Finite Group Databasebeta

← #10690300

Subgroups — #10690300

Order 768.

468 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 93 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#82215C211yesmaximal, characteristic
2#78635C211yesmaximal, characteristic
3C3×#1826C211yesmaximal, characteristic, complemented
4#71584C211yesmaximal, characteristic, complemented
5#69443C211yesmaximal, characteristic
6#67688C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9038—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10#5791C2211yescharacteristic
11C3×#407C2211yescharacteristic
12C3×C4×C22⋊C4C2211yescharacteristic
13C2×#830C2211yescharacteristic
14C2×#830C2211yescharacteristic
15C2×#830C2211yescharacteristic
16#5128—21no—
17#5127—21no—
18#5003—21no—
19#5002—21no—
20#4998—21no—
21#4996—21no—
22#4971C411yescharacteristic, complemented
23#4970C411yescharacteristic, complemented
24#4968C411yescharacteristic, complemented
25#4967C411yescharacteristic, complemented
26#4730—21no—
27#4727—21no—
28#67⋊C8—21no—
29#4720—21no—
30C3×#339—21no—
31C3×#339—21no—
32#4524—21no—
33#4524—21no—
34#4452—21no—
35#4451—21no—
36#3091—31no—
37#2976—31no—
38#1826D311yescharacteristic, complemented
39#1818—31no—
40#1501—31no—
41#1298—31no—
42#1282—31no—
43C23×C3×C4C2311yescharacteristic
44C2×#235D411yescharacteristic
45C2×C3×M16D411yescharacteristic
46C2×C3×M16D411yescharacteristic
47C3×#117—21no—
48C3×#117—21no—
49C3×#117—21no—
50C3×#117—21no—
51C2×C3×C4⋊C4C2×C411yescharacteristic
52C2×C3×C4⋊C4C2×C411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C42—21no—
56C2×C3×C42—21no—
57#953—21no—
58#930C2×C411yescharacteristic
59#930D411yescharacteristic
60#929—21no—
61C4×#67—21no—
62C4×#67—21no—
63#926D411yescharacteristic
64#924C2×C411yescharacteristic
65#924D411yescharacteristic
66C22×C3⋊C8—21no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8D411yescharacteristic
69C22×C3⋊C8D411yescharacteristic
70C2×ΓL(1,𝔽25)—21no—
71C2×D3×C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×#95—21no—
75#830—21no—
76#830—21no—
77#830—21no—
78#830—21no—
79#830—21no—
80#830C2×C411yescharacteristic
81#830C2×C411yescharacteristic
82(C2×D3)⋊C8—21no—
83(C2×D3)⋊C8—21no—
84#815—21no—
85Dic3⋊C8—21no—
86Dic3⋊C8—21no—
87C8×Dic3—21no—
88#810—21no—
89#810—21no—
90#809—21no—
91#809—21no—
92#808—21no—
93#808—21no—
94#807—21no—
95#807—21no—
96#525—31no—
97C2×#117C2×D311yescharacteristic
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#407C2×D311yescharacteristic
113C2×C22⋊C8—31no—
114C2×C22⋊C8—31no—
115C2×C22⋊C8—31no—
116C4×C22⋊C4C2×D311yescharacteristic
117#339—21no—
118#339—21no—
119#336—61no—
120#336—61no—
121#328—61no—
122#327—61no—
123C24×C3#4111yescharacteristic
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4C22×C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C2×#67—21no—
136C2×#67—21no—
137C2×#67—21no—
138#235—41no—
139#235—41no—
140C2×D12C22⋊C411yescharacteristic
141C2×C4×D3—21no—
142C2×Dic6C22⋊C411yescharacteristic
143C3×M16—41no—
144C3×M16—41no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C22⋊C4—22no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C42—41no—
156C3×C42—41no—
157C3×C42—41no—
158C3×C42—41no—
159#217—21no—
160#217—21no—
161#217—21no—
162(C2×D3)⋊C4—21no—
163(C2×D3)⋊C4—21no—
164C4⋊Dic3C22⋊C411yescharacteristic
165C4⋊Dic3C22⋊C411yescharacteristic
166Dic3⋊C4—21no—
167Dic3⋊C4—21no—
168C4×Dic3—21no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—21no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—41no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179C2×C3⋊C8—41no—
180ΓL(1,𝔽25)—81no—
181D3×C8—81no—
182C2×#41—31no—
183C23×C4C22×D311yescharacteristic
184C2×M16—61no—
185C2×M16#6711yescharacteristic
186C2×M16#6711yescharacteristic
187C22×C8—31no—
188C22×C8—31no—
189C22×C8—61no—
190C22×C8—61no—
191C22×C8—61no—
192#123—61no—
193C22⋊Q8—31no—
194C22⋊Q8—31no—
195#121—31no—
196#121—31no—
197#120—61no—
198C4×D4—61no—
199C4×D4—61no—
200#117—21no—
201#117—21no—
202#117—21no—
203#117—21no—
204#117—31no—
205C2×C4⋊C4C4×D311yescharacteristic
206C2×C4⋊C4C4×D311yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42—21no—
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—41no—
238C23×C3—41no—
239C23×C3C4×D411yescharacteristic
240C23×C3#12111yescharacteristic
241C23×C3#12311yescharacteristic
242C22×D3—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
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—81no—
259C2×C3×C4—81no—
260C2×C3×C4C2×C22⋊C411yescharacteristic
261C2×C3×C4C4×D411yescharacteristic
262C2×C3×C4#12011yescharacteristic
263C2×C3×C4#12111yescharacteristic
264#67—41no—
265#67—41no—
266#67—81no—
267#67—81no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271D12—41no—
272C4×D3—41no—
273C4×D3—41no—
274Dic6—41no—
275C3×C8—41no—
276C3×C8—41no—
277C3⋊C8—41no—
278C3⋊C8—41no—
279C3⋊C8—41no—
280C3⋊C8—41no—
281C3⋊C8—81no—
282C24#23711yescharacteristic
283#41—121no—
284#41—121no—
285C2×Q8—31no—
286C2×D4—31no—
287C2×D4—61no—
288C2×D4—61no—
289C2×D4—61no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—61no—
296C22×C4C2×C4×D311yescharacteristic
297C22×C4#23511yescharacteristic
298C22×C4D3×D411yescharacteristic
299C22×C4D3×D411yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4C2×#6711yescharacteristic
302M16—41no—
303M16—41no—
304M16—241no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—61no—
308C2×C8—61no—
309C2×C8—61no—
310C2×C8—61no—
311C2×C8—121no—
312C2×C8—121no—
313C2×C8—121no—
314C2×C8—121no—
315C2×C8—121no—
316C2×C8—121no—
317C2×C8—121no—
318C2×C8—241no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—31no—
324C4⋊C4—31no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C22⋊C4—22no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—61no—
332C22⋊C4—61no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—61no—
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—81no—
350C22×C3—81no—
351C22×C3#38611yescharacteristic
352C22×C3#44311yescharacteristic
353C22×C3#44411yescharacteristic
354C2×D3—41no—
355C2×D3—41no—
356C2×D3—41no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—82no—
361C3×C4—81no—
362C3×C4—81no—
363C3×C4#45811yescharacteristic
364C3×C4#45811yescharacteristic
365Dic3—41no—
366Dic3—41no—
367Dic3—41no—
368C23—21no—
369C23—21no—
370C23—41no—
371C23—41no—
372C23—61no—
373C23#67⋊C411yescharacteristic
374C23#88311yescharacteristic
375C23#93811yescharacteristic
376Q8—121no—
377D4—121no—
378D4—121no—
379D4—121no—
380D4—241no—
381D4—241no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4D3×C22⋊C411yescharacteristic
405C2×C4#88411yescharacteristic
406C2×C4C4×#6711yescharacteristic
407C2×C4#93711yescharacteristic
408C8—41no—
409C8—41no—
410C8—121no—
411C8—121no—
412C8—121no—
413C8—121no—
414C8—241no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3#187611yescharacteristic
420C2×C3#188111yescharacteristic
421C2×C3#189011yescharacteristic
422D3—81no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—81no—
432C22—81no—
433C22—121no—
434C22—121no—
435C22—121no—
436C22#482311yescharacteristic
437C22#482411yescharacteristic
438C22#495611yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—82no—
443C4—81no—
444C4—81no—
445C4—121no—
446C4—121no—
447C4—121no—
448C4#505311yescharacteristic
449C4#505311yescharacteristic
450C3#903811yescharacteristic, complemented, Sylow
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—241no—
456C2#6953811yescharacteristic
457C2#7039011yescharacteristic
458C2#7041611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1826C2
socleC22×C3#386
prime radical#5791C22
cosocleC23×C3×C4C23
hypercenter#1826D3
hypocenterC3#9038
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
3. C3×#1826C2semidirect
4. #71584C2semidirect
6. #67688C2semidirect
7. #67617C2semidirect
22. #4971C4semidirect
23. #4970C4semidirect
24. #4968C4semidirect
25. #4967C4semidirect
38. #1826D3semidirect
450. C3#9038semidirect

Maximal subgroups

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

subgroupindex
#822152
#786352
C3×#18262
#715842
#694432
#676882
#676172
#90383