Finite Group Databasebeta

← #10697581

Subgroups — #10697581

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#83314C211yesmaximal, characteristic
2#83270C211yesmaximal, characteristic
3#73711C211yesmaximal, characteristic, complemented
4C3×#1502C211yesmaximal, characteristic
5#70778C211yesmaximal, characteristic, complemented
6#70767C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#10243—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#951—21no—
11C2×#869—21no—
12C2×#869C2211yescharacteristic
13C2×C4×Dic6C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16#5232—21no—
17#5090—21no—
18#5078—21no—
19#5075—21no—
20#5070—21no—
21#5054—22no—
22#5033—22no—
23#5007—22no—
24#4984—22no—
25#4929—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4661—22no—
29C3×#327—22no—
30#4486—22no—
31#4456—22no—
32#3226—31no—
33#3037—31no—
34#2546—31no—
35#1546—31no—
36#1502D311yescharacteristic
37#1479—31no—
38#1303—31no—
39C2×C3×#41—21no—
40C22×Dic6—21no—
41C22×Dic6D411yescharacteristic
42C3×C4⋊Q8D411yescharacteristic
43C3×#127D411yescharacteristic
44C3×#124—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52#951—42no—
53#951—42no—
54C2×#216—22no—
55C2×#214—22no—
56C2×C4⋊Dic3—24no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4⋊Dic3D412yes—
60C2×Dic3⋊C4—21no—
61C2×C4×Dic3—21no—
62C2×#208—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869—42no—
68#869—42no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C4×Dic6—21no—
73C3×C4⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C4×C8—21no—
76C3×#95D412yes—
77#835—22no—
78#832—22no—
79#832—24no—
80C3⋊#95—24no—
81#830—21no—
82#807—24no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)D412yes—
85C4⋊(C3⋊C8)D412yes—
86C4×C3⋊C8—21no—
87C2×#137—61no—
88#532C2×D311yescharacteristic
89C2×C4⋊Q8—31no—
90C2×C4⋊Q8—61no—
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#423—31no—
102#417—61no—
103#398—62no—
104#391—61no—
105#389—62no—
106C4×C4⋊C4C2×D311yescharacteristic
107#332—62no—
108#327—22no—
109Q8⋊C8—62no—
110C3×#41—41no—
111C3×#41—41no—
112C3×#41—41no—
113C3×#41—41no—
114C2×C3×Q8—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C22×C3×C4—22no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—22no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—41no—
133C2×Dic6—41no—
134C2×Dic6—41no—
135C2×Dic6—41no—
136C2×Dic6—41no—
137C2×Dic6—41no—
138C2×Dic6D812yes—
139C2×C3×C8—41no—
140C3×C4⋊C4—24no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C42—41no—
146C3×C42—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#216—44no—
152#214—44no—
153C4⋊Dic3—22no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—22no—
157C4⋊Dic3—48no—
158C4⋊Dic3—41no—
159C4⋊Dic3—41no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162C4×Dic3—41no—
163C4×Dic3—41no—
164#208—41no—
165#208—41no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×#41—21no—
169C22×Q8—31no—
170C22×Q8—61no—
171#137—122no—
172#137—122no—
173C2×Q16—62no—
174C2×SD16—62no—
175C2×M16—61no—
176C4⋊Q8—61no—
177C4⋊Q8—61no—
178C4⋊Q8—61no—
179C4⋊Q8—61no—
180C4⋊Q8—122no—
181C4⋊Q8—122no—
182C4⋊Q8D1211yescharacteristic
183#127D1211yescharacteristic
184#124—21no—
185#121—21no—
186#121—21no—
187C4×Q8—61no—
188C4×Q8—61no—
189C4×Q8—61no—
190C4×Q8—61no—
191C4×D4—21no—
192C4×D4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—64no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C42—21no—
200C2×C42—61no—
201C2×C42C22×D311yescharacteristic
202#107—64no—
203C4⋊C8—21no—
204C4⋊C8—32no—
205C4⋊C8—32no—
206C4⋊C8—61no—
207#103—62no—
208#102—62no—
209#102—64no—
210C22⋊C8—21no—
211C22⋊C8—61no—
212C4×C8—21no—
213C4×C8—61no—
214#95—64no—
215#95#6712yes—
216C23×C3—41no—
217C23×C3#12011yescharacteristic
218C3×Q8—41no—
219C3×Q8—41no—
220C3×D4—41no—
221C3×D4—41no—
222C3×D4—42no—
223C3×D4—81no—
224C3×D4—81no—
225C3×D4—81no—
226C2×C3×C4—22no—
227C2×C3×C4—24no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—41no—
232C2×C3×C4—81no—
233C2×C3×C4—81no—
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12711yescharacteristic
240C2×C3×C4C2×D811yescharacteristic
241C2×C3×C4#13511yescharacteristic
242C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
243C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
244C2×Dic3—22no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—42no—
248C2×Dic3—41no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252Dic6—22no—
253Dic6—22no—
254Dic6—41no—
255Dic6—41no—
256Dic6—81no—
257Dic6—81no—
258Dic6—81no—
259Dic6—81no—
260C3×C8—81no—
261C3⋊C8—42no—
262C3⋊C8—42no—
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267C2×Q8—21no—
268C2×Q8—32no—
269C2×Q8—61no—
270C2×Q8—61no—
271C2×Q8—61no—
272C2×Q8—61no—
273C2×Q8—61no—
274C2×Q8—121no—
275C2×Q8—121no—
276C2×Q8—121no—
277C2×Q8—121no—
278C2×Q8—121no—
279C2×Q8—121no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—62no—
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4C2×#6711yescharacteristic
292Q16—124no—
293SD16—124no—
294M16—121no—
295M16—121no—
296C2×C8—41no—
297C2×C8—62no—
298C2×C8—62no—
299C4⋊C4—24no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—61no—
303C4⋊C4—62no—
304C4⋊C4—61no—
305C4⋊C4—62no—
306C4⋊C4—128no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C4⋊C4—121no—
310C4⋊C4—121no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—41no—
314C42—41no—
315C42—121no—
316C42—121no—
317C42C2×D1211yescharacteristic
318C42#23511yescharacteristic
319C42D3×D411yescharacteristic
320C42D3×D411yescharacteristic
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3#39211yescharacteristic
327C22×C3#44711yescharacteristic
328C22×C3#44911yescharacteristic
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—44no—
333C3×C4—81no—
334C3×C4#44711yescharacteristic
335C3×C4#44911yescharacteristic
336C3×C4#45612yes—
337C3×C4#46611yescharacteristic
338C3×C4#46911yescharacteristic
339Dic3—42no—
340Dic3—82no—
341Dic3—81no—
342C23—41no—
343C23#93711yescharacteristic
344Q8—41no—
345Q8—41no—
346Q8—62no—
347Q8—62no—
348Q8—121no—
349Q8—121no—
350Q8—241no—
351Q8—241no—
352Q8—241no—
353Q8—241no—
354D4—41no—
355D4—41no—
356D4—42no—
357D4—81no—
358D4—81no—
359D4—81no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—62no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—121no—
375C2×C4—122no—
376C2×C4#88211yescharacteristic
377C2×C4#88211yescharacteristic
378C2×C4#88411yescharacteristic
379C2×C4#88411yescharacteristic
380C2×C4D3×D811yescharacteristic
381C2×C4#91111yescharacteristic
382C2×C4#91411yescharacteristic
383C2×C4#91611yescharacteristic
384C2×C4#93011yescharacteristic
385C2×C4#94011yescharacteristic
386C8—81no—
387C8—122no—
388C8—122no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—161no—
392C2×C3#199911yescharacteristic
393C2×C3#200511yescharacteristic
394C2×C3#200911yescharacteristic
395C22—21no—
396C22—21no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22#496011yescharacteristic
401C22#515611yescharacteristic
402C22#517211yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—44no—
407C4—81no—
408C4—122no—
409C4—242no—
410C4—241no—
411C4#477311yescharacteristic
412C4#477511yescharacteristic
413C4#478111yescharacteristic
414C4#478211yescharacteristic
415C4#482612yes—
416C3#1024311yescharacteristic, complemented, Sylow
417C2—21no—
418C2—21no—
419C2—161no—
420C2#7005311yescharacteristic
421C2#7020411yescharacteristic
422C2#7038911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1502C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1502D3
hypocenterC3#10243
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
3. #73711C2semidirect
5. #70778C2semidirect
6. #70767C2semidirect
7. #67620C2semidirect
416. C3#10243semidirect

Maximal subgroups

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

subgroupindex
#833142
#832702
#737112
C3×#15022
#707782
#707672
#676202
#102433