Finite Group Databasebeta

← #10697554

Subgroups — #10697554

Order 768.

610 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 105 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×#2547C211yesmaximal, characteristic, complemented
2#83215C211yesmaximal, characteristic, complemented
3#82307C211yesmaximal, characteristic
4#71773C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#10239—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#124—21no—
11C2×C3×C4×D4C2211yescharacteristic
12C2×#932—21no—
13#5477C2211yescharacteristic
14C3×#397—22no—
15C3×#391—21no—
16C3×#380—22no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5088—21no—
19#5087—22no—
20#5078—21no—
21#5019—21no—
22#5016—22no—
23#5015—21no—
24#4967—22no—
25#4966—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4547—22no—
30D4⋊(C3⋊C8)—22no—
31#4451—22no—
32#3226—31no—
33#3036—31no—
34#2547D311yescharacteristic, complemented
35#1546—31no—
36#1502—31no—
37#1478—31no—
38#1303—31no—
39C22×C3×Q8—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C2×#235—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×#124—42no—
48C3×#124—42no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D412yes—
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61C2×#214—22no—
62#932—42no—
63#932—42no—
64C2×#213—22no—
65#930—21no—
66#930—21no—
67C2×#208—21no—
68#875—21no—
69#874D411yescharacteristic
70C4×D12—21no—
71C4×D12—21no—
72#869D411yescharacteristic
73C3×#95—24no—
74C3×#95D412yes—
75#830—21no—
76#830—21no—
77#810—22no—
78#809—22no—
79#809—24no—
80#807—24no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4⋊(C3⋊C8)D412yes—
85C4×C3⋊C8—21no—
86C4×C3⋊C8—21no—
87C2×AGL(1,ℤ/8)—61no—
88#532—31no—
89C2×C4⋊Q8C2×D311yescharacteristic
90C2×#124—21no—
91C2×C4×D4C2×D311yescharacteristic
92#500—61no—
93#486—61no—
94#480—62no—
95#474—62no—
96#463—62no—
97#461—61no—
98#459—61no—
99#431—31no—
100#431—31no—
101#423—31no—
102#417—61no—
103#397—22no—
104#391—21no—
105#380—22no—
106C4×C4⋊C4C2×D311yescharacteristic
107#332—62no—
108#327—62no—
109D4⋊C8—62no—
110C24×C3—21no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C2×C3×Q8—41no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4D812yes—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—44no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C2×#67—41no—
133#235—41no—
134#235—41no—
135#235—41no—
136#235—41no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×C4×D3—41no—
141C2×Dic6—21no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—42no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C22⋊C4—48no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C42—41no—
153C3×C42—41no—
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42#4111yescharacteristic
158#214—44no—
159#213—44no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164#208—41no—
165#208—41no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—41no—
169C2×#41—61no—
170C22×Q8—21no—
171C22×D4#6711yescharacteristic
172C23×C4—21no—
173AGL(1,ℤ/8)—122no—
174AGL(1,ℤ/8)—122no—
175C2×SD16—62no—
176C2×D8—62no—
177C2×M16—61no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—31no—
183#127—31no—
184#124—42no—
185#124—42no—
186#124—61no—
187#121—61no—
188#121—61no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—61no—
194C4×D4—61no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4#6711yescharacteristic
198C2×C4⋊C4#6712yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—21no—
201C2×C42—21no—
202C2×C42C22×D311yescharacteristic
203#107—64no—
204C4⋊C8—32no—
205C4⋊C8—32no—
206C4⋊C8—61no—
207C4⋊C8—61no—
208#103—62no—
209#102—62no—
210#102—64no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C4×C8—61no—
214C4×C8—61no—
215#95—24no—
216#95#6712yes—
217C23×C3—22no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3#12011yescharacteristic
223C22×D3—41no—
224C3×Q8—81no—
225C3×Q8—81no—
226C3×Q8—81no—
227C3×Q8—81no—
228C3×D4—22no—
229C3×D4—22no—
230C3×D4—41no—
231C3×D4—41no—
232C2×C3×C4—22no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—82no—
243C2×C3×C4—82no—
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12011yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C2×C3×C4#12711yescharacteristic
250C2×C3×C4C2×D811yescharacteristic
251C2×C3×C4#13511yescharacteristic
252C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
253C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
254#67—81no—
255#67—81no—
256C2×Dic3—41no—
257D12—41no—
258D12—41no—
259D12—42no—
260D12—81no—
261C4×D3—81no—
262C4×D3—81no—
263Dic6—41no—
264Dic6—41no—
265C3⋊C8—42no—
266C3⋊C8—42no—
267C3⋊C8—81no—
268C24—21no—
269#41—121no—
270#41—121no—
271#41—121no—
272#41—121no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—61no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—61no—
288C2×D4—121no—
289C2×D4#21312yes—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—44no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—121no—
297C22×C4C2×#6711yescharacteristic
298C22×C4C2×#6711yescharacteristic
299C22×C4C2×#6711yescharacteristic
300SD16—124no—
301D8—124no—
302M16—121no—
303M16—121no—
304C2×C8—62no—
305C2×C8—62no—
306C2×C8—121no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—61no—
315C4⋊C4—61no—
316C22⋊C4—48no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C42—41no—
322C42—41no—
323C42D3×D411yescharacteristic
324C42D3×D411yescharacteristic
325C42D3×D411yescharacteristic
326C42#23711yescharacteristic
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—44no—
330C22×C3—42no—
331C22×C3—82no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3#39211yescharacteristic
335C22×C3#44711yescharacteristic
336C22×C3#44911yescharacteristic
337C2×D3—81no—
338C2×D3—81no—
339C2×D3—81no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—82no—
344C3×C4—82no—
345C3×C4—81no—
346C3×C4#44711yescharacteristic
347C3×C4#44911yescharacteristic
348C3×C4#45412yes—
349C3×C4#46611yescharacteristic
350C3×C4#46911yescharacteristic
351Dic3—81no—
352C23—22no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—121no—
358C23#95311yescharacteristic
359Q8—81no—
360Q8—81no—
361Q8—81no—
362Q8—81no—
363Q8—121no—
364Q8—121no—
365D4—22no—
366D4—22no—
367D4—41no—
368D4—41no—
369D4—121no—
370D4—121no—
371D4—122no—
372D4—241no—
373D4—241no—
374D4—241no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—121no—
388C2×C4—241no—
389C2×C4—241no—
390C2×C4C2×#21311yescharacteristic
391C2×C4#93211yescharacteristic
392C2×C4#93711yescharacteristic
393C2×C4#93711yescharacteristic
394C2×C4#93811yescharacteristic
395C2×C4#93911yescharacteristic
396C2×C4#93911yescharacteristic
397C2×C4#94011yescharacteristic
398C2×C4#94911yescharacteristic
399C2×C4#95011yescharacteristic
400C8—122no—
401C8—122no—
402C8—241no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—82no—
406C2×C3#199911yescharacteristic
407C2×C3#200311yescharacteristic
408C2×C3#200911yescharacteristic
409D3—161no—
410C22—21no—
411C22—21no—
412C22—44no—
413C22—42no—
414C22—82no—
415C22—81no—
416C22—81no—
417C22—241no—
418C22—241no—
419C22—241no—
420C22#521711yescharacteristic
421C22#522311yescharacteristic
422C22#523811yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—82no—
427C4—82no—
428C4—81no—
429C4—241no—
430C4#503611yescharacteristic
431C4#503711yescharacteristic
432C4#503811yescharacteristic
433C4#503911yescharacteristic
434C4#506212yes—
435C3#1023911yescharacteristic, complemented, Sylow
436C2—21no—
437C2—21no—
438C2—82no—
439C2—481no—
440C2#7168411yescharacteristic
441C2#7173011yescharacteristic
442C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2547C2
socleC22×C3#392
prime radical#82307C2
hypercenter#2547D3
hypocenterC3#10239
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
1. C3×#2547C2semidirect
2. #83215C2semidirect
4. #71773C2semidirect
5. #71597C2semidirect
6. #69469C2semidirect
7. #67620C2semidirect
34. #2547D3semidirect
435. C3#10239semidirect

Maximal subgroups

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

subgroupindex
C3×#25472
#832152
#823072
#717732
#715972
#694692
#676202
#102393