Finite Group Databasebeta

← #10697584

Subgroups — #10697584

Order 768.

578 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 101 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#83239C211yesmaximal, characteristic, complemented
2#83215C211yesmaximal, characteristic
3#79067C211yesmaximal, characteristic
4C3×#1546C211yesmaximal, characteristic, complemented
5#70789C211yesmaximal, characteristic
6#70767C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#10243—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#944—21no—
11C2×#909—21no—
12#5477C2211yescharacteristic
13C2×C4×Dic6C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×#423C2211yescharacteristic
16#5229—21no—
17#5133—21no—
18#5088—21no—
19#5087—22no—
20#5078—21no—
21#4983—22no—
22#4972—22no—
23#4967—22no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26#4849—21no—
27#4848—21no—
28#4845—22no—
29C3×#332—22no—
30#4470—22no—
31#4456—22no—
32#3226—31no—
33#3037—31no—
34#2546—31no—
35#1546D311yescharacteristic, complemented
36#1502—31no—
37#1479—31no—
38#1303—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C22×Dic6D411yescharacteristic
42C2×C3×M16—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D412yes—
49C2×C3×C42C2311yescharacteristic
50#944—42no—
51#944—42no—
52#930—21no—
53#930—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3D411yescharacteristic
56C2×Dic3⋊C4—24no—
57C2×Dic3⋊C4—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—21no—
60#909—42no—
61#909—42no—
62C2×Dic12—22no—
63C2×#204—22no—
64#875—21no—
65#874D411yescharacteristic
66C4×D12—21no—
67C4×D12—21no—
68#869D411yescharacteristic
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C4×Dic6—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8D412yes—
76C3×C22⋊C8—21no—
77C3×C4×C8—21no—
78C3⋊#95—24no—
79C3⋊#95D412yes—
80#830—21no—
81#821—22no—
82#816—22no—
83#809—24no—
84#807—24no—
85C4⋊(C3⋊C8)D412yes—
86C4×C3⋊C8—21no—
87C2×#137—61no—
88#532—31no—
89C2×C4⋊Q8—61no—
90C2×C4⋊Q8C2×D311yescharacteristic
91C2×C4×Q8—31no—
92#500—61no—
93#486—61no—
94#480—62no—
95#474—62no—
96#464—62no—
97#462—61no—
98#460—61no—
99#431—31no—
100#431C2×D311yescharacteristic
101#423C2×D311yescharacteristic
102#417—61no—
103#398—62no—
104#391—61no—
105#389—62no—
106C4×C4⋊C4—31no—
107#332—22no—
108#327—62no—
109Q8⋊C8—62no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C22×C3×C4—22no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C2×#67—41no—
121C22×Dic3—22no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124#235—41no—
125#235—41no—
126#235—41no—
127#235—41no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×C4×D3—41no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—21no—
138C2×Dic6D812yes—
139C3×M16—41no—
140C3×M16—41no—
141C2×C3×C8—22no—
142C2×C3×C8—41no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153C4⋊Dic3—24no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158Dic3⋊C4—48no—
159Dic3⋊C4—41no—
160Dic3⋊C4—41no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163C4×Dic3—41no—
164C4×Dic3—41no—
165C2×C3⋊C8—22no—
166Dic12—44no—
167#204—44no—
168C2×#41—61no—
169C22×Q8—21no—
170C22×Q8—31no—
171#137—122no—
172#137—122no—
173C2×Q16—62no—
174C2×SD16—62no—
175C2×M16—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—31no—
181C4⋊Q8—122no—
182C4⋊Q8—122no—
183#127—31no—
184#124—61no—
185#121—61no—
186#121—61no—
187C4×Q8—61no—
188C4×Q8—61no—
189C4×Q8—61no—
190C4×Q8—61no—
191C4×D4—61no—
192C4×D4—61no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—64no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4D1212yes—
199C2×C42—61no—
200C2×C42—61no—
201C2×C42C22×D311yescharacteristic
202#107—64no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—32no—
206C4⋊C8#6712yes—
207#103—62no—
208#102—64no—
209#102—62no—
210C22⋊C8—21no—
211C22⋊C8—61no—
212C4×C8—21no—
213C4×C8—61no—
214#95—32no—
215#95—64no—
216C23×C3#12011yescharacteristic
217C22×D3—41no—
218C3×Q8—81no—
219C3×Q8—81no—
220C3×Q8—81no—
221C3×Q8—81no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12711yescharacteristic
233C2×C3×C4C2×D811yescharacteristic
234C2×C3×C4#13511yescharacteristic
235C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237#67—81no—
238#67—81no—
239C2×Dic3—24no—
240C2×Dic3—22no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C2×Dic3—42no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248D12—41no—
249D12—41no—
250D12—42no—
251D12—81no—
252C4×D3—81no—
253C4×D3—81no—
254Dic6—22no—
255Dic6—22no—
256Dic6—41no—
257Dic6—41no—
258Dic6—41no—
259Dic6—41no—
260C3×C8—42no—
261C3×C8—81no—
262C3⋊C8—42no—
263#41—121no—
264#41—121no—
265#41—121no—
266#41—121no—
267C2×Q8—32no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—61no—
275C2×Q8—61no—
276C2×Q8—61no—
277C2×Q8—61no—
278C2×Q8—61no—
279C2×Q8—61no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—61no—
283C2×D4—121no—
284C22×C4—22no—
285C22×C4—62no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—121no—
289C22×C4C2×D1211yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292Q16—124no—
293SD16—124no—
294M16—41no—
295M16—41no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—62no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—64no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—128no—
309C4⋊C4—121no—
310C4⋊C4—121no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C42—121no—
314C42—121no—
315C42—121no—
316C42—121no—
317C42#23511yescharacteristic
318C42D3×D411yescharacteristic
319C42D3×D411yescharacteristic
320C42C2×#6711yescharacteristic
321C22×C3—21no—
322C22×C3—21no—
323C22×C3#39211yescharacteristic
324C22×C3#44711yescharacteristic
325C22×C3#44911yescharacteristic
326C2×D3—81no—
327C2×D3—81no—
328C2×D3—81no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—82no—
333C3×C4#44711yescharacteristic
334C3×C4#44911yescharacteristic
335C3×C4#45612yes—
336C3×C4#46611yescharacteristic
337C3×C4#46911yescharacteristic
338Dic3—44no—
339Dic3—42no—
340Dic3—81no—
341Dic3—81no—
342C23—121no—
343C23#88211yescharacteristic
344Q8—62no—
345Q8—62no—
346Q8—81no—
347Q8—81no—
348Q8—81no—
349Q8—81no—
350Q8—121no—
351Q8—121no—
352Q8—121no—
353Q8—121no—
354D4—121no—
355D4—121no—
356D4—122no—
357D4—241no—
358D4—241no—
359D4—241no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—64no—
366C2×C4—62no—
367C2×C4—122no—
368C2×C4—121no—
369C2×C4—122no—
370C2×C4—121no—
371C2×C4—122no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—241no—
375C2×C4—241no—
376C2×C4#88211yescharacteristic
377C2×C4#88411yescharacteristic
378C2×C4#88411yescharacteristic
379C2×C4D3×D811yescharacteristic
380C2×C4#91111yescharacteristic
381C2×C4#91411yescharacteristic
382C2×C4#91611yescharacteristic
383C2×C4#93011yescharacteristic
384C2×C4#93711yescharacteristic
385C2×C4#94011yescharacteristic
386C8—42no—
387C8—81no—
388C8—122no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3#199911yescharacteristic
392C2×C3#200511yescharacteristic
393C2×C3#200911yescharacteristic
394D3—161no—
395C22—21no—
396C22—21no—
397C22—241no—
398C22—241no—
399C22—241no—
400C22#477311yescharacteristic
401C22#478111yescharacteristic
402C22#496011yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—82no—
407C4—124no—
408C4—122no—
409C4—241no—
410C4—241no—
411C4#475212yes—
412C4#477511yescharacteristic
413C4#478211yescharacteristic
414C4#515611yescharacteristic
415C4#517211yescharacteristic
416C3#1024311yescharacteristic, complemented, Sylow
417C2—21no—
418C2—21no—
419C2—481no—
420C2#7011911yescharacteristic
421C2#7020411yescharacteristic
422C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1546D3
hypocenterC3#10243
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
1. #83239C2semidirect
4. C3×#1546C2semidirect
6. #70767C2semidirect
7. #67715C2semidirect
35. #1546D3semidirect
416. C3#10243semidirect

Maximal subgroups

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

subgroupindex
#832392
#832152
#790672
C3×#15462
#707892
#707672
#677152
#102433