Finite Group Databasebeta

← #10714855

Subgroups — #10714855

Order 768.

540 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 79 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#74795C211yesmaximal, characteristic
2C3×#1867C211yesmaximal, characteristic
3#71657C211yesmaximal, characteristic
4#71647C211yesmaximal, characteristic
5#67695C211yesmaximal, characteristic
6#67694C211yesmaximal, characteristic
7#67637C211yesmaximal, characteristic
8#13461—31nomaximal, Sylow
9C3×#518C2211yescharacteristic
10C2×#924—21no—
11C2×#878—21no—
12C2×C3×#99C2211yescharacteristic
13C3×#386C2211yescharacteristic
14C3×#385—21no—
15#5221C2211yescharacteristic
16#5220C2211yescharacteristic
17C2×#834C2211yescharacteristic
18C2×#833C2211yescharacteristic
19#4995—21no—
20#4948—21no—
21#4948—21no—
22#4946—21no—
23#4656—21no—
24#4647—21no—
25C3×#342—21no—
26#4532—21no—
27#4526—21no—
28#2683—31no—
29#1867—31no—
30#1867D311yescharacteristic
31#1856—31no—
32#1393—31no—
33#1390—31no—
34#1390—31no—
35C22×C3×D4C2311yescharacteristic
36C23×C3×C4D411yescharacteristic
37C23×Dic3D411yescharacteristic
38C22×Dic6—21no—
39C3×C4×D4—41no—
40C3×C4×D4—41no—
41C3×#117—21no—
42C2×C3×C4⋊C4—41no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4D411yescharacteristic
48C2×#217—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217D411yescharacteristic
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3D411yescharacteristic
56#924—44no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—41no—
62C2×C4×Dic3D411yescharacteristic
63C2×#208—21no—
64#878—44no—
65C3×#99—21no—
66C3×#99Q812yescomplemented
67C3×#95—21no—
68C3×#95—41no—
69#834—21no—
70#834Q812yescomplemented
71#833—21no—
72#833Q812yescomplemented
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—41no—
76#808—41no—
77#807—41no—
78C2×C22⋊Q8—61no—
79C2×C22⋊Q8—61no—
80#518C2×D311yescharacteristic
81#428—61no—
82C2×#100—31no—
83C2×#99—31no—
84C2×#99C2×D311yescharacteristic
85#396—61no—
86#396—61no—
87#395—61no—
88#390—31no—
89#389—61no—
90#386—31no—
91#386C2×D311yescharacteristic
92#385—21no—
93#385—61no—
94#342—21no—
95#342—61no—
96#339—61no—
97C24×C3C2×D411yescharacteristic
98C24×C3C2×D411yescharacteristic
99C2×C3×D4—41no—
100C2×C3×D4—41no—
101C2×C3×D4C2×Q811yescharacteristic
102C2×C3×D4C2×Q811yescharacteristic
103C2×C3×D4C2×Q811yescharacteristic
104C2×C3×D4#4111yescharacteristic
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—42no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3—42no—
120C22×Dic3—41no—
121C22×Dic3—41no—
122C2×Dic6—42no—
123C2×Dic6—42no—
124C2×Dic6—81no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—82no—
128C3×C22⋊C4—22no—
129C3×C22⋊C4—42no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42—41no—
136#217—22no—
137#217—42no—
138#217—42no—
139#217—42no—
140#217—41no—
141#217—41no—
142C4⋊Dic3—42no—
143C4⋊Dic3—42no—
144C4⋊Dic3—41no—
145Dic3⋊C4—24no—
146Dic3⋊C4—42no—
147Dic3⋊C4—42no—
148Dic3⋊C4—81no—
149Dic3⋊C4—82no—
150C4×Dic3—41no—
151#208—22no—
152#208—41no—
153C2×C3⋊C8—41no—
154C22×Q8—61no—
155C22×D4C22×D311yescharacteristic
156C23×C4—31no—
157C23×C4#6711yescharacteristic
158C2×M16—61no—
159C22⋊Q8—124no—
160C22⋊Q8—124no—
161C4×D4—41no—
162C4×D4—41no—
163#117—21no—
164C2×C4⋊C4—31no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—121no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—31no—
178C2×C22⋊C4—61no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4#6711yescharacteristic
183C2×C42—31no—
184#107—121no—
185#106—121no—
186#100—32no—
187#100—61no—
188#99—21no—
189#99—32no—
190#99—61no—
191#99Dic612yescomplemented
192#95—21no—
193#95—41no—
194#95—61no—
195#95—61no—
196#95—121no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—21no—
200C23×C3—21no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3#12011yescharacteristic
206C23×C3C22⋊Q811yescharacteristic
207C23×C3C22⋊Q811yescharacteristic
208C23×C3C4⋊Q811yescharacteristic
209C23×C3C4⋊Q811yescharacteristic
210C3×D4—41no—
211C3×D4—41no—
212C3×D4—41no—
213C3×D4—41no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4—81no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4C22⋊Q811yescharacteristic
234C2×C3×C4C4⋊Q811yescharacteristic
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—81no—
250C2×Dic3—81no—
251C2×Dic3—81no—
252Dic6—84no—
253C3⋊C8—42no—
254C24D3×D411yescharacteristic
255C24C2×#6711yescharacteristic
256C2×Q8—122no—
257C2×Q8—122no—
258C2×Q8—241no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4C2×Dic611yescharacteristic
262C2×D4#23511yescharacteristic
263C2×D4D3×Q811yescharacteristic
264C2×D4D3×Q811yescharacteristic
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—122no—
279C22×C4—121no—
280C22×C4—121no—
281C22×C4D3×D411yescharacteristic
282M16—62no—
283M16—121no—
284C2×C8—121no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—64no—
288C4⋊C4—82no—
289C4⋊C4—121no—
290C4⋊C4—122no—
291C4⋊C4—122no—
292C4⋊C4—122no—
293C4⋊C4—122no—
294C4⋊C4—241no—
295C4⋊C4—242no—
296C22⋊C4—22no—
297C22⋊C4—42no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—62no—
304C22⋊C4—122no—
305C22⋊C4—122no—
306C22⋊C4—122no—
307C22⋊C4—121no—
308C22⋊C4—121no—
309C42—41no—
310C42—121no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3#39511yescharacteristic
328C22×C3B(4,𝔽2)11yescharacteristic
329C22×C3#45811yescharacteristic
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—42no—
333C3×C4—81no—
334C3×C4—82no—
335C3×C4—81no—
336C3×C4—81no—
337Dic3—42no—
338Dic3—42no—
339Dic3—81no—
340Dic3—82no—
341Dic3—81no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23#88811yescharacteristic
351C23#89611yescharacteristic
352C23#92411yescharacteristic
353C23#93711yescharacteristic
354C23#94411yescharacteristic
355Q8—244no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—41no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—241no—
394C2×C4—241no—
395C2×C4—241no—
396C2×C4#88811yescharacteristic
397C2×C4#89611yescharacteristic
398C8—122no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—41no—
402C2×C3—41no—
403C2×C3—41no—
404C2×C3—41no—
405C2×C3#206211yescharacteristic
406C2×C3#219611yescharacteristic
407C2×C3#220111yescharacteristic
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—81no—
423C22—81no—
424C22#497911yescharacteristic
425C22#503211yescharacteristic
426C22#505311yescharacteristic
427C4—21no—
428C4—21no—
429C4—42no—
430C4—81no—
431C4—82no—
432C4—81no—
433C4—81no—
434C4—122no—
435C4—122no—
436C4—241no—
437C4—242no—
438C4—241no—
439C3#1346111yescharacteristic, complemented, Sylow
440C2—21no—
441C2—21no—
442C2—41no—
443C2—41no—
444C2—41no—
445C2—41no—
446C2#6959411yescharacteristic
447C2#7099611yescharacteristic
448C2#7100311yescharacteristic
subgroupG/H
centerC22#5032
commutator, prime radical, cosocleC22×C3×D4C23
FrattiniC22×D4C22×D3
FittingC3×#1867C2
socleC22×C3B(4,𝔽2)
hypercenter#1867D3
hypocenterC3#13461
LCSG ▷ C22×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1867

Complements

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

NH ≅ G/Ntype
66. C3×#99Q8semidirect
70. #834Q8semidirect
72. #833Q8semidirect
191. #99Dic6semidirect
439. C3#13461semidirect

Maximal subgroups

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

subgroupindex
#747952
C3×#18672
#716572
#716472
#676952
#676942
#676372
#134613