Finite Group Databasebeta

← #10698089

Subgroups — #10698089

Order 768.

698 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 89 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×#2501C211yesmaximal, characteristic, complemented
2#82627C211yesmaximal, characteristic
3#82180C211yesmaximal, characteristic, complemented
4#71673C211yesmaximal, characteristic, complemented
5#71562C211yesmaximal, characteristic, complemented
6#69444C211yesmaximal, characteristic, complemented
7#67618C211yesmaximal, characteristic, complemented
8#10328—31nomaximal, Sylow
9C2×C3×#123—22no—
10C2×C3×#121C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C2×#930C2211yescharacteristic
13C3×#399—22no—
14C3×#390—21no—
15C3×#384—21no—
16C3×C4×C22⋊C4C2211yescharacteristic
17#5217—21no—
18#5213—22no—
19C4×#217C2211yescharacteristic
20#5207C2211yescharacteristic
21#5207C2211yescharacteristic
22#5039—22no—
23#5036—21no—
24#5034—21no—
25#4992—22no—
26#4988—21no—
27#4977—21no—
28#4968—21no—
29#4537—22no—
30#4530—22no—
31#4451—22no—
32#3192—31no—
33#2504—31no—
34#2501D311yescharacteristic, complemented
35#1821—31no—
36#1507—31no—
37#1507—31no—
38#1297—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4D412yes—
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×D12D411yescharacteristic
45C3×#123—48no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C2×C3×C4⋊C4—24no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—24no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C42—21no—
62C2×#217—21no—
63C2×#213—22no—
64#930—21no—
65#930—21no—
66#930—21no—
67#930—21no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—24no—
70C2×C4⋊Dic3D411yescharacteristic
71C2×C4×Dic3—21no—
72C2×#208—21no—
73C2×#208D412yes—
74C3×#95D412yes—
75#832—22no—
76C3⋊#95—24no—
77C3⋊#95D412yes—
78#830—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#809—22no—
83#808—22no—
84C2×#123—22no—
85C2×#121—31no—
86C2×#121C2×D311yescharacteristic
87C22×C4⋊C4C2×D311yescharacteristic
88#481—61no—
89#481—61no—
90#469—62no—
91#447—61no—
92#447—61no—
93#407—31no—
94#407—31no—
95#399—22no—
96#397—62no—
97#390—21no—
98#390—61no—
99#386—62no—
100#384—21no—
101#382—61no—
102C4×C22⋊C4—31no—
103C4×C22⋊C4C2×D311yescharacteristic
104#343—62no—
105#327—62no—
106#327—62no—
107C24×C3—21no—
108C24×C3C2×D411yescharacteristic
109C23×D3—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—42no—
113C2×C3×D4—44no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4—42no—
122C22×C3×C4—44no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C2×#67—44no—
130C2×#67—41no—
131C2×#67—41no—
132C22×Dic3—21no—
133C22×Dic3—22no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—42no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—48no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C22⋊C4—24no—
145C3×C22⋊C4—48no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42—42no—
149#217—42no—
150#213—82no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—84no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4×Dic3—42no—
157#208—42no—
158#208—42no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161C22×D4—21no—
162C22×D4—31no—
163C22×D4—61no—
164C22×D4#6711yescharacteristic
165C23×C4#6712yes—
166C23×C4C22×D311yescharacteristic
167C2×D8—62no—
168C2×M16—32no—
169C2×M16—61no—
170#123—48no—
171#121—21no—
172#121—21no—
173#121—21no—
174#121—21no—
175#121—61no—
176#121—61no—
177#121—61no—
178#121—61no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4#6711yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—64no—
194C2×C42—21no—
195C2×C42—61no—
196#106—62no—
197#102—62no—
198#102—62no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203#95—32no—
204#95—64no—
205#95#6712yes—
206C23×C3—22no—
207C23×C3—22no—
208C23×C3—41no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3—41no—
212C23×C3—42no—
213C23×C3#12011yescharacteristic
214C23×C3#12011yescharacteristic
215C23×C3#12111yescharacteristic
216C22×D3—42no—
217C22×D3—42no—
218C22×D3—41no—
219C22×D3—42no—
220C3×D4—42no—
221C3×D4—81no—
222C3×D4—82no—
223C3×D4—82no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—24no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—44no—
234C2×C3×C4—42no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—44no—
238C2×C3×C4—82no—
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12711yescharacteristic
243#67—82no—
244#67—82no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—41no—
249C2×Dic3—42no—
250D12—42no—
251D12—81no—
252C3⋊C8—42no—
253C3⋊C8—81no—
254C24—21no—
255C24—61no—
256C24C2×#6711yescharacteristic
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—42no—
260C2×D4—44no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—122no—
266C2×D4—124no—
267C2×D4—121no—
268C2×D4—121no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—62no—
277C22×C4—61no—
278C22×C4D3×D411yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4#23711yescharacteristic
282C22×C4C2×#6711yescharacteristic
283C22×C4C2×#6711yescharacteristic
284D8—242no—
285M16—122no—
286M16—122no—
287C2×C8—62no—
288C2×C8—121no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—48no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C22⋊C4—24no—
299C22⋊C4—48no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—122no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C22⋊C4—244no—
306C42—42no—
307C42—122no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—44no—
316C22×C3—84no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3#39211yescharacteristic
321C22×C3#45312yes—
322C22×C3#45412yes—
323C2×D3—84no—
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—44no—
331C3×C4—82no—
332C3×C4—81no—
333C3×C4#45812yes—
334Dic3—82no—
335Dic3—81no—
336C23—22no—
337C23—22no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23—42no—
343C23—122no—
344C23—122no—
345C23—121no—
346C23—122no—
347C23#93711yescharacteristic
348C23#93911yescharacteristic
349C23#95311yescharacteristic
350D4—42no—
351D4—81no—
352D4—82no—
353D4—82no—
354D4—122no—
355D4—241no—
356D4—242no—
357D4—242no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—82no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—121no—
377C2×C4—122no—
378C2×C4#93711yescharacteristic
379C2×C4#93811yescharacteristic
380C2×C4#93911yescharacteristic
381C2×C4#94011yescharacteristic
382C8—122no—
383C8—241no—
384C2×C3—22no—
385C2×C3—22no—
386C2×C3—41no—
387C2×C3—161no—
388C2×C3#200211yescharacteristic
389C2×C3#200311yescharacteristic
390C2×C3#202211yescharacteristic
391D3—161no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—84no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—244no—
405C22—241no—
406C22—241no—
407C22—241no—
408C22#507712yes—
409C22#522311yescharacteristic
410C22#524512yes—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—44no—
415C4—82no—
416C4—81no—
417C4—242no—
418C4—241no—
419C4#505312yes—
420C3#1032811yescharacteristic, complemented, Sylow
421C2—22no—
422C2—22no—
423C2—41no—
424C2—161no—
425C2—481no—
426C2#7170611yescharacteristic
427C2#7176911yescharacteristic
428C2#7186711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2501C2
socleC22×C3#392
prime radical#82627C2
hypercenter#2501D3
hypocenterC3#10328
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2501

Complements

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

NH ≅ G/Ntype
1. C3×#2501C2semidirect
3. #82180C2semidirect
4. #71673C2semidirect
5. #71562C2semidirect
6. #69444C2semidirect
7. #67618C2semidirect
34. #2501D3semidirect
420. C3#10328semidirect

Maximal subgroups

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

subgroupindex
C3×#25012
#826272
#821802
#716732
#715622
#694442
#676182
#103283