Finite Group Databasebeta

← #10698604

Subgroups — #10698604

Order 768.

520 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73802C211yesmaximal, characteristic
2#70479C211yesmaximal, characteristic
3C2×#4482C211yesmaximal, characteristic, complemented
4#69777C211yesmaximal, characteristic, complemented
5C2×#4459C211yesmaximal, characteristic
6#69466C211yesmaximal, characteristic, complemented
7C3×#1279C211yesmaximal, characteristic, complemented
8#10416—31nomaximal, Sylow
9C2×#875—21no—
10C2×C4×D12C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15C2×#821—21no—
16#4987—21no—
17#4987—21no—
18C2×#807—21no—
19#4935C2211yescharacteristic
20#4929—21no—
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22#4674—22no—
23#4551—21no—
24#4551—22no—
25#4529—21no—
26#4482—22no—
27#4459—22no—
28#2547—31no—
29#1990—31no—
30#1947—31no—
31#1802—31no—
32C2×#332—31no—
33C2×D4⋊C8—31no—
34#1279D311yescharacteristic, complemented
35C22×D12D411yescharacteristic
36C22×C4×D3—21no—
37C22×Dic6—21no—
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C42—21no—
43C2×C3×C42C2311yescharacteristic
44C2×(C2×D3)⋊C4—22no—
45C2×(C2×D3)⋊C4—22no—
46C2×(C2×D3)⋊C4—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3D411yescharacteristic
52C2×C4⋊Dic3D412yes—
53C22×C3⋊C8—21no—
54#875—42no—
55#875—42no—
56C4×D12—21no—
57C4×D12—21no—
58C4×D12—21no—
59C4×D12—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4×C8—21no—
67C3×C4×C8—21no—
68C3×#95—21no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71#821—42no—
72#807—42no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D412yes—
75C2×C4⋊Q8—31no—
76C2×#124—61no—
77C2×C4×D4—31no—
78C2×#107—61no—
79C2×C4⋊C8—31no—
80C2×C4⋊C8C2×D311yescharacteristic
81C2×#102—61no—
82C2×C4×C8C2×D311yescharacteristic
83#397—62no—
84#391—61no—
85#384—31no—
86#380—61no—
87#380—61no—
88C4×C4⋊C4C2×D311yescharacteristic
89#340—61no—
90#340—62no—
91#340—61no—
92#332—62no—
93D4⋊C8—62no—
94C23×D3—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×Dic3—21no—
101C22×Dic3—22no—
102C22×Dic3—41no—
103C2×D12—21no—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12D812yes—
109C2×C4×D3—41no—
110C2×C4×D3—41no—
111C2×C4×D3—41no—
112C2×C4×D3—41no—
113C2×C4×D3—41no—
114C2×C4×D3—41no—
115C2×Dic6—41no—
116C2×Dic6—41no—
117C2×Dic6—41no—
118C2×Dic6—41no—
119C2×Dic6—41no—
120C2×Dic6—41no—
121C2×C3×C8—22no—
122C2×C3×C8—41no—
123C2×C3×C8—41no—
124C2×C3×C8—41no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C42—41no—
132C3×C42—41no—
133C3×C42C2×D411yescharacteristic
134C3×C42#4111yescharacteristic
135C3×C42#4111yescharacteristic
136C3×C42#4111yescharacteristic
137(C2×D3)⋊C4—44no—
138(C2×D3)⋊C4—44no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141C4⋊Dic3—22no—
142C4⋊Dic3—21no—
143C4⋊Dic3—22no—
144C4⋊Dic3—21no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4⋊Dic3—42no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C22×Q8—61no—
156C22×D4—31no—
157C23×C4—61no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—61no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165#124—122no—
166#124—122no—
167C4×D4—61no—
168C4×D4—61no—
169C4×D4—61no—
170C4×D4—61no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—32no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4D1211yescharacteristic
178C2×C4⋊C4D1211yescharacteristic
179C2×C22⋊C4—62no—
180C2×C22⋊C4—62no—
181C2×C22⋊C4—61no—
182C2×C42—21no—
183C2×C42C22×D311yescharacteristic
184#107—122no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—32no—
188C4⋊C8—61no—
189#102—122no—
190C4×C8—21no—
191C4×C8—21no—
192#95—21no—
193#95—62no—
194#95—62no—
195C23×C3#12011yescharacteristic
196C22×D3—22no—
197C22×D3—41no—
198C22×D3—41no—
199C22×D3—41no—
200C22×D3—41no—
201C22×D3—41no—
202C22×D3—41no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12111yescharacteristic
214C2×C3×C4#12311yescharacteristic
215C2×C3×C4#12311yescharacteristic
216C2×C3×C4#12411yescharacteristic
217C2×C3×C4C2×D811yescharacteristic
218C2×C3×C4#13511yescharacteristic
219C2×C3×C4#13511yescharacteristic
220C2×C3×C4#13511yescharacteristic
221C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
222C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
223C2×C3×C4#13711yescharacteristic
224C2×C3×C4#13711yescharacteristic
225C2×Dic3—42no—
226C2×Dic3—41no—
227C2×Dic3—42no—
228C2×Dic3—41no—
229C2×Dic3—42no—
230C2×Dic3—41no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233C2×Dic3—81no—
234D12—22no—
235D12—22no—
236D12—41no—
237D12—41no—
238C4×D3—81no—
239C4×D3—81no—
240C4×D3—81no—
241C4×D3—81no—
242Dic6—81no—
243Dic6—81no—
244Dic6—81no—
245Dic6—81no—
246C3×C8—42no—
247C3×C8—81no—
248C3⋊C8—42no—
249C24—61no—
250C2×Q8—121no—
251C2×Q8—121no—
252C2×Q8—121no—
253C2×Q8—121no—
254C2×Q8—121no—
255C2×Q8—121no—
256C2×D4—32no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—61no—
265C22×C4—62no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—121no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4—121no—
273C22×C4C2×D1211yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4D3×D411yescharacteristic
276C2×C8—22no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—62no—
283C2×C8—121no—
284C2×C8—121no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—61no—
290C4⋊C4—62no—
291C4⋊C4—61no—
292C4⋊C4—62no—
293C4⋊C4—122no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C4⋊C4—121no—
300C22⋊C4—124no—
301C22⋊C4—124no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C42—41no—
305C42—41no—
306C42#23511yescharacteristic
307C42#23511yescharacteristic
308C42D3×D411yescharacteristic
309C42#23911yescharacteristic
310C22×C3#39411yescharacteristic
311C22×C3#44711yescharacteristic
312C22×C3#44911yescharacteristic
313C22×C3#45211yescharacteristic
314C22×C3#45211yescharacteristic
315C22×C3#45411yescharacteristic
316C22×C3#45611yescharacteristic
317C2×D3—42no—
318C2×D3—42no—
319C2×D3—42no—
320C2×D3—81no—
321C2×D3—81no—
322C2×D3—81no—
323C2×D3—81no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4#46311yescharacteristic
329C3×C4#46411yescharacteristic
330C3×C4#46611yescharacteristic
331C3×C4#46811yescharacteristic
332C3×C4#48011yescharacteristic
333C3×C4#48211yescharacteristic
334C3×C4#48811yescharacteristic
335C3×C4#48911yescharacteristic
336Dic3—82no—
337Dic3—81no—
338Dic3—161no—
339C23—62no—
340C23—121no—
341C23—121no—
342C23—121no—
343C23—121no—
344C23—121no—
345C23—121no—
346C23#88211yescharacteristic
347Q8—241no—
348Q8—241no—
349Q8—241no—
350Q8—241no—
351D4—62no—
352D4—62no—
353D4—121no—
354D4—121no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—121no—
364C2×C4—122no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—122no—
368C2×C4—122no—
369C2×C4—241no—
370C2×C4—241no—
371C2×C4—241no—
372C2×C4—241no—
373C2×C4—241no—
374C2×C4—241no—
375C2×C4—241no—
376C2×C4#87511yescharacteristic
377C2×C4#88411yescharacteristic
378C2×C4#88411yescharacteristic
379C2×C4#89411yescharacteristic
380C2×C4#89411yescharacteristic
381C2×C4#89511yescharacteristic
382C2×C4#90411yescharacteristic
383C2×C4#90411yescharacteristic
384C2×C4#90811yescharacteristic
385C2×C4#90911yescharacteristic
386C2×C4D3×D811yescharacteristic
387C2×C4#91411yescharacteristic
388C2×C4#91611yescharacteristic
389C2×C4#91811yescharacteristic
390C8—42no—
391C8—81no—
392C8—122no—
393C2×C3#162511yescharacteristic
394C2×C3#163811yescharacteristic
395C2×C3#164711yescharacteristic
396C2×C3#165211yescharacteristic
397C2×C3#203311yescharacteristic
398C2×C3#203811yescharacteristic
399C2×C3#204211yescharacteristic
400D3—82no—
401C22—122no—
402C22—122no—
403C22—122no—
404C22—241no—
405C22—241no—
406C22—241no—
407C22—241no—
408C22#467211yescharacteristic
409C22#473311yescharacteristic
410C22#473411yescharacteristic
411C22#477311yescharacteristic
412C22#480811yescharacteristic
413C22#482611yescharacteristic
414C22#482711yescharacteristic
415C4—21no—
416C4—21no—
417C4—81no—
418C4—81no—
419C4—241no—
420C4—242no—
421C4—481no—
422C4#469711yescharacteristic
423C4#471411yescharacteristic
424C4#477511yescharacteristic
425C4#481211yescharacteristic
426C4#484411yescharacteristic
427C4#484511yescharacteristic
428C4#486611yescharacteristic
429C4#488211yescharacteristic
430C3#1041611yescharacteristic, complemented, Sylow
431C2—242no—
432C2#6818011yescharacteristic
433C2#6818411yescharacteristic
434C2#6824711yescharacteristic
435C2#6827411yescharacteristic
436C2#6863011yescharacteristic
437C2#6873411yescharacteristic
438C2#6882211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenter#1279D3
hypocenterC3#10416
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
3. C2×#4482C2semidirect
4. #69777C2semidirect
6. #69466C2semidirect
7. C3×#1279C2semidirect
34. #1279D3semidirect
430. C3#10416semidirect

Maximal subgroups

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

subgroupindex
#738022
#704792
C2×#44822
#697772
C2×#44592
#694662
C3×#12792
#104163