Finite Group Databasebeta

← #10690301

Subgroups — #10690301

Order 768.

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

normal subgroups only

This group has 468 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
1#82919C211yesmaximal, characteristic
2#82913C211yesmaximal, characteristic
3C3×#1501C211yesmaximal, characteristic
4#69911C211yesmaximal, characteristic, complemented
5#69442C211yesmaximal, characteristic, complemented
6#67680C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9038—31nomaximal, Sylow
9C3×#525C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13#5234—21no—
14#5233—21no—
15#5207C2211yescharacteristic
16C2×#830C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5045—21no—
19#5044C411yescharacteristic, complemented
20#5043C411yescharacteristic, complemented
21#5035—21no—
22#5034C411yescharacteristic, complemented
23#5033C411yescharacteristic, complemented
24#5012—21no—
25#5011—21no—
26#5011—21no—
27D4×C3⋊C8—21no—
28#5000—21no—
29#4998—21no—
30C3×#328—21no—
31C3×#327—21no—
32#4550—21no—
33#4550—21no—
34#4523—21no—
35#4523—21no—
36#3091—31no—
37#2976—31no—
38#1826—31no—
39#1818—31no—
40#1501D311yescharacteristic
41#1298—31no—
42#1282—31no—
43C2×C3×#41D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C22×C3×C8—21no—
46C3×#123—21no—
47C3×C22⋊Q8C2×C411yescharacteristic
48C3×C22⋊Q8D411yescharacteristic
49C3×#121C2×C411yescharacteristic
50C3×#121D411yescharacteristic
51C3×#120—21no—
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×#117D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C42—21no—
57C2×#217—21no—
58#926—21no—
59#926—21no—
60#926—21no—
61#926—21no—
62C2×C4⋊Dic3C2×C411yescharacteristic
63C2×C4⋊Dic3C2×C411yescharacteristic
64C2×C4×Dic3—21no—
65C2×#208—21no—
66C2×#208D411yescharacteristic
67C2×#208D411yescharacteristic
68C22×C3⋊C8—21no—
69C22×C3⋊C8—21no—
70C22×C3⋊C8D411yescharacteristic
71C22×C3⋊C8D411yescharacteristic
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×#95—21no—
75#835—21no—
76#835—21no—
77#832—21no—
78#832—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84#830—21no—
85#830—21no—
86#830C2×C411yescharacteristic
87#830C2×C411yescharacteristic
88#808—21no—
89#808—21no—
90#807—21no—
91#807—21no—
92C4⋊(C3⋊C8)—21no—
93C4⋊(C3⋊C8)—21no—
94#803—21no—
95C4×C3⋊C8—21no—
96#525C2×D311yescharacteristic
97C2×#117—31no—
98#485—61no—
99#484—31no—
100#483—31no—
101#482—61no—
102#481—31no—
103#480—31no—
104#436—61no—
105#435—61no—
106#435—61no—
107C8×D4—61no—
108#433—61no—
109#427—61no—
110#416—61no—
111#408—61no—
112#407—31no—
113C2×C22⋊C8—31no—
114C2×C22⋊C8—31no—
115C2×C22⋊C8C2×D311yescharacteristic
116C4×C22⋊C4C2×D311yescharacteristic
117#339—61no—
118#339—61no—
119#336—61no—
120#336—61no—
121#328—21no—
122#327—21no—
123C24×C3#4111yescharacteristic
124C3×#41—41no—
125C3×#41—41no—
126C2×C3×Q8C22⋊C411yescharacteristic
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4C22⋊C411yescharacteristic
131C22×C3×C4—21no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4C22×C411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×Dic3—21no—
142C22×Dic3—21no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C2×C3×C8—41no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4C22⋊C411yescharacteristic
149C3×C4⋊C4C22⋊C411yescharacteristic
150C3×C22⋊C4—21no—
151C3×C22⋊C4—21no—
152C3×C22⋊C4—21no—
153C3×C22⋊C4—21no—
154C3×C22⋊C4—21no—
155C3×C22⋊C4—22no—
156C3×C22⋊C4—41no—
157C3×C42—21no—
158C3×C42—41no—
159C3×C42—41no—
160#217—41no—
161#217—41no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4×Dic3—41no—
167C4×Dic3—41no—
168#208—41no—
169#208—41no—
170#208—81no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179C2×C3⋊C8—41no—
180C2×C3⋊C8—41no—
181C2×C3⋊C8—81no—
182C2×#41D1211yescharacteristic
183C23×C4C22×D311yescharacteristic
184C2×M16—31no—
185C2×M16—31no—
186C2×M16—61no—
187C22×C8—21no—
188C22×C8—31no—
189C22×C8—31no—
190C22×C8—61no—
191C22×C8—61no—
192#123—21no—
193C22⋊Q8C4×D311yescharacteristic
194C22⋊Q8#6711yescharacteristic
195#121C4×D311yescharacteristic
196#121#6711yescharacteristic
197#120—21no—
198C4×D4—21no—
199C4×D4—21no—
200#117—61no—
201#117—61no—
202#117—61no—
203#117—61no—
204#117D1211yescharacteristic
205C2×C4⋊C4—31no—
206C2×C4⋊C4—31no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—61no—
209C2×C42—21no—
210C2×C42—61no—
211#107—61no—
212#107—61no—
213#106—61no—
214#106—61no—
215C4⋊C8—61no—
216C4⋊C8—61no—
217#103—61no—
218#103—61no—
219#102—61no—
220#102—61no—
221C22⋊C8—21no—
222C22⋊C8—21no—
223C22⋊C8—31no—
224C22⋊C8—31no—
225C22⋊C8—61no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228C22⋊C8—61no—
229C22⋊C8—61no—
230C22⋊C8—61no—
231C22⋊C8—61no—
232#97—61no—
233C4×C8—61no—
234#95—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3C4×D411yescharacteristic
241C23×C3#12111yescharacteristic
242C23×C3#12311yescharacteristic
243C3×Q8—41no—
244C3×D4—41no—
245C3×D4—41no—
246C3×D4—41no—
247C3×D4—81no—
248C3×D4—81no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—81no—
265C2×C3×C4—81no—
266C2×C3×C4C2×C22⋊C411yescharacteristic
267C2×C3×C4C4×D411yescharacteristic
268C2×C3×C4#12011yescharacteristic
269C2×C3×C4#12111yescharacteristic
270C2×Dic3—42no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C3×C8—81no—
276C3⋊C8—41no—
277C3⋊C8—41no—
278C3⋊C8—41no—
279C3⋊C8—41no—
280C3⋊C8—41no—
281C3⋊C8—41no—
282C24#23711yescharacteristic
283#41—41no—
284#41—41no—
285C2×Q8(C2×D3)⋊C411yescharacteristic
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4(C2×D3)⋊C411yescharacteristic
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4C2×C4×D311yescharacteristic
297C22×C4C2×D1211yescharacteristic
298C22×C4D3×D411yescharacteristic
299C22×C4D3×D411yescharacteristic
300C22×C4#23711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302M16—121no—
303M16—121no—
304M16—241no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—61no—
309C2×C8—61no—
310C2×C8—61no—
311C2×C8—61no—
312C2×C8—61no—
313C2×C8—61no—
314C2×C8—121no—
315C2×C8—121no—
316C2×C8—121no—
317C2×C8—121no—
318C2×C8—241no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4(C2×D3)⋊C411yescharacteristic
326C4⋊C4(C2×D3)⋊C411yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—22no—
333C22⋊C4—41no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C42—21no—
337C42—41no—
338C42—41no—
339C42—121no—
340C42—121no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3#38611yescharacteristic
355C22×C3#44311yescharacteristic
356C22×C3#44411yescharacteristic
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—41no—
361C3×C4—41no—
362C3×C4—41no—
363C3×C4—82no—
364C3×C4#45811yescharacteristic
365C3×C4#45811yescharacteristic
366Dic3—81no—
367Dic3—81no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—41no—
372C23—41no—
373C23#67⋊C411yescharacteristic
374C23#88611yescharacteristic
375C23#93911yescharacteristic
376Q8—41no—
377D4—41no—
378D4—41no—
379D4—41no—
380D4—81no—
381D4—81no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—122no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4#67⋊C411yescharacteristic
405C2×C4#88211yescharacteristic
406C2×C4C2×(C2×D3)⋊C411yescharacteristic
407C2×C4#93911yescharacteristic
408C8—81no—
409C8—121no—
410C8—121no—
411C8—121no—
412C8—121no—
413C8—121no—
414C8—121no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3—81no—
420C2×C3#187611yescharacteristic
421C2×C3#188111yescharacteristic
422C2×C3#189011yescharacteristic
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—81no—
436C22#482311yescharacteristic
437C22#482411yescharacteristic
438C22#495811yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—82no—
446C4—241no—
447C4—241no—
448C4#474211yescharacteristic
449C4#474211yescharacteristic
450C3#903811yescharacteristic, complemented, Sylow
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2#7002311yescharacteristic
457C2#7029311yescharacteristic
458C2#7037711yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1501C2
socleC22×C3#386
prime radicalC3×#525C22
cosocleC23×C3×C4C23
hypercenter#1501D3
hypocenterC3#9038
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1501

Complements

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

NH ≅ G/Ntype
4. #69911C2semidirect
5. #69442C2semidirect
6. #67680C2semidirect
7. #67617C2semidirect
19. #5044C4semidirect
20. #5043C4semidirect
22. #5034C4semidirect
23. #5033C4semidirect
450. C3#9038semidirect

Maximal subgroups

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

subgroupindex
#829192
#829132
C3×#15012
#699112
#694422
#676802
#676172
#90383