Finite Group Databasebeta

← #10714864

Subgroups — #10714864

Order 768.

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

normal subgroups only

This group has 476 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#76405C211yesmaximal, characteristic
2C3×#1856C211yesmaximal, characteristic
3#71657C211yesmaximal, characteristic
4#71657C211yesmaximal, characteristic
5#67698C211yesmaximal, characteristic
6#67698C211yesmaximal, characteristic
7#67641C211yesmaximal, characteristic
8#13462—31nomaximal, Sylow
9C3×#518C2211yescharacteristic
10C3×#428—21no—
11C2×C3×#100C2211yescharacteristic
12C3×#390C2211yescharacteristic
13#5220C2211yescharacteristic
14#5220C2211yescharacteristic
15C2×#834C2211yescharacteristic
16C2×#834C2211yescharacteristic
17#4982—21no—
18#4949—21no—
19#4948—21no—
20#4946—21no—
21#4946—21no—
22#4946—21no—
23#4942—21no—
24#4933—21no—
25C3×#339—21no—
26#4526—21no—
27#4526—21no—
28#2748—31no—
29#1867—31no—
30#1867—31no—
31#1856D311yescharacteristic
32#1393—31no—
33#1390—31no—
34#1390—31no—
35C22×C3×D4C2311yescharacteristic
36C23×Dic3D411yescharacteristic
37C23×Dic3D411yescharacteristic
38C2×C3×M16—21no—
39C3×C4×D4—41no—
40C3×C4×D4—41no—
41C3×#117—21no—
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C42D411yescharacteristic
48C2×#217—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217D411yescharacteristic
53C2×#217D411yescharacteristic
54C2×C4⋊Dic3—21no—
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—41no—
60C2×Dic3⋊C4—41no—
61C3×#107—41no—
62C3×#106—41no—
63C3×#100—21no—
64C3×#100Q812yes—
65#834—21no—
66#834—21no—
67#834Q812yes—
68#834Q812yes—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—41no—
75C3⋊#95—41no—
76#518C2×D311yescharacteristic
77#428—21no—
78C2×#100C2×D311yescharacteristic
79C2×#99—31no—
80C2×#99—31no—
81#399—61no—
82#398—61no—
83#396—61no—
84#390C2×D311yescharacteristic
85#387—61no—
86#386—31no—
87#386—31no—
88#385—61no—
89#385—61no—
90#385—61no—
91#380—61no—
92#342—61no—
93#342—61no—
94#339—21no—
95C24×C3C2×D411yescharacteristic
96C24×C3C2×D411yescharacteristic
97C2×C3×D4—41no—
98C2×C3×D4—41no—
99C2×C3×D4C2×Q811yescharacteristic
100C2×C3×D4C2×Q811yescharacteristic
101C2×C3×D4C2×Q811yescharacteristic
102C2×C3×D4#4111yescharacteristic
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4C2×D411yescharacteristic
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—42no—
115C22×Dic3—41no—
116C22×Dic3—41no—
117C22×Dic3—41no—
118C22×Dic3—41no—
119C22×Dic3—81no—
120C3×M16—22no—
121C3×M16—41no—
122C2×C3×C8—41no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C22⋊C4—81no—
130C3×C22⋊C4—81no—
131C3×C42—41no—
132C3×C42—41no—
133#217—22no—
134#217—22no—
135#217—42no—
136#217—41no—
137#217—41no—
138#217—41no—
139#217—41no—
140#217—81no—
141C4⋊Dic3—81no—
142Dic3⋊C4—42no—
143Dic3⋊C4—81no—
144Dic3⋊C4—81no—
145Dic3⋊C4—81no—
146Dic3⋊C4—82no—
147Dic3⋊C4—82no—
148C22×D4C22×D311yescharacteristic
149C23×C4—31no—
150C23×C4—31no—
151C2×M16—21no—
152C4×D4—41no—
153C4×D4—41no—
154#117—21no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—121no—
161C2×C4⋊C4—121no—
162C2×C4⋊C4#6711yescharacteristic
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—31no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—61no—
171C2×C22⋊C4—61no—
172C2×C22⋊C4—61no—
173C2×C42#6711yescharacteristic
174#107—41no—
175#106—41no—
176#100—21no—
177#100Dic612yes—
178#99—32no—
179#99—32no—
180#99—61no—
181#99—61no—
182#95—61no—
183#95—61no—
184#95—61no—
185#95—61no—
186#95—61no—
187#95—121no—
188#95—121no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3#12011yescharacteristic
198C23×C3C22⋊Q811yescharacteristic
199C23×C3C22⋊Q811yescharacteristic
200C23×C3C4⋊Q811yescharacteristic
201C23×C3C4⋊Q811yescharacteristic
202C3×D4—41no—
203C3×D4—41no—
204C3×D4—41no—
205C3×D4—41no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—81no—
217C2×C3×C4—81no—
218C2×C3×C4—81no—
219C2×C3×C4—81no—
220C2×C3×C4C22⋊Q811yescharacteristic
221C2×C3×C4C4⋊Q811yescharacteristic
222C2×Dic3—22no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—42no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—81no—
239C2×Dic3—81no—
240C2×Dic3—81no—
241C2×Dic3—81no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244C3×C8—42no—
245C24D3×D411yescharacteristic
246C24D3×D411yescharacteristic
247C2×D4—41no—
248C2×D4—41no—
249C2×D4C2×Dic611yescharacteristic
250C2×D4#23511yescharacteristic
251C2×D4D3×Q811yescharacteristic
252C2×D4D3×Q811yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—122no—
264C22×C4—121no—
265C22×C4—121no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—241no—
269C22×C4C2×#6711yescharacteristic
270M16—22no—
271M16—41no—
272C2×C8—41no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—41no—
276C4⋊C4—122no—
277C4⋊C4—241no—
278C4⋊C4—241no—
279C4⋊C4—241no—
280C4⋊C4—241no—
281C4⋊C4—242no—
282C4⋊C4—242no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—62no—
287C22⋊C4—62no—
288C22⋊C4—81no—
289C22⋊C4—81no—
290C22⋊C4—122no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C22⋊C4—121no—
295C22⋊C4—241no—
296C42—41no—
297C42—41no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3#39511yescharacteristic
315C22×C3B(4,𝔽2)11yescharacteristic
316C22×C3#45811yescharacteristic
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—81no—
320C3×C4—81no—
321C3×C4—81no—
322C3×C4—81no—
323Dic3—42no—
324Dic3—81no—
325Dic3—81no—
326Dic3—82no—
327Dic3—82no—
328Dic3—81no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23#88811yescharacteristic
338C23#88811yescharacteristic
339C23#89611yescharacteristic
340C23#89611yescharacteristic
341C23#93711yescharacteristic
342D4—41no—
343D4—41no—
344D4—41no—
345D4—41no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—62no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—122no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—241no—
377C2×C4—241no—
378C2×C4—241no—
379C2×C4—241no—
380C2×C4—241no—
381C2×C4—241no—
382C2×C4#92411yescharacteristic
383C2×C4#94411yescharacteristic
384C8—42no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—41no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3#206211yescharacteristic
392C2×C3#219711yescharacteristic
393C2×C3#220011yescharacteristic
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—81no—
409C22—81no—
410C22#497911yescharacteristic
411C22#503211yescharacteristic
412C22#505311yescharacteristic
413C4—21no—
414C4—21no—
415C4—81no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4—122no—
420C4—241no—
421C4—241no—
422C4—242no—
423C4—242no—
424C4—241no—
425C3#1346211yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—41no—
429C2—41no—
430C2—41no—
431C2—41no—
432C2#6959411yescharacteristic
433C2#7098511yescharacteristic
434C2#7099011yescharacteristic
subgroupG/H
centerC22#5032
commutator, prime radical, cosocleC22×C3×D4C23
FrattiniC22×D4C22×D3
FittingC3×#1856C2
socleC22×C3B(4,𝔽2)
hypercenter#1856D3
hypocenterC3#13462
LCSG ▷ C22×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1856

Complements

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

NH ≅ G/Ntype
425. C3#13462semidirect

Maximal subgroups

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

subgroupindex
#764052
C3×#18562
#716572
#676982
#676412
#134623