Finite Group Databasebeta

← #10691514

Subgroups — #10691514

Order 768.

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

normal subgroups only

This group has 478 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
1C2×#4737C211yesmaximal, characteristic
2#78584C211yesmaximal, characteristic, complemented
3#69921C211yesmaximal, characteristic
4#69444C211yesmaximal, characteristic
5#69442C211yesmaximal, characteristic, complemented
6C3×#1282C211yesmaximal, characteristic, complemented
7#67679C211yesmaximal, characteristic, complemented
8#9242—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C2×#926C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×C4×C22⋊C4C2211yescharacteristic
14#5207C2211yescharacteristic
15#5207C2211yescharacteristic
16C2×#821C411yescharacteristic, complemented
17C2×#821C411yescharacteristic, complemented
18C2×#818C411yescharacteristic, complemented
19C2×#817C411yescharacteristic, complemented
20#5102—21no—
21#5101—21no—
22#4737—21no—
23#4737—21no—
24#4737—21no—
25#4737—21no—
26#4720—22no—
27#4719—21no—
28#4718—21no—
29#4523—21no—
30#4523—21no—
31#4468—21no—
32#4468—21no—
33#4451—22no—
34C2×#482—31no—
35#2924—31no—
36#1826—31no—
37#1507—31no—
38#1471—31no—
39#1293—31no—
40#1282D311yescharacteristic, complemented
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×D12C2×C411yescharacteristic
44C22×C3×C8—21no—
45C22×C3×C8C2×C411yescharacteristic
46C22×C3×C8C2×C411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C42—21no—
49C2×#217—21no—
50#930—21no—
51#930—21no—
52#930—21no—
53#930—21no—
54C2×(C2×D3)⋊C4—21no—
55#926—21no—
56#926—21no—
57#926D411yescharacteristic
58#926D411yescharacteristic
59#926D411yescharacteristic
60#926D411yescharacteristic
61C2×C4⋊Dic3C2×C411yescharacteristic
62C2×C4⋊Dic3C2×C411yescharacteristic
63C2×C4⋊Dic3C2×C411yescharacteristic
64C2×C4×Dic3—21no—
65C2×#208—21no—
66C2×#208—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8D411yescharacteristic
71C3×C22⋊C8D411yescharacteristic
72C3×#95—21no—
73#830—21no—
74#830—21no—
75#830—21no—
76#830D412yes—
77#821—21no—
78#821—21no—
79#821—21no—
80#821—21no—
81#818—21no—
82#818—21no—
83#817—21no—
84#817—21no—
85#815—22no—
86Dic3⋊C8—22no—
87Dic3⋊C8—22no—
88C8×Dic3—22no—
89C2×#121—31no—
90C2×#117—31no—
91#482—61no—
92#482—61no—
93#482—61no—
94#482—61no—
95#433—62no—
96#432—61no—
97#431—61no—
98C2×#107—31no—
99C2×#106—31no—
100#424—61no—
101C2×#102—31no—
102C2×#102—31no—
103#407—31no—
104#407—31no—
105C2×C22⋊C8C2×D311yescharacteristic
106C2×C22⋊C8C2×D311yescharacteristic
107#405—61no—
108C4×C22⋊C4C2×D311yescharacteristic
109#339—61no—
110#339—61no—
111#327—62no—
112#323—61no—
113#323—61no—
114C24×C3#4111yescharacteristic
115C23×D3—21no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—41no—
126C2×#67—41no—
127C2×#67—82no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—41no—
134C2×D12—41no—
135C2×C3×C8—21no—
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C2×C3×C8—21no—
139C2×C3×C8—41no—
140C2×C3×C8—41no—
141C2×C3×C8—41no—
142C2×C3×C8—41no—
143C3×C22⋊C4—22no—
144C3×C22⋊C4—41no—
145C3×C42—41no—
146C3×C42—41no—
147#217—21no—
148#217—21no—
149#217—21no—
150#217—21no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3C22⋊C411yescharacteristic
158C4⋊Dic3C22⋊C411yescharacteristic
159C4⋊Dic3C22⋊C411yescharacteristic
160C4⋊Dic3C22⋊C411yescharacteristic
161C4×Dic3—21no—
162C4×Dic3—21no—
163C4×Dic3—21no—
164C4×Dic3—21no—
165#208—41no—
166#208—41no—
167#208—41no—
168#208—41no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—41no—
171C22×D4—31no—
172C22×D4—61no—
173C23×C4C22×D311yescharacteristic
174C2×M16—61no—
175C2×M16—61no—
176C22×C8—21no—
177C22×C8C4×D311yescharacteristic
178C22×C8C4×D311yescharacteristic
179#121—61no—
180#121—61no—
181#121—61no—
182#121—61no—
183#117—31no—
184#117—31no—
185#117—31no—
186#117—31no—
187#117—61no—
188#117—61no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—31no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—61no—
195C2×C42—21no—
196C2×C42—61no—
197#107—61no—
198#107—61no—
199#106—61no—
200#106—61no—
201C4⋊C8—62no—
202C4⋊C8—62no—
203#102—61no—
204#102—61no—
205#102—61no—
206#102—61no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—32no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8D1211yescharacteristic
215C22⋊C8D1211yescharacteristic
216#97—62no—
217C4×C8—62no—
218#95—21no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3C4×D411yescharacteristic
224C23×C3#12111yescharacteristic
225C23×C3#12311yescharacteristic
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—81no—
230C22×D3—81no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—81no—
242C2×C3×C4—81no—
243C2×C3×C4C2×C22⋊C411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12111yescharacteristic
247#67—82no—
248#67—82no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—81no—
257C2×Dic3—81no—
258D12—41no—
259D12—41no—
260D12—81no—
261C3×C8—41no—
262C3×C8—41no—
263C3×C8—41no—
264C3×C8—41no—
265C3⋊C8—42no—
266C3⋊C8—81no—
267C24—61no—
268C24#23511yescharacteristic
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—242no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4C2×C4×D311yescharacteristic
283C22×C4C2×D1211yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4#23911yescharacteristic
288M16—121no—
289M16—121no—
290M16—121no—
291M16—121no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—62no—
301C2×C8—121no—
302C4⋊C4—31no—
303C4⋊C4—31no—
304C4⋊C4—31no—
305C4⋊C4—31no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C22⋊C4—22no—
311C22⋊C4—41no—
312C22⋊C4—61no—
313C22⋊C4—61no—
314C22⋊C4—61no—
315C22⋊C4—61no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C42—41no—
319C42—41no—
320C42—61no—
321C42—61no—
322C42—61no—
323C42—61no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3#38611yescharacteristic
335C22×C3#44311yescharacteristic
336C22×C3#44411yescharacteristic
337C2×D3—81no—
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—161no—
341C2×D3—161no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—82no—
346C3×C4#41011yescharacteristic
347C3×C4#41011yescharacteristic
348Dic3—41no—
349Dic3—41no—
350Dic3—41no—
351Dic3—41no—
352Dic3—81no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23—121no—
358C23—121no—
359C23—121no—
360C23—241no—
361C23—241no—
362C23C4×D1211yescharacteristic
363C23#88411yescharacteristic
364C23#89411yescharacteristic
365D4—121no—
366D4—121no—
367D4—241no—
368D4—242no—
369D4—242no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4D3×C22⋊C411yescharacteristic
392C2×C4#88211yescharacteristic
393C2×C4#89311yescharacteristic
394C2×C4#89511yescharacteristic
395C8—41no—
396C8—41no—
397C8—41no—
398C8—41no—
399C8—122no—
400C8—241no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—41no—
404C2×C3—41no—
405C2×C3#187611yescharacteristic
406C2×C3#189911yescharacteristic
407C2×C3#190311yescharacteristic
408D3—161no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—241no—
420C22—241no—
421C22—241no—
422C22—481no—
423C22—481no—
424C22#466511yescharacteristic
425C22#470011yescharacteristic
426C22#471711yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—82no—
431C4—121no—
432C4—121no—
433C4—121no—
434C4—121no—
435C4—241no—
436C4#474011yescharacteristic
437C4#474011yescharacteristic
438C3#924211yescharacteristic, complemented, Sylow
439C2—21no—
440C2—21no—
441C2—41no—
442C2—41no—
443C2—481no—
444C2#6869511yescharacteristic
445C2#6895411yescharacteristic
446C2#6903811yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1282C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1282D3
hypocenterC3#9242
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
2. #78584C2semidirect
5. #69442C2semidirect
6. C3×#1282C2semidirect
7. #67679C2semidirect
16. C2×#821C4semidirect
17. C2×#821C4semidirect
18. C2×#818C4semidirect
19. C2×#817C4semidirect
40. #1282D3semidirect
438. C3#9242semidirect

Maximal subgroups

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

subgroupindex
C2×#47372
#785842
#699212
#694442
#694422
C3×#12822
#676792
#92423