Finite Group Databasebeta

← #10695837

Subgroups — #10695837

Order 768.

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

normal subgroups only

This group has 602 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#82722C211yesmaximal, characteristic
2#82616C211yesmaximal, characteristic, complemented
3#78624C211yesmaximal, characteristic, complemented
4C3×#1510C211yesmaximal, characteristic, complemented
5#69534C211yesmaximal, characteristic, complemented
6#69531C211yesmaximal, characteristic, complemented
7#67686C211yesmaximal, characteristic, complemented
8#9983—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#926C2211yescharacteristic
12C3×#407C2211yescharacteristic
13C3×#381C2211yescharacteristic
14#5223—21no—
15#5223—22no—
16#5215—21no—
17#5211C2211yescharacteristic
18#5207C2211yescharacteristic
19#5131—21no—
20#5121—21no—
21#5035—22no—
22#5034—21no—
23#5033—21no—
24#4992—21no—
25#4964—21no—
26#4960—21no—
27#4953—21no—
28#4737—22no—
29#4736—21no—
30#4735—21no—
31C3×#327—22no—
32#4527—22no—
33#4451—22no—
34#3094—31no—
35#3094—31no—
36#2815—31no—
37#1820—31no—
38#1510—31no—
39#1510D311yescharacteristic, complemented
40#1306—31no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×#67—21no—
45C22×#67—22no—
46C22×D12D411yescharacteristic
47C2×C3×M16—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×#217—21no—
56C2×#217—22no—
57C2×#217D412yes—
58#930—21no—
59#930—21no—
60#930—21no—
61#930—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—22no—
64#926—21no—
65#926—21no—
66#926—21no—
67#926—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D411yescharacteristic, complemented
70C2×C4⋊Dic3D411yescharacteristic, complemented
71C2×C4×Dic3—21no—
72C2×#208—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8D412yescomplemented
75C3×#95—21no—
76C3×#95—21no—
77#832—22no—
78#832—22no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82#830—21no—
83#830D412yescomplemented
84#821—22no—
85#821—22no—
86#818—22no—
87#817—22no—
88#808—22no—
89#807—22no—
90C2×#121—31no—
91C2×#121C2×D311yescharacteristic
92C2×#117—31no—
93#482—62no—
94#482—62no—
95#481—61no—
96#481—61no—
97#480—61no—
98#480—61no—
99#428—61no—
100#428—61no—
101#418—61no—
102#418—61no—
103#407—31no—
104#407C2×D311yescharacteristic
105#397—61no—
106#397—61no—
107#392—61no—
108#392—62no—
109#392—61no—
110#381—31no—
111#381C2×D311yescharacteristic
112#342—62no—
113#327—22no—
114#327—62no—
115C24×C3—21no—
116C24×C3C2×D411yescharacteristic
117C23×D3—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—42no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—82no—
124C22×C3×C4—21no—
125C22×C3×C4—42no—
126C22×C3×C4—41no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C2×#67—41no—
134C2×#67—41no—
135C2×#67—82no—
136C2×#67—82no—
137C2×#67—82no—
138C2×#67—82no—
139C22×Dic3—22no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C2×D12—21no—
144C2×D12—21no—
145C2×D12—42no—
146C3×M16—42no—
147C2×C3×C8—22no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—21no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—42no—
153#217—41no—
154#217—41no—
155#217—42no—
156#217—82no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—82no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4×Dic3—41no—
167C4×Dic3—41no—
168#208—42no—
169C2×C3⋊C8—22no—
170C22×D4—21no—
171C22×D4—31no—
172C22×D4—61no—
173C22×D4—62no—
174C22×D4D1211yescharacteristic
175C23×C4C22×D311yescharacteristic
176C2×M16—21no—
177C2×M16—61no—
178#121—21no—
179#121—21no—
180#121—21no—
181#121—21no—
182#121—61no—
183#121—61no—
184#121—61no—
185#121—61no—
186#117—61no—
187#117—61no—
188#117—61no—
189#117—61no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4D1211yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—32no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—62no—
200C2×C22⋊C4—62no—
201C2×C42—61no—
202#107—62no—
203#107—62no—
204#106—62no—
205#106—62no—
206#102—62no—
207#102—62no—
208#102—62no—
209#102—62no—
210C22⋊C8—21no—
211C22⋊C8—32no—
212C22⋊C8—61no—
213C22⋊C8#6712yescomplemented
214#95—21no—
215#95—21no—
216#95—62no—
217#95—61no—
218#95—61no—
219C23×C3—22no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—42no—
225C23×C3—41no—
226C23×C3—42no—
227C23×C3#12011yescharacteristic
228C23×C3#12111yescharacteristic
229C23×C3#12711yescharacteristic
230C22×D3—41no—
231C22×D3—41no—
232C22×D3—42no—
233C22×D3—41no—
234C22×D3—42no—
235C3×D4—42no—
236C3×D4—81no—
237C3×D4—82no—
238C3×D4—82no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—81no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254#67—82no—
255#67—82no—
256#67—88no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—41no—
261C2×Dic3—41no—
262C2×Dic3—41no—
263C2×Dic3—42no—
264C2×Dic3—81no—
265C2×Dic3—81no—
266D12—42no—
267D12—81no—
268C3×C8—42no—
269C3⋊C8—42no—
270C24—21no—
271C24—61no—
272C24D3×D411yescharacteristic
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—61no—
279C2×D4—61no—
280C2×D4—82no—
281C2×D4—122no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—242no—
285C2×D4—242no—
286C2×D4—242no—
287C2×D4—242no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—62no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4C2×D1211yescharacteristic
296C22×C4#23511yescharacteristic
297C22×C4D3×D411yescharacteristic
298C22×C4D3×D411yescharacteristic
299C22×C4D3×D411yescharacteristic
300C22×C4C2×#6711yescharacteristic
301M16—42no—
302M16—122no—
303C2×C8—22no—
304C2×C8—62no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—122no—
321C22⋊C4—242no—
322C22⋊C4—242no—
323C42—121no—
324C42—121no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—82no—
334C22×C3—82no—
335C22×C3—82no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3#39211yescharacteristic
340C22×C3#45812yes—
341C2×D3—82no—
342C2×D3—82no—
343C2×D3—81no—
344C2×D3—81no—
345C2×D3—81no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—81no—
351C3×C4—161no—
352Dic3—82no—
353Dic3—81no—
354Dic3—81no—
355Dic3—81no—
356C23—22no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—121no—
365C23—122no—
366C23—121no—
367C23—121no—
368C23—122no—
369C23#88211yescharacteristic
370C23#88411yescharacteristic
371C23#94011yescharacteristic
372D4—42no—
373D4—81no—
374D4—82no—
375D4—82no—
376D4—122no—
377D4—241no—
378D4—242no—
379D4—242no—
380D4—248no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—122no—
399C2×C4—241no—
400C2×C4—241no—
401C2×C4#88211yescharacteristic
402C2×C4#88411yescharacteristic
403C2×C4#93011yescharacteristic
404C2×C4#93711yescharacteristic
405C8—42no—
406C8—122no—
407C2×C3—22no—
408C2×C3—41no—
409C2×C3—41no—
410C2×C3—161no—
411C2×C3#200811yescharacteristic
412C2×C3#200811yescharacteristic
413C2×C3#202211yescharacteristic
414D3—161no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—82no—
424C22—82no—
425C22—82no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—242no—
430C22—242no—
431C22—241no—
432C22—241no—
433C22—241no—
434C22#474212yes—
435C22#496011yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—81no—
441C4—161no—
442C4—242no—
443C4—241no—
444C4—241no—
445C4—241no—
446C3#998311yescharacteristic, complemented, Sylow
447C2—22no—
448C2—41no—
449C2—41no—
450C2—161no—
451C2—481no—
452C2#7003111yescharacteristic
453C2#7033111yescharacteristic
454C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1510C2
socleC22×C3#392
prime radical#82722C2
hypercenter#1510D3
hypocenterC3#9983
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1510

Complements

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

NH ≅ G/Ntype
2. #82616C2semidirect
3. #78624C2semidirect
4. C3×#1510C2semidirect
5. #69534C2semidirect
6. #69531C2semidirect
7. #67686C2semidirect
39. #1510D3semidirect
69. C2×C4⋊Dic3D4semidirect
70. C2×C4⋊Dic3D4semidirect
74. C3×C22⋊C8D4semidirect
83. #830D4semidirect
213. C22⋊C8#67semidirect
446. C3#9983semidirect

Maximal subgroups

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

subgroupindex
#827222
#826162
#786242
C3×#15102
#695342
#695312
#676862
#99833