Finite Group Databasebeta

← #10695801

Subgroups — #10695801

Order 768.

544 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×#3094C211yesmaximal, characteristic, complemented
2#82625C211yesmaximal, characteristic
3#82217C211yesmaximal, characteristic, complemented
4#71676C211yesmaximal, characteristic, complemented
5#71584C211yesmaximal, characteristic, complemented
6#69970C211yesmaximal, characteristic, complemented
7#67687C211yesmaximal, characteristic, complemented
8#9977—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C3×#482—22no—
13C3×#481—21no—
14C3×#480—21no—
15C3×#428—21no—
16C3×#418—21no—
17C3×#407C2211yescharacteristic
18#5217—21no—
19#5207C2211yescharacteristic
20C2×#830C2211yescharacteristic
21C2×#830C2211yescharacteristic
22#5038—22no—
23#5036—21no—
24#5033—21no—
25#5003—22no—
26#5001—21no—
27#4995—21no—
28#4968—21no—
29#4967—21no—
30#4967—21no—
31#4525—22no—
32#4524—22no—
33#4468—22no—
34#3186—31no—
35#3094—31no—
36#3094D311yescharacteristic, complemented
37#1818—31no—
38#1471—31no—
39#1471—31no—
40#1293—31no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×#67—21no—
45C22×D12D411yescharacteristic
46C2×C3×M16—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#117—21no—
52C3×#117—21no—
53C3×#117—21no—
54C3×#117—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C42—21no—
61C2×#213—22no—
62#930—21no—
63#930—21no—
64#930—21no—
65#930—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×C4⋊Dic3D411yescharacteristic
68C2×#208—21no—
69C22×C3⋊C8—21no—
70C22×C3⋊C8D412yes—
71C3×#107—22no—
72C3×#106—22no—
73C3×#102—22no—
74C3×#102—22no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8D412yes—
77#832—22no—
78#830—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830D412yes—
84#809—22no—
85#809—22no—
86#809—22no—
87#808—22no—
88#807—22no—
89#807—22no—
90C2×#121—31no—
91C2×#121C2×D311yescharacteristic
92C2×#117C2×D311yescharacteristic
93#482—22no—
94#482—62no—
95#481—21no—
96#481—61no—
97#480—21no—
98#480—61no—
99#480—61no—
100#480—61no—
101#466—62no—
102#447—61no—
103#447—61no—
104#428—21no—
105#428—61no—
106#418—21no—
107#418—61no—
108#407—31no—
109#407C2×D311yescharacteristic
110C2×C22⋊C8—31no—
111C2×C22⋊C8—31no—
112#336—62no—
113#323—62no—
114#323—62no—
115C24×C3—21no—
116C24×C3#4111yescharacteristic
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—21no—
126C22×C3×C4—21no—
127C22×C3×C4—42no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C2×#67—41no—
135C2×#67—41no—
136C2×#67—82no—
137C22×Dic3—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12—42no—
141C3×M16—42no—
142C2×C3×C8—22no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C42—41no—
154C3×C42—41no—
155#213—82no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—41no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160#208—42no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—42no—
164C2×C3⋊C8—42no—
165C2×C3⋊C8—41no—
166C22×D4—21no—
167C22×D4—31no—
168C22×D4—61no—
169C22×D4#6711yescharacteristic
170C23×C4C22×D311yescharacteristic
171C2×D8—62no—
172C2×M16—21no—
173C2×M16—61no—
174C22×C8—32no—
175C22×C8—61no—
176#121—21no—
177#121—21no—
178#121—21no—
179#121—21no—
180#121—61no—
181#121—61no—
182#121—61no—
183#121—61no—
184#117—21no—
185#117—21no—
186#117—21no—
187#117—21no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4#6711yescharacteristic
190C2×C4⋊C4#6711yescharacteristic
191C2×C4⋊C4#6711yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—61no—
195C2×C42—21no—
196#107—22no—
197#107—62no—
198#107—62no—
199#106—22no—
200#106—62no—
201#102—22no—
202#102—22no—
203#102—62no—
204#102—62no—
205#102—62no—
206#102—62no—
207C22⋊C8—21no—
208C22⋊C8—32no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8#6712yes—
215C23×C3—22no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—42no—
219C23×C3—41no—
220C23×C3—81no—
221C23×C3—81no—
222C23×C3#12111yescharacteristic
223C23×C3#12111yescharacteristic
224C23×C3#12111yescharacteristic
225C22×D3—41no—
226C22×D3—42no—
227C22×D3—81no—
228C22×D3—81no—
229C3×D4—42no—
230C3×D4—42no—
231C3×D4—82no—
232C3×D4—82no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—42no—
244C2×C3×C4—81no—
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12011yescharacteristic
248C2×C3×C4#12711yescharacteristic
249#67—82no—
250#67—82no—
251C2×Dic3—41no—
252C2×Dic3—81no—
253D12—42no—
254D12—42no—
255C3×C8—42no—
256C3⋊C8—42no—
257C3⋊C8—42no—
258C3⋊C8—81no—
259C24—21no—
260C24—61no—
261C24#23711yescharacteristic
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—42no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—61no—
268C2×D4—61no—
269C2×D4—82no—
270C2×D4—122no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—242no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—61no—
279C22×C4D3×D411yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282C22×C4C2×#6711yescharacteristic
283C22×C4C2×#6711yescharacteristic
284C22×C4C2×#6711yescharacteristic
285D8—242no—
286M16—42no—
287M16—122no—
288C2×C8—22no—
289C2×C8—62no—
290C2×C8—62no—
291C2×C8—122no—
292C2×C8—122no—
293C2×C8—121no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—121no—
307C22⋊C4—121no—
308C42—41no—
309C42—41no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—82no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3—162no—
323C22×C3#39211yescharacteristic
324C22×C3#47212yes—
325C2×D3—81no—
326C2×D3—81no—
327C2×D3—81no—
328C2×D3—162no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—81no—
333C3×C4—81no—
334C3×C4—81no—
335C3×C4#45812yes—
336Dic3—81no—
337C23—22no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—41no—
342C23—81no—
343C23—81no—
344C23—122no—
345C23—121no—
346C23—241no—
347C23—241no—
348C23#93811yescharacteristic
349C23#93911yescharacteristic
350C23#93911yescharacteristic
351D4—42no—
352D4—42no—
353D4—82no—
354D4—82no—
355D4—122no—
356D4—122no—
357D4—242no—
358D4—242no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—121no—
372C2×C4—241no—
373C2×C4#93711yescharacteristic
374C2×C4#93711yescharacteristic
375C2×C4#94011yescharacteristic
376C2×C4#95311yescharacteristic
377C8—42no—
378C8—122no—
379C8—122no—
380C8—241no—
381C2×C3—22no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—161no—
385C2×C3#201711yescharacteristic
386C2×C3#201811yescharacteristic
387C2×C3#202211yescharacteristic
388D3—161no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—82no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—162no—
402C22—241no—
403C22—241no—
404C22—241no—
405C22—482no—
406C22#519912yes—
407C22#522311yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—241no—
415C4#505312yes—
416C3#997711yescharacteristic, complemented, Sylow
417C2—22no—
418C2—41no—
419C2—41no—
420C2—161no—
421C2—481no—
422C2#7170611yescharacteristic
423C2#7209511yescharacteristic
424C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3094C2
socleC22×C3#392
prime radical#82625C2
hypercenter#3094D3
hypocenterC3#9977
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3094

Complements

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

NH ≅ G/Ntype
1. C3×#3094C2semidirect
3. #82217C2semidirect
4. #71676C2semidirect
5. #71584C2semidirect
6. #69970C2semidirect
7. #67687C2semidirect
36. #3094D3semidirect
416. C3#9977semidirect

Maximal subgroups

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

subgroupindex
C3×#30942
#826252
#822172
#716762
#715842
#699702
#676872
#99773