Finite Group Databasebeta

← #10697579

Subgroups — #10697579

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×#3037C211yesmaximal, characteristic
2#83270C211yesmaximal, characteristic
3#82475C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic
5#71621C211yesmaximal, characteristic, complemented
6#70506C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#10243—31nomaximal, Sylow
9C2×C3×#137—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C2×#944—21no—
13C2×#869C2211yescharacteristic
14C3×#464—22no—
15C3×#462—21no—
16C3×#460—21no—
17C3×#423C2211yescharacteristic
18C3×#417—21no—
19#5078—21no—
20#5070—21no—
21#5033—22no—
22#5007—22no—
23#4981—22no—
24#4973—21no—
25C4×C4⋊Dic3C2211yescharacteristic
26#4928—22no—
27#4925C2211yescharacteristic
28#4925C2211yescharacteristic
29#4537—22no—
30#4535—22no—
31#4482—22no—
32#3226—31no—
33#3037D311yescharacteristic
34#2546—31no—
35#1546—31no—
36#1502—31no—
37#1479—31no—
38#1303—31no—
39C2×C3×#41—21no—
40C22×C3×Q8D411yescharacteristic
41C22×Dic6—21no—
42C3×#137—42no—
43C3×#137—42no—
44C2×C3×Q16—22no—
45C2×C3×SD16—22no—
46C2×C3×M16—21no—
47C3×C4⋊Q8D411yescharacteristic
48C3×#127D411yescharacteristic
49C3×#124—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62#944—42no—
63#944—42no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3D412yes—
67C2×Dic3⋊C4—24no—
68C2×C4×Dic3—21no—
69#869—21no—
70#869—21no—
71#869—21no—
72#869—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8D412yes—
75C3×#103—22no—
76C3×#102—22no—
77#832—24no—
78C3⋊#95—24no—
79C3⋊#95D412yes—
80#830—21no—
81#830—21no—
82#807—24no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)D412yes—
85C4×C3⋊C8—21no—
86C4×C3⋊C8—21no—
87C2×#137—21no—
88#532C2×D311yescharacteristic
89C2×C4⋊Q8—31no—
90C2×C4⋊Q8—61no—
91C2×C4×Q8C2×D311yescharacteristic
92#500—61no—
93#486—61no—
94#480—62no—
95#474—62no—
96#464—22no—
97#462—21no—
98#460—21no—
99#431—31no—
100#431—31no—
101#423C2×D311yescharacteristic
102#417—21no—
103#398—62no—
104#391—61no—
105#389—62no—
106C4×C4⋊C4—31no—
107#332—62no—
108#327—62no—
109Q8⋊C8—62no—
110C3×#41—41no—
111C3×#41—41no—
112C3×#41—41no—
113C3×#41—41no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8D812yes—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—41no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—41no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×Dic3—22no—
132C22×Dic3—22no—
133C2×Dic6—41no—
134C2×Dic6—41no—
135C2×Dic6—41no—
136C2×Dic6—41no—
137C2×Dic6—41no—
138C2×Dic6—41no—
139C3×Q16—44no—
140C3×SD16—44no—
141C3×M16—41no—
142C3×M16—41no—
143C2×C3×C8—22no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
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
158C4⋊Dic3—24no—
159C4⋊Dic3—22no—
160C4⋊Dic3—22no—
161C4⋊Dic3—41no—
162C4⋊Dic3—41no—
163Dic3⋊C4—48no—
164C4×Dic3—41no—
165C4×Dic3—41no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—41no—
168C2×#41—21no—
169C22×Q8—61no—
170C22×Q8#6711yescharacteristic
171#137—42no—
172#137—42no—
173C2×Q16—22no—
174C2×SD16—22no—
175C2×M16—21no—
176C4⋊Q8—61no—
177C4⋊Q8—61no—
178C4⋊Q8—61no—
179C4⋊Q8—61no—
180C4⋊Q8—122no—
181C4⋊Q8—122no—
182C4⋊Q8#6711yescharacteristic
183#127#6711yescharacteristic
184#124—21no—
185#121—21no—
186#121—21no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×D4—21no—
192C4×D4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—32no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—64no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4#6711yescharacteristic
199C2×C42—21no—
200C2×C42—61no—
201C2×C42C22×D311yescharacteristic
202#107—64no—
203C4⋊C8—21no—
204C4⋊C8—32no—
205C4⋊C8—61no—
206C4⋊C8#6712yes—
207#103—22no—
208#102—22no—
209#102—64no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C4×C8—61no—
213C4×C8—61no—
214#95—32no—
215#95—64no—
216C23×C3—41no—
217C23×C3#12011yescharacteristic
218C3×Q8—22no—
219C3×Q8—22no—
220C3×Q8—41no—
221C3×Q8—41no—
222C3×Q8—41no—
223C3×Q8—41no—
224C3×D4—41no—
225C3×D4—41no—
226C3×D4—42no—
227C3×D4—81no—
228C3×D4—81no—
229C3×D4—81no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12711yescharacteristic
246C2×C3×C4C2×D811yescharacteristic
247C2×C3×C4#13511yescharacteristic
248C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
249C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
250C2×Dic3—24no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256Dic6—81no—
257Dic6—81no—
258Dic6—81no—
259Dic6—81no—
260C3×C8—42no—
261C3⋊C8—42no—
262C3⋊C8—81no—
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—121no—
274C2×Q8—121no—
275C2×Q8—121no—
276C2×Q8—121no—
277C2×Q8—121no—
278C2×Q8—121no—
279C2×Q8#21312yes—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—62no—
288C22×C4—62no—
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4C2×#6711yescharacteristic
292Q16—44no—
293SD16—44no—
294M16—41no—
295M16—41no—
296C2×C8—22no—
297C2×C8—62no—
298C2×C8—121no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—64no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C4⋊C4—128no—
309C4⋊C4—121no—
310C4⋊C4—121no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—41no—
314C42—41no—
315C42—121no—
316C42—121no—
317C42D3×D411yescharacteristic
318C42#23711yescharacteristic
319C42C2×#6711yescharacteristic
320C42C2×#6711yescharacteristic
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—42no—
333C3×C4—81no—
334C3×C4—81no—
335C3×C4#44711yescharacteristic
336C3×C4#44911yescharacteristic
337C3×C4#45612yes—
338C3×C4#46611yescharacteristic
339C3×C4#46911yescharacteristic
340Dic3—44no—
341Dic3—82no—
342C23—41no—
343C23#93711yescharacteristic
344Q8—22no—
345Q8—22no—
346Q8—41no—
347Q8—41no—
348Q8—41no—
349Q8—41no—
350Q8—241no—
351Q8—241no—
352Q8—241no—
353Q8—241no—
354D4—42no—
355D4—41no—
356D4—41no—
357D4—81no—
358D4—81no—
359D4—81no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—64no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4C2×#21311yescharacteristic
377C2×C4#93211yescharacteristic
378C2×C4#93711yescharacteristic
379C2×C4#93811yescharacteristic
380C2×C4#93911yescharacteristic
381C2×C4#93911yescharacteristic
382C2×C4#94011yescharacteristic
383C2×C4#94911yescharacteristic
384C2×C4#95011yescharacteristic
385C2×C4#95311yescharacteristic
386C8—42no—
387C8—122no—
388C8—241no—
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#503611yescharacteristic
401C22#503711yescharacteristic
402C22#522311yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—42no—
407C4—81no—
408C4—81no—
409C4—124no—
410C4—242no—
411C4#503811yescharacteristic
412C4#503911yescharacteristic
413C4#520512yes—
414C4#521711yescharacteristic
415C4#523811yescharacteristic
416C3#1024311yescharacteristic, complemented, Sylow
417C2—21no—
418C2—21no—
419C2—161no—
420C2#7168411yescharacteristic
421C2#7210811yescharacteristic
422C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3037C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3037D3
hypocenterC3#10243
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
3. #82475C2semidirect
5. #71621C2semidirect
6. #70506C2semidirect
7. #67715C2semidirect
416. C3#10243semidirect

Maximal subgroups

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

subgroupindex
C3×#30372
#832702
#824752
#718112
#716212
#705062
#677152
#102433