Finite Group Databasebeta

← #10707828

Subgroups — #10707828

Order 768.

484 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 59 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×#2752C211yesmaximal, characteristic, complemented
2#69714C211yesmaximal, characteristic, complemented
3#69696C211yesmaximal, characteristic
4#69539C211yesmaximal, characteristic, complemented
5#67651C211yesmaximal, characteristic, complemented
6#67645C211yesmaximal, characteristic, complemented
7#67627C211yesmaximal, characteristic, complemented
8#12191—31nomaximal, Sylow
9C2×#929—21no—
10#5493C2211yescharacteristic
11C3×#399C2211yescharacteristic
12C3×#397—21no—
13C3×#394—21no—
14C3×#392C2211yescharacteristic
15C3×#390—21no—
16C3×#388—21no—
17C3×#387—21no—
18C3×#386—21no—
19C3×#381C2211yescharacteristic
20#5217—21no—
21#4968—21no—
22C2×#806—21no—
23#4526C2211yescharacteristic
24#4525C2211yescharacteristic
25#4525C2211yescharacteristic
26#4451—21no—
27#4451—21no—
28#2752D311yescharacteristic, complemented
29#2022—31no—
30#1613—31no—
31#1595—31no—
32#1352—31no—
33#1343—31no—
34#1317—31no—
35C22×C3×D4—21no—
36C22×C3×D4D411yescharacteristic
37C23×C3×C4D411yescharacteristic
38C22×#67—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4D411yescharacteristic
49C2×C3×C22⋊C4C2311yescharacteristic
50C2×C3×C42—21no—
51C2×#217—21no—
52C2×#213—41no—
53#930—21no—
54#930—21no—
55#929—42no—
56#929—42no—
57C2×(C2×D3)⋊C4—41no—
58C2×Dic3⋊C4—41no—
59#886—41no—
60#882—41no—
61C3×#95—21no—
62C3×#95—21no—
63C3×#95—21no—
64C3×#95—21no—
65C3×#95D411yescharacteristic
66C3×#95D411yescharacteristic
67#832—41no—
68#830—21no—
69#830—21no—
70#809—41no—
71#808—41no—
72#806—42no—
73#534—31no—
74C2×#123—61no—
75#481—61no—
76#447—61no—
77C2×#99—61no—
78#399C2×D311yescharacteristic
79#397—21no—
80#394—21no—
81#392C2×D311yescharacteristic
82#390—21no—
83#388—21no—
84#387—21no—
85#386—21no—
86#381C2×D311yescharacteristic
87#342—31no—
88#327—61no—
89#327—61no—
90#323—31no—
91#323—31no—
92C24×C3—21no—
93C24×C3C2×D411yescharacteristic
94C23×D3—21no—
95C2×C3×D4—41no—
96C2×C3×D4—42no—
97C2×C3×D4—41no—
98C2×C3×D4—41no—
99C2×C3×D4—81no—
100C2×C3×D4—81no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—41no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C2×#67—41no—
113C2×#67—41no—
114C2×#67—41no—
115C2×#67—41no—
116C2×#67—41no—
117C2×#67—41no—
118C22×Dic3—21no—
119C22×Dic3—41no—
120C2×D12—21no—
121C2×D12—21no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—42no—
124C3×C4⋊C4—81no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—42no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—81no—
129C3×C22⋊C4—81no—
130C3×C22⋊C4—81no—
131C3×C22⋊C4—81no—
132C3×C22⋊C4—81no—
133C3×C42—81no—
134#217—22no—
135#217—41no—
136#213—82no—
137(C2×D3)⋊C4—41no—
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—44no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142Dic3⋊C4—44no—
143C2×C3⋊C8—41no—
144C2×C3⋊C8—41no—
145C22×D4—21no—
146C22×D4—61no—
147C22×D4#6711yescharacteristic
148C23×C4#6711yescharacteristic
149C2×D8—121no—
150#123—121no—
151#123—122no—
152#123—122no—
153#121—61no—
154#121—61no—
155#120—121no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—121no—
159C2×C4⋊C4#6711yescharacteristic
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—121no—
168C2×C22⋊C4#6711yescharacteristic
169C2×C22⋊C4C22×D311yescharacteristic
170C2×C42—21no—
171#106—121no—
172#102—121no—
173#102—121no—
174#99—122no—
175C22⋊C8—61no—
176C22⋊C8—61no—
177#95—21no—
178#95—21no—
179#95—21no—
180#95—21no—
181#95#6711yescharacteristic
182#95#6711yescharacteristic
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—41no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3#12011yescharacteristic
195C23×C3#13511yescharacteristic
196C23×C3AGL(1,ℤ/8)11yescharacteristic
197C22×D3—41no—
198C22×D3—41no—
199C22×D3—41no—
200C22×D3—81no—
201C22×D3—81no—
202C3×D4—42no—
203C3×D4—82no—
204C3×D4—82no—
205C3×D4—81no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—81no—
227#67—81no—
228#67—81no—
229#67—81no—
230#67—81no—
231C2×Dic3—41no—
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—81no—
236C2×Dic3—81no—
237D12—81no—
238D12—81no—
239C3⋊C8—81no—
240C3⋊C8—81no—
241C24—21no—
242C24—61no—
243C24C2×#6711yescharacteristic
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—81no—
251C2×D4—81no—
252C2×D4—121no—
253C2×D4—121no—
254C2×D4—121no—
255C2×D4—121no—
256C2×D4—121no—
257C2×D4—121no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—61no—
268C22×C4—121no—
269C22×C4C2×#6711yescharacteristic
270C22×C4C2×#6711yescharacteristic
271D8—242no—
272C2×C8—121no—
273C2×C8—121no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—81no—
279C4⋊C4—124no—
280C22⋊C4—41no—
281C22⋊C4—42no—
282C22⋊C4—41no—
283C22⋊C4—62no—
284C22⋊C4—81no—
285C22⋊C4—81no—
286C22⋊C4—81no—
287C22⋊C4—81no—
288C22⋊C4—81no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—124no—
292C22⋊C4—121no—
293C42—81no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—81no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3—81no—
306C22×C3—81no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3#44911yescharacteristic
311C22×C3#45311yescharacteristic
312C22×C3B(4,𝔽2)11yescharacteristic
313C2×D3—82no—
314C2×D3—81no—
315C2×D3—81no—
316C2×D3—81no—
317C2×D3—161no—
318C3×C4—42no—
319C3×C4—41no—
320C3×C4—41no—
321C3×C4—81no—
322C3×C4—81no—
323C3×C4—81no—
324C3×C4—81no—
325Dic3—82no—
326Dic3—81no—
327C23—21no—
328C23—21no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—121no—
339C23—121no—
340C23—121no—
341C23—241no—
342C23—241no—
343C23#94911yescharacteristic
344C23#95011yescharacteristic
345C23#95311yescharacteristic
346D4—42no—
347D4—82no—
348D4—82no—
349D4—81no—
350D4—241no—
351D4—241no—
352D4—241no—
353D4—241no—
354D4—241no—
355D4—241no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—122no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—241no—
382C2×C4—241no—
383C8—241no—
384C8—241no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—41no—
388C2×C3—41no—
389C2×C3—161no—
390C2×C3#160011yescharacteristic
391C2×C3#219811yescharacteristic
392C2×C3#220011yescharacteristic
393D3—161no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—242no—
411C22—241no—
412C22—241no—
413C22—241no—
414C22—481no—
415C22#523811yescharacteristic
416C22#524111yescharacteristic
417C22#524411yescharacteristic
418C4—42no—
419C4—41no—
420C4—41no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4—242no—
426C4—241no—
427C3#1219111yescharacteristic, complemented, Sylow
428C2—21no—
429C2—21no—
430C2—41no—
431C2—41no—
432C2—161no—
433C2—481no—
434C2#7188411yescharacteristic
435C2#7194611yescharacteristic
436C2#7194711yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC2×C3×C22⋊C4C23
FrattiniC2×C22⋊C4C22×D3
FittingC3×#2752C2
socleC22×C3B(4,𝔽2)
prime radical#69696C2
hypercenter#2752D3
hypocenterC3#12191
LCSG ▷ C2×C3×C22⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2752

Complements

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

NH ≅ G/Ntype
1. C3×#2752C2semidirect
2. #69714C2semidirect
4. #69539C2semidirect
5. #67651C2semidirect
6. #67645C2semidirect
7. #67627C2semidirect
28. #2752D3semidirect
427. C3#12191semidirect

Maximal subgroups

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

subgroupindex
C3×#27522
#697142
#696962
#695392
#676512
#676452
#676272
#121913