no—
47C3×#124—42no—
48C3×#121—22no—
49C3×#121—22no—
50C2×C3×C4⋊C4C2×C411yescharacteristic
51C2×C3×C22⋊C4—21no—
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×C42C2×C411yescharacteristic
58C2×C3×C42C2×C411yescharacteristic
59C2×#217—22no—
60C2×#208—22no—
61C2×#208—22no—
62C2×#208—42no—
63C2×#208—42no—
64C22×C3⋊C8—22no—
65C3×#95—21no—
66C3×#95—21no—
67C3×#95—21no—
68#834—42no—
69#834—42no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72#830—24no—
73#830—22no—
74#830D414yescomplemented
75C22×#41C2×D311yescharacteristic
76C22×M16—61no—
77C2×#127Dic311yescharacteristic, complemented
78C2×#124Dic311yescharacteristic, complemented
79C2×#121C2×D311yescharacteristic
80C2×#99—61no—
81C2×#99—61no—
82#408—64no—
83#407—32no—
84#407—61no—
85#392—21no—
86#392—21no—
87#391Dic311yescharacteristic, complemented
88#390Dic311yescharacteristic, complemented
89#386—21no—
90#386—21no—
91#381—32no—
92C4×C22⋊C4C2×D311yescharacteristic
93#327—64no—
94C24×C3—21no—
95C24×C3C22⋊C411yescharacteristic
96C24×C3C22⋊C411yescharacteristic
97C24×C3C2×D411yescharacteristic
98C3×#41—44no—
99C3×#41—44no—
100C3×#41—48no—
101C2×C3×Q8—22no—
102C2×C3×Q8—22no—
103C2×C3×Q8—41no—
104C2×C3×D4—22no—
105C2×C3×D4—22no—
106C2×C3×D4—22no—
107C2×C3×D4—22no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—42no—
117C2×C3×D4—42no—
118C2×C3×D4—42no—
119C2×C3×D4—42no—
120C2×C3×D4—42no—
121C2×C3×D4—81no—
122C2×C3×D4—81no—
123C2×C3×D4—81no—
124C2×C3×D4—81no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4C22⋊C411yescharacteristic
139C22×C3×C4C22⋊C411yescharacteristic
140C22×C3×C4C22×C411yescharacteristic
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4#4112yes—
144C22×Dic3—42no—
145C3×C4⋊C4—22no—
146C3×C22⋊C4—42no—
147C3×C22⋊C4—42no—
148C3×C22⋊C4—42no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—81no—
152C3×C22⋊C4—81no—
153C3×C42—41no—
154C3×C42—41no—
155#217—42no—
156#217—42no—
157#208—44no—
158#208—42no—
159#208—42no—
160C2×C3⋊C8—24no—
161C2×C3⋊C8—44no—
162C2×#41—22no—
163C2×#41—22no—
164C2×#41—42no—
165C2×#41—42no—
166C22×Q8C2×Dic311yescharacteristic
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4C2×Dic311yescharacteristic
171C22×D4C2×Dic311yescharacteristic
172C22×D4#6711yescharacteristic
173C22×D4#6711yescharacteristic
174C23×C4#6711yescharacteristic
175C23×C4#6711yescharacteristic
176C23×C4C22×D311yescharacteristic
177C2×M16—62no—
178C2×M16—62no—
179C2×M16—122no—
180C2×M16—122no—
181C22×C8—62no—
182#127—42no—
183#124—42no—
184#121—22no—
185#121—22no—
186C2×C4⋊C4C2×Dic311yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—62no—
194C2×C42C2×Dic311yescharacteristic
195C2×C42C2×Dic311yescharacteristic
196#99—122no—
197#99—122no—
198C22⋊C8—34no—
199C22⋊C8—64no—
200C22⋊C8—62no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—62no—
205#95—62no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3#9912yes—
227C23×C3C2×C22⋊C411yescharacteristic
228C23×C3#12112yes—
229C3×Q8—44no—
230C3×D4—44no—
231C3×D4—44no—
232C3×D4—44no—
233C3×D4—82no—
234C3×D4—84no—
235C3×D4—84no—
236C3×D4—84no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—24no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—22no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
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#9912yes—
263C2×C3×C4C4×D412yes—
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12311yescharacteristic
266C2×Dic3—82no—
267C2×Dic3—82no—
268C2×Dic3—82no—
269C3⋊C8—44no—
270C24—21no—
271C24#21711yescharacteristic
272C24#21711yescharacteristic
273C24C2×#6711yescharacteristic
274#41—44no—
275#41—44no—
276#41—48no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—81no—
298C2×D4—81no—
299C2×D4—81no—
300C2×D4—81no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—122no—
315C22×C4#21711yescharacteristic
316C22×C4#21711yescharacteristic
317C22×C4D3×D412yes—
318C22×C4#23712yes—
319C22×C4C22×Dic311yescharacteristic
320C22×C4C2×#6711yescharacteristic
321M16—124no—
322M16—122no—
323M16—122no—
324C2×C8—64no—
325C2×C8—124no—
326C4⋊C4—22no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—81no—
333C22⋊C4—81no—
334C22⋊C4—122no—
335C22⋊C4—122no—
336C42—41no—
337C42—41no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—42no—
345C22×C3—42no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—81no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3—81no—
361C22×C3—81no—
362C22×C3#38611yescharacteristic
363C22×C3C2×#9911yescharacteristic
364C22×C3C2×#9911yescharacteristic
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—42no—
370C3×C4—41no—
371C3×C4—81no—
372C3×C4—81no—
373Dic3—162no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23#83412yes—
395C23#93912yes—
396C23C2×#21711yescharacteristic
397Q8—44no—
398D4—44no—
399D4—44no—
400D4—44no—
401D4—82no—
402D4—84no—
403D4—84no—
404D4—84no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—242no—
431C2×C4—242no—
432C2×C4—242no—
433C2×C4#83412yes—
434C2×C4D4×Dic312yes—
435C2×C4#93511yescharacteristic
436C2×C4#93711yescharacteristic
437C8—124no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—42no—
441C2×C3—42no—
442C2×C3—81no—
443C2×C3—161no—
444C2×C3#188411yescharacteristic
445C2×C3#189811yescharacteristic
446C2×C3#191011yescharacteristic
447C22—21no—
448C22—21no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—42no—
454C22—42no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22C2×#83411yescharacteristic
472C22C2×#83411yescharacteristic
473C22#522011yescharacteristic
474C4—21no—
475C4—21no—
476C4—22no—
477C4—22no—
478C4—42no—
479C4—41no—
480C4—81no—
481C4—81no—
482C4—482no—
483C3#944511yescharacteristic, complemented, Sylow
484C2—21no—
485C2—21no—
486C2—42no—
487C2—42no—
488C2—81no—
489C2—161no—
490C2#7167011yescharacteristic
491C2#7181711yescharacteristic
492C2#7185411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2431C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#2431D3
hypocenterC3#9445
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2431

Complements

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

NH ≅ G/Ntype
2. #84074C2semidirect
3. #72139C2semidirect
4. #71674C2semidirect
5. #67616C2semidirect
8. C2×C3×#127C4semidirect
9. C2×C3×#124C4semidirect
14. C3×#391C4semidirect
15. C3×#390C4semidirect
74. #830D4semidirect
77. C2×#127Dic3semidirect
78. C2×#124Dic3semidirect
87. #391Dic3semidirect
88. #390Dic3semidirect
483. C3#9445semidirect

Maximal subgroups

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

subgroupindex
C3×#24312
#840742
#721392
#716742
#676162
#94453