Finite Group Databasebeta

← #10697582

Subgroups — #10697582

Order 768.

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

normal subgroups only

This group has 578 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×#2546C211yesmaximal, characteristic, complemented
2#83215C211yesmaximal, characteristic
3#82441C211yesmaximal, characteristic
4#71773C211yesmaximal, characteristic, complemented
5#71621C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic, complemented
8#10243—31nomaximal, Sylow
9C2×C3×C4⋊Q8—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C2×#942—21no—
13#5477C2211yescharacteristic
14C3×#398—22no—
15C3×#391—21no—
16C3×#389—22no—
17C3×C4×C4⋊C4C2211yescharacteristic
18#5088—21no—
19#5087—22no—
20#5078—21no—
21#5031—21no—
22#5028—22no—
23#5027—21no—
24#4967—22no—
25#4966—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4547—22no—
30#4535—22no—
31#4451—22no—
32#3226—31no—
33#3037—31no—
34#2546D311yescharacteristic, complemented
35#1546—31no—
36#1502—31no—
37#1479—31no—
38#1303—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8D411yescharacteristic
41C2×#235—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—42no—
47C3×C4⋊Q8—42no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C4⋊C4D412yes—
58C2×C3×C42—21no—
59C2×C3×C42—21no—
60C2×C3×C42C2311yescharacteristic
61C2×#216—22no—
62#942—42no—
63#942—42no—
64C2×#215—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×#137—61no—
88#532—31no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8C2×D311yescharacteristic
91C2×C4×Q8C2×D311yescharacteristic
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#431—31no—
101#423—31no—
102#417—61no—
103#398—22no—
104#391—21no—
105#389—22no—
106C4×C4⋊C4C2×D311yescharacteristic
107#332—62no—
108#327—62no—
109Q8⋊C8—62no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—41no—
116C2×C3×Q8—41no—
117C2×C3×Q8—41no—
118C2×C3×Q8—41no—
119C2×C3×Q8—41no—
120C2×C3×Q8—41no—
121C2×C3×Q8D812yes—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C2×#67—41no—
130#235—41no—
131#235—41no—
132#235—41no—
133#235—41no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—21no—
137C2×C4×D3—41no—
138C2×Dic6—21no—
139C3×C4⋊C4—24no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—48no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C42—41no—
150C3×C42—41no—
151C3×C42—41no—
152C3×C42—41no—
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C3×C42#4111yescharacteristic
157#216—44no—
158#215—44no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163#208—41no—
164#208—41no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—41no—
168C2×#41—61no—
169C22×Q8—21no—
170C22×Q8#6711yescharacteristic
171#137—122no—
172#137—122no—
173C2×Q16—62no—
174C2×SD16—62no—
175C2×M16—61no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—31no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183#127—31no—
184#124—61no—
185#121—61no—
186#121—61no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×D4—61no—
192C4×D4—61no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4#6711yescharacteristic
198C2×C4⋊C4#6712yes—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42C22×D311yescharacteristic
202#107—64no—
203C4⋊C8—32no—
204C4⋊C8—32no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207#103—62no—
208#102—62no—
209#102—64no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C4×C8—61no—
213C4×C8—61no—
214#95—24no—
215#95#6712yes—
216C23×C3#12011yescharacteristic
217C22×D3—41no—
218C3×Q8—22no—
219C3×Q8—22no—
220C3×Q8—41no—
221C3×Q8—41no—
222C3×Q8—81no—
223C3×Q8—81no—
224C3×Q8—81no—
225C3×Q8—81no—
226C2×C3×C4—22no—
227C2×C3×C4—24no—
228C2×C3×C4—22no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12711yescharacteristic
245C2×C3×C4C2×D811yescharacteristic
246C2×C3×C4#13511yescharacteristic
247C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
248C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
249#67—81no—
250#67—81no—
251C2×Dic3—41no—
252D12—41no—
253D12—41no—
254D12—42no—
255D12—81no—
256C4×D3—81no—
257C4×D3—81no—
258Dic6—41no—
259Dic6—41no—
260C3⋊C8—42no—
261C3⋊C8—42no—
262C3⋊C8—81no—
263#41—121no—
264#41—121no—
265#41—121no—
266#41—121no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—61no—
279C2×Q8#21312yes—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—61no—
283C2×D4—121no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—121no—
289C22×C4C2×#6711yescharacteristic
290C22×C4C2×#6711yescharacteristic
291C22×C4C2×#6711yescharacteristic
292Q16—124no—
293SD16—124no—
294M16—121no—
295M16—121no—
296C2×C8—62no—
297C2×C8—62no—
298C2×C8—121no—
299C4⋊C4—24no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—48no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42D3×D411yescharacteristic
318C42D3×D411yescharacteristic
319C42D3×D411yescharacteristic
320C42#23711yescharacteristic
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—44no—
333C3×C4—42no—
334C3×C4—82no—
335C3×C4—81no—
336C3×C4#44711yescharacteristic
337C3×C4#44911yescharacteristic
338C3×C4#45612yes—
339C3×C4#46611yescharacteristic
340C3×C4#46911yescharacteristic
341Dic3—81no—
342C23—121no—
343C23#95311yescharacteristic
344Q8—22no—
345Q8—22no—
346Q8—41no—
347Q8—41no—
348Q8—81no—
349Q8—81no—
350Q8—81no—
351Q8—81no—
352Q8—121no—
353Q8—121no—
354D4—121no—
355D4—121no—
356D4—122no—
357D4—241no—
358D4—241no—
359D4—241no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—121no—
374C2×C4—241no—
375C2×C4—241no—
376C2×C4C2×#21311yescharacteristic
377C2×C4#93211yescharacteristic
378C2×C4#93711yescharacteristic
379C2×C4#93711yescharacteristic
380C2×C4#93811yescharacteristic
381C2×C4#93911yescharacteristic
382C2×C4#93911yescharacteristic
383C2×C4#94011yescharacteristic
384C2×C4#94911yescharacteristic
385C2×C4#95011yescharacteristic
386C8—122no—
387C8—122no—
388C8—241no—
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#521711yescharacteristic
401C22#522311yescharacteristic
402C22#523811yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—44no—
407C4—42no—
408C4—82no—
409C4—81no—
410C4—241no—
411C4#503611yescharacteristic
412C4#503711yescharacteristic
413C4#503811yescharacteristic
414C4#503911yescharacteristic
415C4#509512yes—
416C3#1024311yescharacteristic, complemented, Sylow
417C2—21no—
418C2—21no—
419C2—481no—
420C2#7168411yescharacteristic
421C2#7179511yescharacteristic
422C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2546C2
socleC22×C3#392
prime radical#5477C22
hypercenter#2546D3
hypocenterC3#10243
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2546

Complements

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

NH ≅ G/Ntype
1. C3×#2546C2semidirect
4. #71773C2semidirect
5. #71621C2semidirect
7. #67620C2semidirect
34. #2546D3semidirect
416. C3#10243semidirect

Maximal subgroups

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

subgroupindex
C3×#25462
#832152
#824412
#717732
#716212
#694692
#676202
#102433