Finite Group Databasebeta

← #10698091

Subgroups — #10698091

Order 768.

698 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 87 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#82180C211yesmaximal, characteristic, complemented
2#82177C211yesmaximal, characteristic, complemented
3#78641C211yesmaximal, characteristic
4C3×#1821C211yesmaximal, characteristic, complemented
5#69977C211yesmaximal, characteristic, complemented
6#69977C211yesmaximal, characteristic, complemented
7#67683C211yesmaximal, characteristic, complemented
8#10328—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C2×#929—22no—
13C3×#407C2211yescharacteristic
14C3×#407C2211yescharacteristic
15#5213—22no—
16C4×#217C2211yescharacteristic
17C4×#217C2211yescharacteristic
18#5135—22no—
19#4993—22no—
20#4992—22no—
21#4988—21no—
22#4988—21no—
23#4977—21no—
24#4975—21no—
25#4735—21no—
26#4735—21no—
27#4732—21no—
28#4732—21no—
29C3×#343—22no—
30#4470—22no—
31#4470—22no—
32#3192—31no—
33#2504—31no—
34#2501—31no—
35#1821D311yescharacteristic, complemented
36#1507—31no—
37#1507—31no—
38#1297—31no—
39C23×C3×C4D412yes—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C22×#67—21no—
43C22×D12D411yescharacteristic
44C22×D12D411yescharacteristic
45C2×C3×M16—21no—
46C2×C3×M16D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×#217—21no—
54C2×#217—21no—
55#930—21no—
56#930—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60#930—21no—
61#930—21no—
62#930—21no—
63#929—48no—
64C2×(C2×D3)⋊C4—24no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—24no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D411yescharacteristic
70C2×Dic3⋊C4—24no—
71C2×C4×Dic3—21no—
72C2×C4×Dic3—21no—
73C2×D24—22no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3⋊#95—24no—
79C3⋊#95D412yes—
80C3⋊#95D412yes—
81#821—22no—
82#821—22no—
83#817—22no—
84C2×#123—62no—
85C2×#121—31no—
86C2×#121—31no—
87C22×C4⋊C4C2×D311yescharacteristic
88#481—61no—
89#481—61no—
90#469—62no—
91#447—61no—
92#447—61no—
93#407C2×D311yescharacteristic
94#407C2×D311yescharacteristic
95#399—62no—
96#397—62no—
97#390—61no—
98#390—61no—
99#386—62no—
100#384—61no—
101#382—61no—
102C4×C22⋊C4—31no—
103C4×C22⋊C4—31no—
104#343—22no—
105#327—62no—
106#327—62no—
107C24×C3C2×D411yescharacteristic
108C23×D3—21no—
109C23×D3—21no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—42no—
114C22×C3×C4—44no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×#67—44no—
122C2×#67—44no—
123C2×#67—41no—
124C2×#67—41no—
125C2×#67—41no—
126C2×#67—41no—
127C22×Dic3—22no—
128C22×Dic3—21no—
129C22×Dic3—22no—
130C22×Dic3—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—42no—
136C2×D12—42no—
137C3×M16—42no—
138C3×M16—42no—
139C2×C3×C8—22no—
140C2×C3×C8—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145#217—24no—
146#217—42no—
147(C2×D3)⋊C4—48no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—84no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157Dic3⋊C4—48no—
158C4×Dic3—42no—
159C4×Dic3—42no—
160D24—82no—
161C22×D4—31no—
162C22×D4—31no—
163C22×D4—61no—
164C22×D4—61no—
165C23×C4D1212yes—
166C23×C4C22×D311yescharacteristic
167C2×D8—62no—
168C2×M16—21no—
169C2×M16#6712yes—
170#123—128no—
171#121—61no—
172#121—61no—
173#121—61no—
174#121—61no—
175#121—61no—
176#121—61no—
177#121—61no—
178#121—61no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—64no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—64no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—64no—
194C2×C42—61no—
195C2×C42—61no—
196#106—62no—
197#102—62no—
198#102—62no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203#95—32no—
204#95—32no—
205#95—64no—
206C23×C3—22no—
207C23×C3—22no—
208C23×C3—41no—
209C23×C3#12011yescharacteristic
210C23×C3#12011yescharacteristic
211C23×C3#12111yescharacteristic
212C22×D3—42no—
213C22×D3—42no—
214C22×D3—42no—
215C22×D3—42no—
216C22×D3—41no—
217C22×D3—41no—
218C22×D3—42no—
219C22×D3—42no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—44no—
227C2×C3×C4—42no—
228C2×C3×C4—44no—
229C2×C3×C4—82no—
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12711yescharacteristic
234#67—82no—
235#67—82no—
236#67—82no—
237#67—82no—
238C2×Dic3—24no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248D12—42no—
249D12—42no—
250D12—81no—
251D12—81no—
252C3×C8—42no—
253C3×C8—81no—
254C24—61no—
255C24—61no—
256C24C2×D1211yescharacteristic
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—122no—
262C2×D4—122no—
263C2×D4—124no—
264C2×D4—124no—
265C2×D4—121no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—121no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—62no—
275C22×C4—62no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4#23511yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4C2×#6711yescharacteristic
284D8—242no—
285M16—42no—
286M16—42no—
287C2×C8—22no—
288C2×C8—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—128no—
298C22⋊C4—64no—
299C22⋊C4—128no—
300C22⋊C4—122no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C22⋊C4—244no—
306C42—122no—
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#39211yescharacteristic
317C22×C3#45312yes—
318C22×C3#45412yes—
319C2×D3—84no—
320C2×D3—84no—
321C2×D3—81no—
322C2×D3—81no—
323C2×D3—81no—
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—82no—
331C3×C4#45812yes—
332Dic3—44no—
333Dic3—82no—
334Dic3—81no—
335Dic3—81no—
336C23—22no—
337C23—22no—
338C23—41no—
339C23—122no—
340C23—122no—
341C23—122no—
342C23—122no—
343C23—121no—
344C23—121no—
345C23—122no—
346C23—122no—
347C23#88211yescharacteristic
348C23#88211yescharacteristic
349C23#93011yescharacteristic
350D4—122no—
351D4—122no—
352D4—241no—
353D4—241no—
354D4—242no—
355D4—242no—
356D4—242no—
357D4—242no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—64no—
368C2×C4—82no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—122no—
378C2×C4#88411yescharacteristic
379C2×C4#88411yescharacteristic
380C2×C4#93711yescharacteristic
381C2×C4#94011yescharacteristic
382C8—42no—
383C8—81no—
384C2×C3—22no—
385C2×C3—22no—
386C2×C3—41no—
387C2×C3#200211yescharacteristic
388C2×C3#200311yescharacteristic
389C2×C3#202211yescharacteristic
390D3—161no—
391D3—161no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—244no—
401C22—244no—
402C22—241no—
403C22—241no—
404C22—241no—
405C22—241no—
406C22—241no—
407C22—241no—
408C22#474712yes—
409C22#475312yes—
410C22#496011yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—82no—
415C4—124no—
416C4—242no—
417C4—241no—
418C4—241no—
419C4#505312yes—
420C3#1032811yescharacteristic, complemented, Sylow
421C2—22no—
422C2—22no—
423C2—41no—
424C2—481no—
425C2—481no—
426C2#6953911yescharacteristic
427C2#7008211yescharacteristic
428C2#7011611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1821C2
socleC22×C3#392
prime radical#78641C2
hypercenter#1821D3
hypocenterC3#10328
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
1. #82180C2semidirect
2. #82177C2semidirect
4. C3×#1821C2semidirect
5. #69977C2semidirect
6. #69977C2semidirect
7. #67683C2semidirect
35. #1821D3semidirect
420. C3#10328semidirect

Maximal subgroups

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

subgroupindex
#821802
#821772
#786412
C3×#18212
#699772
#676832
#103283