Finite Group Databasebeta

← #10692973

Subgroups — #10692973

Order 768.

556 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 109 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#82197C211yesmaximal, characteristic
2#78602C211yesmaximal, characteristic
3C3×#1824C211yesmaximal, characteristic, complemented
4#71554C211yesmaximal, characteristic, complemented
5#69530C211yesmaximal, characteristic
6#67676C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9473—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×C4×#67C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13#5213—21no—
14#5212C2211yescharacteristic
15#5212C2211yescharacteristic
16#5210—21no—
17C2×#830C2211yescharacteristic
18#5111—22no—
19#4993C412yescomplemented
20#4987—21no—
21#4985C412yescomplemented
22#4972—21no—
23C2×#806—21no—
24#4730—22no—
25#4726—21no—
26#4719—21no—
27C3×#340—22no—
28#4529—22no—
29#4527—21no—
30#2973—31no—
31#2487—31no—
32#1824—31no—
33#1824D311yescharacteristic, complemented
34#1799—31no—
35#1289—31no—
36#1289—31no—
37C23×C3×C4—21no—
38C23×C3×C4C2311yescharacteristic
39C22×#67D411yescharacteristic
40C22×C4×D3D411yescharacteristic
41C22×C3×C8D412yes—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4C2×C412yes—
49C2×C3×C22⋊C4—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217D411yescharacteristic
53C4×#67—21no—
54C4×#67—21no—
55C4×#67—21no—
56C4×#67—21no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4C2×C411yescharacteristic
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×Dic3⋊C4—22no—
62C2×Dic3⋊C4C2×C411yescharacteristic
63C2×C4×Dic3D411yescharacteristic
64C22×C3⋊C8D412yes—
65C2×ΓL(1,𝔽25)—22no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95—21no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95—21no—
72C3⋊#95C2×C412yes—
73#830—21no—
74#830—21no—
75(C2×D3)⋊C8—22no—
76#815—22no—
77Dic3⋊C8—22no—
78#806—42no—
79C2×C4×D4—31no—
80C22×C4⋊C4C2×D311yescharacteristic
81#435—62no—
82#435—62no—
83#432—61no—
84C2×#99—61no—
85#407—61no—
86C2×C22⋊C8—31no—
87C2×C22⋊C8C2×D311yescharacteristic
88#400—32no—
89#399—32no—
90#389—61no—
91#386—61no—
92#385—61no—
93#380—31no—
94#380—31no—
95#380—61no—
96#380C2×D311yescharacteristic
97#342—61no—
98#340—22no—
99#340—62no—
100C24×C3C2×D411yescharacteristic
101C23×D3C22⋊C411yescharacteristic
102C22×C3×C4—22no—
103C22×C3×C4—42no—
104C22×C3×C4—41no—
105C22×C3×C4—41no—
106C22×C3×C4—41no—
107C22×C3×C4—82no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C2×#67—21no—
115C2×#67—21no—
116C2×#67—21no—
117C2×#67—21no—
118C2×#67—21no—
119C2×#67—21no—
120C22×Dic3—22no—
121C22×Dic3—41no—
122C22×Dic3C22⋊C411yescharacteristic
123C22×Dic3C22⋊C411yescharacteristic
124C22×Dic3C22⋊C411yescharacteristic
125C2×C4×D3—21no—
126C2×C4×D3—21no—
127C2×C4×D3—42no—
128C2×C3×C8—22no—
129C2×C3×C8—42no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—42no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137#217—21no—
138#217—21no—
139#217—41no—
140#217—41no—
141#217—41no—
142#217—41no—
143(C2×D3)⋊C4—21no—
144(C2×D3)⋊C4—21no—
145(C2×D3)⋊C4—82no—
146C4⋊Dic3—42no—
147C4⋊Dic3—42no—
148Dic3⋊C4—21no—
149Dic3⋊C4—21no—
150Dic3⋊C4—82no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—42no—
155ΓL(1,𝔽25)—82no—
156C22×D4—31no—
157C23×C4—21no—
158C23×C4—31no—
159C23×C4C22×D311yescharacteristic
160C2×M16—62no—
161C22×C8—32no—
162C22×C8#6712yes—
163C4×D4—61no—
164C4×D4—61no—
165C4×D4—61no—
166C4×D4—61no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—62no—
177C2×C4⋊C4C4×D312yes—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—31no—
180C2×C22⋊C4—31no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—62no—
184C2×C42—31no—
185C4⋊C8—62no—
186#99—122no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—62no—
190C22⋊C8—61no—
191C22⋊C8—61no—
192#97—62no—
193#95—21no—
194#95—32no—
195#95—62no—
196#95—62no—
197#95—61no—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3C2×C22⋊C411yescharacteristic
201C23×C3#12111yescharacteristic
202C23×C3#12111yescharacteristic
203C23×C3AGL(1,ℤ/8)11yescharacteristic
204C23×C3#13711yescharacteristic
205C23×C3#13711yescharacteristic
206C23×C3#13711yescharacteristic
207C22×D3—21no—
208C22×D3—21no—
209C22×D3—21no—
210C22×D3—41no—
211C22×D3—41no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—82no—
222C2×C3×C4—82no—
223C2×C3×C4—81no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12311yescharacteristic
230#67—41no—
231#67—41no—
232#67—41no—
233#67—41no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—81no—
245C2×Dic3—81no—
246C2×Dic3—81no—
247C4×D3—42no—
248C4×D3—81no—
249C3×C8—42no—
250C3⋊C8—42no—
251C24—31no—
252C24D3×D411yescharacteristic
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259C22×C4—22no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—62no—
270C22×C4—82no—
271C22×C4—122no—
272C22×C4—121no—
273C22×C4C2×C4×D311yescharacteristic
274C22×C4#23511yescharacteristic
275C22×C4D3×D411yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4#23711yescharacteristic
278C22×C4C2×#6711yescharacteristic
279M16—242no—
280C2×C8—22no—
281C2×C8—42no—
282C2×C8—62no—
283C2×C8—122no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—122no—
292C4⋊C4—122no—
293C4⋊C4—242no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C22⋊C4—242no—
305C42—61no—
306C42—61no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3#38611yescharacteristic
318C22×C3#41711yescharacteristic
319C22×C3#41911yescharacteristic
320C22×C3#46811yescharacteristic
321C22×C3#46811yescharacteristic
322C22×C3#47011yescharacteristic
323C22×C3#47011yescharacteristic
324C2×D3—42no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—81no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—82no—
333C3×C4—161no—
334C3×C4#45812yes—
335Dic3—41no—
336Dic3—41no—
337Dic3—41no—
338Dic3—82no—
339Dic3—161no—
340C23—41no—
341C23—41no—
342C23—61no—
343C23—61no—
344C23—61no—
345C23—121no—
346C23—121no—
347C23D3×C22⋊C411yescharacteristic
348C23#88411yescharacteristic
349C23#91411yescharacteristic
350C23#91511yescharacteristic
351C23#91811yescharacteristic
352C23#91811yescharacteristic
353C23#93811yescharacteristic
354D4—121no—
355D4—121no—
356D4—121no—
357D4—121no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—122no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—241no—
384C2×C4—241no—
385C2×C4—241no—
386C2×C4—241no—
387C2×C4#67⋊C411yescharacteristic
388C2×C4#88311yescharacteristic
389C2×C4C4×#6711yescharacteristic
390C2×C4#93711yescharacteristic
391C8—42no—
392C8—122no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3#189011yescharacteristic
400C2×C3#189411yescharacteristic
401C2×C3#191011yescharacteristic
402D3—81no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—122no—
414C22—121no—
415C22—121no—
416C22—121no—
417C22—241no—
418C22#480211yescharacteristic
419C22#480311yescharacteristic
420C22#481211yescharacteristic
421C22#481611yescharacteristic
422C22#495611yescharacteristic
423C22#517411yescharacteristic
424C22#518911yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—82no—
429C4—121no—
430C4—121no—
431C4—121no—
432C4—161no—
433C4—242no—
434C4—481no—
435C4#505312yes—
436C3#947311yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—241no—
444C2#6953811yescharacteristic
445C2#7027111yescharacteristic
446C2#7033011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1824C2
socleC22×C3#386
prime radicalC22×#67D4
cosocleC23×C3×C4C23
hypercenter#1824D3
hypocenterC3#9473
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
3. C3×#1824C2semidirect
4. #71554C2semidirect
6. #67676C2semidirect
7. #67625C2semidirect
19. #4993C4semidirect
21. #4985C4semidirect
33. #1824D3semidirect
436. C3#9473semidirect

Maximal subgroups

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

subgroupindex
#821972
#786022
C3×#18242
#715542
#695302
#676762
#676252
#94733